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Triangle Congruence
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Triangle Congruence

Key Terms and Concepts

IMP Toolkit: Triangle Congruence x 1

Triangle Congruence Toolkit

Congruence

Figures are called congruent if one can be mapped onto the other using rigid transformations. The three rigid transformations that define congruence are: ____________________ ____________________ ____________________ Corresponding or matching parts of congruent figures have the same measurements. This is a defining characteristic of congruent figures. When making a congruence statement, corresponding parts should match. For example: ∆𝐴𝐵𝐶 ≅ ∆𝑋𝑌𝑍 means that <A and <X are congruent, corresponding parts, 𝐴𝐶̅̅ ̅̅ and 𝑋𝑍̅̅ ̅̅ are congruent, corresponding parts, etc There are also three triangle congruence theorems which give us other ways to show triangles are congruent. These are: _________ which means ____________________________________________________________________ _________ which means ____________________________________________________________________ _________ which means ____________________________________________________________________

SSS

A compass and ruler can be used to draw a triangle given the three side measurements. ___________ ___________ ___________

My steps are:

_________________________

_________________________

_________________________

_________________________

_________________________

_________________________

Name:____________________________________Date:_________________Period:___________

IMP Toolkit: Triangle Congruence x 2

SAS

A ruler and protractor can be used to draw a triangle given two side measurements and the included angle.

___________ ___________ ___________

My steps are:

_________________________

_________________________

_________________________

_________________________

_________________________

_________________________

ASA

A ruler and protractor can be used to draw a triangle given two angle measurements and the included side.

___________ ___________ ___________

My steps are:

_________________________

_________________________

_________________________

_________________________

_________________________

_________________________

Triangle Inequality

The Triangle Inequality Theorem helps me know if three measurements could make a triangle. It says:

Essential Lessons• Where are the Congruent Halves?

• Congruent Figures• Drawing Triangles with Given Sides

• My 1st Triangle Congruence Th eorem• Testing Other Congruence Th eorems

• Looking for more Congruence Th eorems• Formal Language

References:1) “Progressions Documents for the Common Core Math Standards.” The University of Arizona.Brookhill Founda on, 2007.

2) “Mathema cs Framework Chapters.” Curriculum Frameworks (CA Dept of Educa on). 2013.

Triangle CongruenceStorybook

The Math Behind CongruenceRigid Mo on preserves congruence. The transforma ons that are rigid mo ons are transla ons, rota ons and refl ec ons. Rigid mo ons preserve distance as well as angle measure and thus shapes transformed by a rigid mo on remain congruent. If a shape undergoes a transforma on using only rigid mo ons, the image will be congruent to the pre-image. Conversely, if two shapes are congruent, then a series of rigid mo ons can be found that maps one fi gure onto another. This defi ni on of congruence applies to all fi gures.

Triangle Inequality TheoremThe Triangle Inequality Theorem states that in order for three lengths to form a triangle, the sum of the shorter two sides must be greater than the length of the longest side. Algebraically, if sides a and b are the shorter sides, then a̅ + b > c ̅.

This theorem comes about by understanding that if the sum of the two shorter sides is equal to or less than the length of the longest side, no angle can be formed. As shown below, with a side length of 1 unit and a side length of 2 units, together these two sides will be the same length as the longest side of 3 units. When the sides form an angle greater than 0°, you can see a gap form and thus the side lengths won’t meet and there is no triangle.

Construc ng a Triangle given Three Side LengthsTo construct a triangle using a ruler and compass given three side lengths, we fi rst measure and draw a line segment represen ng the length of the base. Next, we open the compass to the distance represen ng the second side length. With the point of the compass on the le endpoint of the segment drawn, we make an arc (or semi-circle) near where a vertex will be. We repeat this process from the right end point, this me opening the compass to the length of the third side length. The intersec on of the two arcs represents the third vertex, and segments are then drawn connec ng this vertex to each end point. If there is no intersec on point the three sides cannot form a triangle.

Construc ng a Triangle given one or two angle measures and/or one or two side lengthsThe steps taken to construct triangles given other criteria (such as SAS, or AAS) are similar to the construc on given three side lengths, with the diff erence being needing to use a protractor to measure the given angle measure(s). Below are steps taken to draw a triangle when given Angle-Side-Side.

1. Draw a segment for the base that is the fi rst length given.

Essential Question # 1How can rigid motions be used to determine if two fi gures are congruent?

Essential Question # 2What are the minimum criteria needed to know

if two triangles are congruent?

Essential Question # 3If two fi gures are

congruent, what can you conclude about their

corresponding sides and corresponding angles?

The Unit!2. Then, use your protractor and ruler to draw the angle a ached to

the le endpoint of the base. Remember, we only know the angle on the le side, so we do not know the length of this side of the triangle. Make sure you draw it long in case it needs to be long. Use an arrow to show that we know the direc on, but not the length.

3. Finally, swing an arc for the second given length a ached to the right endpoint of the base.

Example: Construct a triangle with the following: 30°, 7 cm, 4 cm (note the angle is not between the sides).

Corresponding Pairs of Sides and Corresponding Pairs of Angles in Congruent TrianglesUsing a sequence of rigid mo ons to map a triangle onto another triangle results in the two triangles being congruent; one can visually see which angles on the fi rst triangle match which angles on the second. These are known as corresponding angles and can be shown to have equal angle measures in congruent fi gures. This mapping also allows one to see which pairs of line segments in each fi gure correspond and that these segments are equal in length. Thus, students come to see that in congruent fi gures, corresponding pairs of sides are congruent and corresponding pairs of angles are congruent. For example, in the fi gures to the right, two trapezoids are shown. The second trapezoid can be refl ected over a horizontal line of refl ec on (shown above) to map onto the fi rst trapezoid. When this is done, the two

trapezoids are shown to be congruent and one can see which angle measures correspond in each fi gure as well as which side lengths correspond. In this case, AC ~ FH ; this is oneexample of a pair of corresponding sides. Angle A corresponds to angle F, and thus they are an example of a pair of corresponding angles.

Criteria for Triangle CongruenceThere are three diff erent sets of criteria for congruence of triangles that follow from the defi ni on of congruence in terms of rigid mo on: Knowing the measures of all three side lengths (Side-Side-Side) (SSS), knowing the

• Rigid Mo on• Translate• Rotate• Refl ect• Corresponding Angles• Corresponding Sides • Side-Side-Side (SSS)• Side-Angle-Side (SAS)• Angle-Side-Angle (ASA)• Theorem• Proof• Counterexample• Congruence• Acute Triangle• Right Triangle• Obtuse Triangle• Scalene Triangle• Isosceles Triangle• Equilateral Triangle

Real-World Application

• Construc on

• Bridges

• Roof Trusses

• Anima on

• Video Games

The Math Behind CongruenceRigid Mo on preserves congruence. The transforma ons that are rigid mo ons are transla ons, rota ons and refl ec ons. Rigid mo ons preserve distance as well as angle measure and thus shapes transformed by a rigid mo on remain congruent. If a shape undergoes a transforma on using only rigid mo ons, the image will be congruent to the pre-image. Conversely, if two shapes are congruent, then a series of rigid mo ons can be found that maps one fi gure onto another. This defi ni on of congruence applies to all fi gures.

Triangle Inequality TheoremThe Triangle Inequality Theorem states that in order for three lengths to form a triangle, the sum of the shorter two sides must be greater than the length of the longest side. Algebraically, if sides a and b are the shorter sides, then a̅ + b > c ̅.

This theorem comes about by understanding that if the sum of the two shorter sides is equal to or less than the length of the longest side, no angle can be formed. As shown below, with a side length of 1 unit and a side length of 2 units, together these two sides will be the same length as the longest side of 3 units. When the sides form an angle greater than 0°, you can see a gap form and thus the side lengths won’t meet and there is no triangle.

Construc ng a Triangle given Three Side LengthsTo construct a triangle using a ruler and compass given three side lengths, we fi rst measure and draw a line segment represen ng the length of the base. Next, we open the compass to the distance represen ng the second side length. With the point of the compass on the le endpoint of the segment drawn, we make an arc (or semi-circle) near where a vertex will be. We repeat this process from the right end point, this me opening the compass to the length of the third side length. The intersec on of the two arcs represents the third vertex, and segments are then drawn connec ng this vertex to each end point. If there is no intersec on point the three sides cannot form a triangle.

Construc ng a Triangle given one or two angle measures and/or one or two side lengthsThe steps taken to construct triangles given other criteria (such as SAS, or AAS) are similar to the construc on given three side lengths, with the diff erence being needing to use a protractor to measure the given angle measure(s). Below are steps taken to draw a triangle when given Angle-Side-Side.

1. Draw a segment for the base that is the fi rst length given.

Essential Question # 1How can rigid motions be used to determine if two fi gures are congruent?

Essential Question # 2What are the minimum criteria needed to know

if two triangles are congruent?

Essential Question # 3If two fi gures are

congruent, what can you conclude about their

corresponding sides and corresponding angles?

The Unit!2. Then, use your protractor and ruler to draw the angle a ached to

the le endpoint of the base. Remember, we only know the angleon the le side, so we do not know the length of this side of thetriangle. Make sure you draw it long in case it needs to be long.Use an arrow to show that we know the direc on, but not thelength.

3. Finally, swing an arc for the second given length a ached to theright endpoint of the base.

Example: Construct a triangle with the following: 30°, 7 cm, 4 cm (note the angle is not between the sides).

Corresponding Pairs of Sides and Corresponding Pairs of Angles in Congruent TrianglesUsing a sequence of rigid mo ons to map a triangle onto another triangle results in the two triangles being congruent; one can visually see which angles on the fi rst triangle match which angles on the second. These are known as corresponding angles and can be shown to have equal angle measures in congruent fi gures. This mapping also allows one to see which pairs of line segments in each fi gure correspond and that these segments are equal in length. Thus, students come to see that in congruent fi gures, corresponding pairs of sides are congruent and corresponding pairs of angles are congruent. For example, in the fi gures to the right, two trapezoids are shown.

The second trapezoid can be refl ected over a horizontal line of refl ec on (shown above) to map onto the fi rst trapezoid. When this is done, the two

trapezoids are shown to be congruent and one can see which angle measures correspond in each fi gure as well as which side lengths correspond. In this case, AC ~ FH ; this is oneexample of a pair of corresponding sides. Angle A corresponds to angle F, and thus they are an example of a pair of corresponding angles.

Criteria for Triangle CongruenceThere are three diff erent sets of criteria for congruence of triangles that follow from the defi ni on of congruence in terms of rigid mo on: Knowing the measures of all three side lengths (Side-Side-Side) (SSS), knowing the

• Rigid Mo on• Translate• Rotate• Refl ect• Corresponding Angles• Corresponding Sides• Side-Side-Side (SSS)• Side-Angle-Side (SAS)• Angle-Side-Angle (ASA)• Theorem• Proof• Counterexample• Congruence• Acute Triangle• Right Triangle• Obtuse Triangle• Scalene Triangle• Isosceles Triangle• Equilateral Triangle

Real-World Application

• Construc on

• Bridges

• Roof Trusses

• Anima on

• Video Games

The Math Behind measures of any two angles and a side length (Angle-Side-Angle (ASA), Angle-Angle-Side (AAS) or Side-Angle-Angle (SAA)), and knowing the measure of two side lengths and the included angle (Side-Angle-Side (SAS)).

SSS tells us that if we know the measures of all three sides of a triangle, all triangles with these same three side lengths will be congruent. This is shown to be true by construc ng all 6 possible triangles given the three side lengths (changing the order of which is the base side and which are the right and le sides). The original triangle drawn can be shown to map onto each of the other fi ve other triangles through a sequence of rigid mo ons; thus, all triangles having the exact same three side lengths are congruent.

Ex: Some possible triangles given the measures of the sides to be 8cm, 4 cm and 6 cm. Similar reasoning can be used to show triangles with these three given side lengths to be congruent through a sequence of rigid mo ons.

Common CongruenceA common misconcep on is that congruent fi gures must look exactly the same upon fi rst glance. If congruence is fi rst defi ned as same size and same shape, it can be hard for students to recognize congruent fi gures in which the pre-image has undergone a series of transforma ons, such as the one shown below. If students come to defi ne congruence as a fi gure that maps onto itself a er a series of transla ons, refl ec ons and rota ons, they will avoid this misconcep on.

Congruence is only for trianglesHistorically, congruence has been taught using criteria for triangles (SSS, SAS, ASA), and thus units have focused solely upon triangle congruence. If students defi ne congruence primarily through the lens of SSS, ASA and SAS, they are likely to see congruence as only for triangles. However, congruence needs to be understood in terms of rigid mo ons, as this understanding allows us to defi ne congruence for any fi gures. The congruence “theorems” should be seen as a consequence of the rigid mo ons rather than serve as the primary means of proving congruence. This will help students see these theorems as characteris cs of congruent triangles, while being able to understand that any shape can be shown to be congruent to another through a sequence of rigid mo ons.

Any three lengths will form a triangleAs students have yet to study side lengths of triangles, many will naively believe that any three side lengths will form a triangle. Allowing students an opportunity to explore this with spaghe will help clear up major misconcep ons. Further, by allowing students to predict how o en three randomly generated numbers represen ng side lengths will form a triangle and then le ng them perform the experiment, they will

1) Make sense of problems and

persevere in solving them.

2) Reason abstractly and quantitatively.

3) Construct viable arguments and critique the reasoning of others.

4) Model with mathematics.

5) Use appropriate tools strategically.

6) Attend to precision.

7) Look for and make use of structure.

8) Look for and express regularity in repeated reasoning.

Math Practice Standards

OverviewIMPORTANT IN INTEGRATED MATH Iand then narrow their focus to triangles to understand criteria specifi c to congruent triangles. This approach allows more students to be successful in geometry, as they can focus on the big ideas and visual means by which to prove congruence. This also prepares students for the world they will live and work in, as transformational geometry serves as the basis of most computer animation as well as things we experience in our everyday lives such as the construction of bridges and earthquake-resistant buildings.

This Topic Fit?also investigate when three given side lengths will comprise a triangle and discover the triangle inequality theorem. Grade 8 students begin a formal study of transformations. They use patty paper or software to perform the transformations: translations, refl ections, rotations and dilations. Grade 8 students come to understand that rigid motions preserve side length and angle measure, and that parallel line segments remain parallel under rigid motion. Students defi ne congruent fi gures as those that can be mapped onto one another through a sequence of rigid motions.

HIGH SCHOOLIn Math II, students will use congruence criteria (along with similarity criteria) to solve problems and prove relationships in geometric fi gures.

CoherenceConnections to other

Math I TopicsThis unit builds directly upon work done with

transformations. Students use rigid motions to prove two shapes are congruent.

Students use the defi nition of congruence in terms of rigid

motions to show that triangles are congruent if and only

if corresponding sides and corresponding pairs of angles

are congruent. Students explain how the criteria for triangle congruence (ASA, SAS and SSS) follow from

the defi nition of congruence in terms of rigid motions. This unit extends work done with constructions in the previous

unit and supports work students will do in coordinate geometry, in particular with

additional constructions and with rotations of slope

triangles in showing the product of the slopes of perpendicular lines is -1.

UnitWHY TRIANGLE CONGRUENCE IS

This unit present congruence with a diff erent approach than what most adults will remember about triangle congruence, namely memorizing countless theorems and fi guring out which to use in a 2-column proof. Rather than triangle criteria serving as the means by which we prove congruence, this unit takes the approach of defi ning congruence in terms of rigid motions (transformations, refl ections and rotations). In learning to defi ne congruence this way, students can understand congruence as applied to all fi gures

Where Does GRADES K-5In grades K-5, students develop precise defi nitions of polygons and characteristics of them. Beginning in grade 2, students learn to use a ruler to measure distance and draw segments of a certain length. In grade 4, students come to understand degrees as angular measure and learn to build and use a protractor to measure angles. Grade 4 students also come to understand precise names of triangles, using adjectives to describe their angle measures and side lengths.

GRADES 7-8Grade 7 students draw fi gures with given conditions (using a ruler or straightedge and a protractor). They study these conditions with three side lengths to determine when a unique triangle, no triangle or multiple triangles could be drawn. Grade 7 students

Types OfKnowledge• MEMORIZATION

(QUICK RECALL)

• PROCEDURAL (FOLLOW STEPS/ DO SOMETHING)

• CONCEPTUAL (UNDERSTND BIG IDEA/ EXPLAIN/ DERIVE)

• RELATIONAL (APPLY/ ANALYZE/EVALUATE)

SBACClaims

• CLAIM #1: CONCEPTS AND PROCEDURES (40% STUDENT SCORE)

• CLAIM #2: PROBLEM SOLVING (20% STUDENT SCORE)

• CLAIM #3: COMMUNICATING AND REASONING (20% STUDENT SCORE)

• CLAIM #4: MODELING AND DATA ANALYSIS (20% STUDENT SCORE)

The Unit! (Cont.)ASA tells us that if we know the measures of two angles of a triangle and the included side (the side length between those angles), all triangles with these same two angle measures and same included side length will be congruent. Again, these criteria are true as any triangle you construct given the ASA criteria can be mapped onto the original triangle through a sequence of rigid mo ons; thus, all triangles having the same two angle measures and the same included side length between are congruent.

SAS tells us that if we know the measures of two sides of a triangle and the measure of the included angle, all triangles with these same two side lengths and same included angle measure will be congruent. As before, these criteria are true as any triangle you construct given the SAS criteria can be mapped onto the original triangle through a sequence of rigid mo ons; thus, all triangles having the same two side lengths and the same included angle are congruent.

In the cases of other criteria for triangles (such as SSA), two diff erent triangles can be constructed with the given criteria for which the original triangle cannot map onto the second triangle, and thus these criteria cannot demonstrate congruence.

Student Talk Strategies

Misconceptionshave a meaningful experience on which to recall this major concept later on. Finally, for numbers that are close to forming a triangle, encourage discussion and debate to get at the reasoning that if the two shorter side lengths have a sum equal to the longer side, the two shorter sides will be the exact length of the longer side and thus there can not be any angle formed that will allow the shorter sides to stay touching; therefore, those lengths cannot form a triangle.

Knowing the three side lengths of a triangle makes it easy to drawMany students think that knowing three side lengths means they can easily take a ruler to draw this triangle. Students soon learn that the angle measures ma er, and thus, using a compass makes construc ng triangles easy and precise. Allowing students to try to draw a triangle with three given side lengths without the use of a compass will help the students understand the role of a compass in construc ng triangles.

ProofIn recent mes, tradi onal math textbooks have focused on proof using 2-columns and relying on students’ ability to memorize countless theorems. While proof is only introduced in this course, it is crucial to develop the correct understanding, especially when it comes to proving triangle congruence. Students need to understand that describing a sequence of rigid mo ons that map one fi gure onto another is a proof of congruence. A focus in this unit on paragraph proof lets students experience success with proof and builds upon the means students used to explain and jus fy their thinking in K-8 mathema cs. It should be understood that there are mul ple ways to “prove” things in geometry, and two-column proofs are not necessary at this level nor are they “more rigorous”; rather, they have o en served as the gatekeeper from allowing many students to be successful in geometry and move onto advanced mathema cs.

• Report to a partnerEach student reports his/her own answer to a peer. The students listen to their partner’s response. (“Turn to a partner on your le .” “Now turn to a partner on your right” etc.)

• Give one get oneA er brainstorming ideas, students circulate among other students sharing one idea and ge ng one. Students fold paper lengthwise they label the le side “give one” and the right side “get one”.

• Think, Pair, ShareStudents think about a topic suggested by the teacher. Pairs discuss the topic. Students individually share informa on from their discussion with the class.

• Inside-outside circleTwo concentric circles of students stand or sit, facing one another. The teacher poses a ques on to the class, and the partner responds. At a signal, the outer of inner circle or outer circle rotates and the conversa on con nues.

• Appointment clockPartnering to make future discussion/work appointments.

• JigsawGroup of students assigned a por on of a text; teach that por on to the remainder of the class.

IMP IMP Activity: Where are the Congruent Halves? Teacher Directions 1

Where are the Congruent Halves?

Directions:

1. Each figure can be made from 2 congruent figures.

2. Find the congruent figures by drawing in a segment(s).

3. Use patty paper to verify that your answer is correct for the difficult problems.

4. The solution for problem # 1 is given to the right.

1)

2)

3)

Definition of Congruent Figures: Two figures are congruent if and only if there is a sequence of

reflections, translations, and/or rotations that maps one figure onto the other.

TASK B:Describe a sequence of transformations that would map each half to its congruent half.

Label one figure A and the other figure B before you describe the transformation.

Problem #1

Problem #2 Problem # 3

Objective: Students will use their understanding of rotations, reflections and translations to find a

way to cut a figure into two congruent parts. Students will prove congruence through rigid

transformations.

IMP IMP Activity: Where are the Congruent Halves? Teacher Directions 2

Find the congruent figures by drawing in a segment(s). Some hints have been provided for you.

Describe a sequence of transformations that would map each half to its congruent half. Label

one figure A and the other figure B before you describe the transformation.

4)

5) 6)

4)

5) 6)

7)

8)

9)

7) 8) 9)

IMP IMP Activity: Where are the Congruent Halves? Teacher Directions 3

Analysis:

1. Which problem did you find most challenging? Why?

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

2. Group each of the shapes into categories. Record the number of each problem that you

grouped together. Explain your reasoning for each of your groups.

Problems: __________________________________________________________________

Reason:

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

3. What does it mean for two figures to be congruent?

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

4. What transformations can be used to prove congruence?

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

IMP Activity: Congruent Figures 1

Congruent Figures

Today we will take a look at the congruent halves. Draw in the segment that will cut the

Pentagon ABCDE into two congruent halves. Label your new point F. Then trace each half onto

its own piece of patty paper. Make sure to label all the vertices.

1. What is the shape of each half of the pentagon? Justify your reasoning.

______________________________________________________________________________

______________________________________________________________________________

2. Explain how you know the two figures are congruent.

______________________________________________________________________________

______________________________________________________________________________

3. Draw each figure below such that the orientation of the figures looks exactly the same. Hint:

you may have to reflect (flip over your patty paper) one of your figures to do this. Make sure to

label the vertices on your picture.

Figure 1. Figure 2.

Objective: Students will identify relationships between corresponding parts within congruent

figures, i.e., corresponding parts of congruent figures are congruent and if two sides of a figure are parallel, then the corresponding sides in a congruent figure are also parallel.

IMP Activity: Congruent Figures 2

4. Name all the segments used to create figure 1. _____________________________________

5. Name all the segments used to create figure 2._____________________________________

Corresponding Sides are sides that match when one figure is transformed onto another using a

combination of translations, reflections or rotations.

6. Identify the corresponding sides in Figure 1 and Figure 2.

𝐴𝐵̅̅ ̅̅ 𝑐𝑜𝑟𝑟𝑒𝑠𝑝𝑜𝑛𝑑𝑠 𝑡𝑜 ______ _______ 𝑐𝑜𝑟𝑟𝑒𝑠𝑝𝑜𝑛𝑑𝑠 𝑡𝑜 ______

_______ 𝑐𝑜𝑟𝑟𝑒𝑠𝑝𝑜𝑛𝑑𝑠 𝑡𝑜 ______ _______ 𝑐𝑜𝑟𝑟𝑒𝑠𝑝𝑜𝑛𝑑𝑠 𝑡𝑜 ______

7. Compare the lengths of the corresponding sides in Figure 1 and Figure 2. What do you

conclude?

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

8. Write a congruence statement for each of the corresponding sides in Figure 1 and Figure 2.

𝐴𝐵̅̅ ̅̅ ≅ ______ _______ ≅ ______

_______ ≅ ______ _______ ≅ ______

9. What can you conclude about the corresponding sides of congruent figures?

______________________________________________________________________________

______________________________________________________________________________

10. Do you think this holds true for all congruent figures? Justify your reasoning.

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

IMP Activity: Congruent Figures 3

11. Name all the interior angles used to create Figure 1. ________________________________

12. Name all the interior angles used to create Figure 2.________________________________

13. By constructing CF on the original pentagon, what happened to angle\

C? __________________________________________________

14. How many angles have point C as the vertex? _______________

15. Name all the angles with C as the vertex? _________________

__________________________________________________________________

Corresponding Angles are angles that match when one figure is transformed onto another

using a combination of translations, reflections or rotations.

16. Identify the corresponding angles in Figure 1 and Figure 2.

∠𝐴 𝑐𝑜𝑟𝑟𝑒𝑠𝑝𝑜𝑛𝑑𝑠 𝑡𝑜 ______

17. Compare the measurements of the corresponding sides in Figure 1 and Figure 2. What do

you conclude?

______________________________________________________________________________

______________________________________________________________________________

18. Write a congruence statement for each of the corresponding angles in Figure 1 and Figure 2

∠A ≅ ______

19. What can you conclude about the corresponding angles of congruent figures?

______________________________________________________________________________

______________________________________________________________________________

20. Do you think this holds true for all congruent figures? Justify your reasoning.

______________________________________________________________________________

______________________________________________________________________________

IMP Activity: Congruent Figures 4

21. Earlier we said that Figure 1 was a trapezoid. How did we know this?

_____________________________________________________________________________

22. Identify the parallel sides in Figure 1. ___________________________________________

23. Are the corresponding sides in Figure 2 also parallel?_______________________________

24. Do you think that corresponding sides of parallel lines will also be parallel in all congruent

figures? Justify your reasoning.

______________________________________________________________________________

______________________________________________________________________________

25. In general, write three things you learned about congruent figures and their corresponding

sides and angles.

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

26. Two of the following congruence statements for Figures 1 and 2 are incorrect. Circle the

correct congruence statement for the figures 1 and 2. Explain why you picked that one.

𝐴𝐵𝐶F ≅ FCDE 𝑜𝑟 𝐴𝐵𝐶F ≅ CDEF 𝑜𝑟 𝐴𝐵𝐶F ≅ EDCF

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

27. Sally now says, “I can write a congruence statement for Figure 1 and Figure 2 many different

ways.” Johnny says, “No, there is only one correct way to write a congruence statement for

Figure 1 and Figure 2.” Who is correct?

_____________________________________________________________________________

_____________________________________________________________________________

Practice #1

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Name___________________________________

Period____Date________________Congruence and Triangles

Complete each congruence statement by naming the corresponding angle or side.

1) ∆DEF ≅ ∆KJI

D

E

F

K

J

I

FD ≅ ?

2) ∆BAC ≅ ∆LMN

B

A

C

LM

N

∠A ≅ ?

3) ∆TUV ≅ ∆GFE

T

U

V

GF

E

∠U ≅ ?

4) ∆WVU ≅ ∆GHI

W V

U G H

I

∠W ≅ ?

5) ∆ZXY ≅ ∆ZXC

Y

ZX

C

∠Y ≅ ?

6) ∆DEF ≅ ∆DSR

E F

D

SR

∠F ≅ ?

Write a statement that indicates that the triangles in each pair are congruent.

7)

J

KI

T R

S

8)

BC

D

G

H I

-1-

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9) R

PQ

D

10)

TR

SI

11)

VW

XC

D

E12)

ST

U CD

E

Mark the angles and sides of each pair of triangles to indicate that they are congruent.

13) ∆BDC ≅ ∆MLK

B

D

C

M

L

K

14) ∆GFE ≅ ∆LKM

G

F

EL

KM

15) ∆MKL ≅ ∆STL

M

K

L

S

T

16) ∆HIJ ≅ ∆JTS

H

I

J

T

S

17) ∆CDB ≅ ∆CDL

B

C

D

L

18) ∆JIK ≅ ∆JCD

I K

J

C

D

-2-

IMP Pre-Unit: Classifying Triangles 1

Classifying Triangles Here are examples of seven types of triangles. Look at them closely. Use your protractor and ruler to draw a larger example of each one. Label each figure with its name and the measures of its sides and angles.

Right Scalene

Acute Isosceles Equilateral Acute Scalene

Right Isosceles

Obtuse Scalene Obtuse Isosceles

Objective: Students will draw triangles using a protractor and ruler. Students review types

of triangles and characteristics of each.

IMP Pre-Unit: Classifying Triangles 2

There are three adjectives that can be used to describe the sides of a triangle. These are scalene, isosceles and equilateral. Use the examples from the previous page to help you write definitions for these three words when they are applied to triangles. Scalene: __________________________________________________ Isosceles: _________________________________________________ Equilateral: ________________________________________________ There are three adjectives that can be used to describe the angles of a triangle. These are acute, obtuse and right. Use the examples from the previous page to help you write definitions for these three words when they are applied to triangles. Acute: __________________________________________________ Obtuse: _________________________________________________ Right: ___________________________________________________ When your team has drafted definitions, use a dictionary to check and make revisions. Complete the table below sketching example triangles in each box that is possible.

Scalene Isosceles Equilateral

Acute

Obtuse

Right

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Kuta Software - Infinite Geometry Name___________________________________

Period____Date________________Classifying Triangles

Classify each triangle by each angles and sides. Base your decision on the actual lengths of the sides and

the measures of the angles.

1) 2)

3) 4)

5) 6)

-1-

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Classify each triangle by each angles and sides.

7)

8.6

8.6

8.660°

60°60°

8)

8.7

7.4

6.1

57°

44°79°

9)

11.213.2

7

90°

32°

58°

10)

4.5

2.5

2.5

26°

26°128°

11)

3

4.8

4.872°

72° 36°

12)

4.8

6.8

4.8

45°45°

Classify each triangle by each angles and sides. Equal sides and equal angles, if any, are indicated in

each diagram.

13) 14)

-2-

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15) 16)

17) 18)

Sketch an example of the type of triangle described. Mark the triangle to indicate what information is

known. If no triangle can be drawn, write "not possible."

19) acute isosceles 20) right scalene

21) right isosceles 22) right equilateral

23) acute scalene 24) obtuse scalene

25) right obtuse 26) equilateral

-3-

1 Investigating Triangles

Investigating Triangles

TASK A: Using your ruler and protractor, draw any type of triangle. Label the vertices A, B and C. Then label

the side opposite A as a, the side opposite B as b and the side opposite C as c.

Complete the following: I drew a _______________________________(scalene, isosceles, equilateral, acute,

right, obtuse) triangle because the sides are ________________________________ and the angles

______________________.

TASK B: Using your protractor and ruler, measure and label all the angles and side lengths of your triangle.

What is the sum of all the angles in the triangle? _______________________

Compare with two neighbors. Were the measures of their angles the same as yours? _______

What was the sum of the measures of their angles? _____________________

What conclusion can you draw about the sum of all the angles of any triangle?

__________________________________________________________________________________________

__________________________________________________________________________________________

Complete the following:

The longest side of my triangle was _______________. The largest angle was _____________.

The shortest side of my triangle was _______________. The smallest angle was ____________.

Compare your answers with a neighbor. Is there a relationship between the largest angles and the longest side and the smallest angle and the shortest side? Explain.

__________________________________________________________________________________________

__________________________________________________________________________________________

__________________________________________________________________________________________

__________________________________________________________________________________________

Objective: Students will review the triangle sum theorem and be introduced to relationships within a triangle.

2 Investigating Triangles Teacher Directions

Teacher Directions: Investigating Triangles

Objective: Students will review the triangle sum theorem and be introduced to relationships within a

triangle.

Materials:

Ruler – 1 per student

Protractor – 1 per student

Activity Notes:

TASK A: Students should create a triangle and be able to label it according to its side lengths and angle measures. It also allows students to practice using a protractor.

Triangle Angle Sum Theorem states: The sum of the three angles in any triangle sum to 180 degrees

Inequalities in One Triangle:

If two sides of a triangle are not congruent, then the larger angle lies opposite the longer side.

If two angles of a triangle are not congruent, then the longer side lies opposite the larger angle.

These concepts will help in establishing base angles are congruent in isosceles triangles.

Triangle Sum Theorem

Find the value of x and the measure of each angle.

7)

8)

9)

10)

Name: ___________________________________ Date: ____________ Period: __________

Practice: Inequalities in Triangles and Triangle Sum

Name: _________________________

IMP Activity: Drawing Triangles with Given Sides 1

Drawing Triangles with Given Sides

1. Using a ruler and pencil draw a triangle ABC with sides measuring exactly 4, 5 and 6 cm.

Report your steps: _____________________________ _____________________________ _____________________________ How many tries did it take? _____________________________ How accurate is your triangle? _____________________________ What step is challenging? Why? _____________________________ _____________________________

2. Now, using a using a compass and ruler, Construct a triangle with side lengths 4, 5, and 6 cm:

List the steps: _____________________________ _____________________________ _____________________________ _____________________________ _____________________________ _____________________________ _____________________________

Objective: Students experience and practice constructing triangles with given side lengths. Students

are given three side lengths and investigate whether or not a triangle can be created with these measurements.

IMP Activity: Drawing Triangles with Given Sides 2

Task B: Using a ruler and compass, construct the following:

3. Construct a triangle with sides measuring exactly 3, 5 and 7 cm.

4. Construct a triangle with sides measuring exactly 3, 4 and 5 cm.

5. Construct a triangle with sides measuring exactly 4, 6, and 6 cm.

6. Construct a triangle with sides measuring exactly 5, 5 and 5 cm.

IMP Activity: Drawing Triangles with Given Sides 3

7. Describe each of the following triangles using two of the following terms: Scalene, Isosceles,

Equilateral, Acute, Right, or Obtuse.

Triangle # 3: _____________________________ Triangle # 4: _________________________

Triangle # 5: ____________________________ Triangle # 6: __________________________

8. Jill is having trouble constructing a triangle with sides measuring 1, 5 and 7 cm. See if you

can construct this triangle.

9. Compare your conclusion with a neighbor. Record your findings.

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

10. Give 3 examples of three side lengths that couldn’t be used to construct a triangle

1) _______________________ 2) ___________________ 3)__________________

11. Is there a way to know if you could construct a triangle just given any three sides lengths

without trying to construct the triangle? Explain.

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

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Kuta Software - Infinite Geometry Name___________________________________

Period____Date________________The Triangle Inequality Theorem

State if the three numbers can be the measures of the sides of a triangle.

1) 7, 5, 4 2) 3, 6, 2

3) 5, 2, 4 4) 8, 2, 8

5) 9, 6, 5 6) 5, 8, 4

7) 4, 7, 8 8) 11, 12, 9

9) 3, 10, 8 10) 1, 13, 13

11) 2, 15, 16 12) 10, 18, 10

Two sides of a triangle have the following measures. Find the range of possible measures for the third

side.

13) 9, 5 14) 5, 8

15) 6, 10 16) 6, 9

17) 11, 8 18) 14, 11

IMP Activity: My 1stTriangle Congruence Theorem

1

My 1st Triangle Congruence Theorem

TASK A: Using just these three lengths, 8 cm, 4 cm and 6 cm, we can make constructions in a variety of ways. We could choose differing lengths as the base, the left side and the right side. Use the chart below to organize all the possible ways to construct a triangle using these three lengths.

1. Do you think you have found all the possible combinations? Explain how you know.

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

2. From your experience, describe the steps to create a triangle with given side lengths.

______________________________________________________________________________

______________________________________________________________________________

________________________________________________________________________________________________________

Base Left side Right side

Original Triangle

8 cm 4 cm 6 cm

base

left right

Objective: Students will construct a triangle with given side lengths. Students will use rigid

transformations to prove the SSS ≅ theorem.

IMP Activity: My 1stTriangle Congruence Theorem

2

Use a compass and ruler with the steps to construct the “original” triangle from the chart, with a base of 8 cm, a left side of 4 cm and a right side of 6 cm. Then, carefully create each triangle listed in the chart on the previous page. Lengths have been provided for your convenience.

Original Triangle

Triangle 2

Triangle 3

Triangle 4

Triangle 5

Triangle 6

IMP Activity: My 1stTriangle Congruence Theorem

3

TASK C: Next, use a sheet of patty paper to trace the “original triangle” from the previous page. Determine if the original triangle is congruent to each of the other triangles and describe the transformation that proves they are congruent.

Original Triangle to Triangle 2:

______________________________________________________________________________

Original Triangle to Triangle 3:

______________________________________________________________________________

Original Triangle to Triangle 4:

______________________________________________________________________________

Original Triangle to Triangle 5:

______________________________________________________________________________

Original Triangle to Triangle 6:

______________________________________________________________________________

3. After completing the activity, how many different triangles can you make given 3 sidelengths? ______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

4. What can you conclude about triangles that have the same side lengths?______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

The Side-Side-Side Congruence Theorem states that if the three sides of one triangle are congruent to the three sides of a second triangle, then the two

triangles are congruent.

IMP Activity: Testing Other Triangle Congruence Theorems

1

Testing Other Triangle Congruence Theorems We learned that all triangles with the same three side lengths are congruent(Side-Side-Side≅Theorem). Today we are going to check if any other rules will also show triangle congruence. Let’s check Angle-Side-Side. Here, the order prescribes the location of the measurements. Angle – Side – Side is different than Side – Angle – Side. Here are the steps to draw a triangle when given Angle-Side-Side:

1. Draw a segment for the base that is the first length given.

2. Then, use your protractor and ruler to draw the angle attached to the left endpoint of

the base. Remember, we only know the angle on the left side, so we do not know the

length of this side of the triangle. Make sure you draw it long in case it needs to be long.

Use an arrow to show that we know the direction, but not the length.

3. Finally, swing an arc for the second given length attached to the right endpoint of the

base.

TASK A Construct a triangle with the given the Angle – Side – Side measurements:

1. Use a compass, ruler, and protractor with these steps to draw a triangle with (30°, 7 cm,

4 cm).

angle

The angle is not in

between the two known

sides. the first given side

the second given side

(angle, side, side)

Objective: Students will justify their reasoning as to whether or not Angle – Side – Side is a

congruence theorem.

IMP Activity: Testing Other Triangle Congruence Theorems

2

2. Is there anything interesting about this construction? In other words, is there anything

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

3. Did everyone draw the same triangle? If not, what do you think this means?

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

4. Write some ideas and pictures that come from your class discussion here

5. Let’s try another Angle-Side-Side drawing using the same measurements in a different

order. This time draw a triangle with 30°, 4 cm, 7 cm.

6. How many triangles were you able to draw this way?

________________________________________________________________________

IMP Activity: Testing Other Triangle Congruence Theorems

3

7. Is the triangle you drew in #5 congruent to either of the two triangles that were possible

in #1? Remember that congruent means that one shape can be moved using

translations, rotations or reflections to match exactly on top of the other. Use

appropriate tools to check and justify your answer.

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

8. Would anything be different for Side-Side-Angle? Try drawing 4 cm, 7 cm, 30°.

9. Prepare some thoughts for your class discussion:

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

10. Write your class conclusions here:

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

11. Is there an Angle – Side –Side ≅ theorem? Explain your answer.

________________________________________________________________________

________________________________________________________________________

______________________________________________________________________

IMP Activity: Looking for More Triangle Congruence Theorems 1

Looking for More Triangle Congruence Theorems

So far we have learned that when given 3 sides lengths, we can construct exactly 1 triangle. Therefore, if three sides of one triangle are the same measurements as three sides of another triangle, the triangles are congruent (Side – Side - Side≅ Theorem). Then we tried to see what would happen if we were given 1 angle measure and 2 consecutive side lengths. We found that we could create 2 different triangles. So, Angle – Side – Side is NOT a prescription for congruent triangles. Today we will try again using 1 angle measure and 2 side lengths, but this time the angle will be between the two sides. Let’s investigate Side-Angle-Side. Here are the steps to use when given Side-Angle-Side:

1. Draw a segment for the base that is one of the lengths given.

2. Use your protractor to draw a ray with the given angle attached to one of the endpoints.

3. Use your compass to swing an arc for the second length attached to the same endpoint.

4. Finally, connect the endpoint of the base to the intersection of the arc and the ray.

1. Construct the Side-Angle-Side combination of 4 cm , 20°, 5 cm. Find how many different

ways there are to draw the triangle. Show your drawings here.

Objective: Students will investigate Side- Angle – Side and Angle-Side-Angle ≅ theorems.

1. Base that is one of the given lengths

2. measure angle with protractor 3. arc that is the second length

Side – Angle – Side

The angle is BETWEEN the

two known sides. It is also

called the “included” angle.

4. connect the endpoint of the base to the intersection

IMP Activity: Looking for More Triangle Congruence Theorems 2

2. Are any of the triangles you have drawn congruent to each other? Justify your answer. ______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

3. What conclusion can you make about Side-Angle-Side? ______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

So far we have found 2 triangle congruence theorems.

1. Side – Side – Side

2. Side – Angle – Side

We have also learned that Angle – Side – Side is NOT a congruence theorem. Lets try: Angle – Side - Angle TASK B When you are given the measurements for Angle-Side-Angle, list the steps needed to draw the triangle:

First,_________________________________________________________________

_____________________________________________________________________

_____________________________________________________________________

_____________________________________________________________________

The Side-Angle-Side Congruence Theorem states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent.

IMP Activity: Looking for More Triangle Congruence Theorems 3

4. For the Angle-Side-Angle combination of 30°, 6 cm, 60°, find how many different ways there

are to draw the triangle. Show your drawings here.

5. Are any of the triangles you have drawn congruent to each other? Justify your answer. ______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

6. What conclusion can you make about Angle-Side-Angle? ______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

The Angle-Side-Angle Congruence Theorem states that if two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the two triangles are congruent.

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Name___________________________________

Period____Date________________SSS and SAS Congruence

State if the two triangles are congruent. If they are, state how you know.

1) 2)

3) 4)

5) 6)

7) 8)

9) 10)

-1-

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State what additional information is required in order to know that the triangles are congruent for the

reason given.

11) SAS

J

H

I

E

G

12) SAS

ML

K

G

HI

13) SSS

YZ

XD

14) SSS

R S

T XY

Z

15) SAS

UV

WX

YZ

16) SSS

E

FG

Y

XW

17) SAS

E F

G

Q

18) SAS

TR

S

D B

-2-

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Name___________________________________

Period____Date________________ASA and AAS Congruence

State if the two triangles are congruent. If they are, state how you know.

1) 2)

3) 4)

5) 6)

7) 8)

9) 10)

-1-

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State what additional information is required in order to know that the triangles are congruent for the

reason given.

11) ASA

E

C D

Q

12) ASA

KL

M

U

TS

13) ASA

TR

S

E

C

14) ASA

U

W

V

M

K

15) AAS

E

D

C

T

16) AAS

XY

Z L

M

N

17) ASA

IG

HV

18) AAS

J

K L

F

-2-

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Name___________________________________

Period____Date________________SSS, SAS, ASA, and AAS Congruence

State if the two triangles are congruent. If they are, state how you know.

1) 2)

3) 4)

5) 6)

7) 8)

9) 10)

-1-

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State what additional information is required in order to know that the triangles are congruent for the

reason given.

11) ASA

S

U T

D

12) SAS

W

X

V

K

13) SAS

AB

C

JK

L

14) ASA

D

E

F J

K

L

15) SAS

H

IJ

R

ST

16) ASA

M

L

K

S T

U

17) SSS

R

SQ

D

18) SAS

W

U

V

M

K

-2-

IMP Lesson: Bringing It All Together on Triangle Congruence 1

Bringing It All Together on Triangle Congruence

1. Two triangles have all three sides congruent to each other. Draw and label a picture of

two triangles with the given properties.

2. Is that enough to say the triangles are congruent? What do we call this situation? Answer with

complete sentences. ____________________________ ____________________________

____________________________ ____________________________ _____________________________

3. Two triangles have two sides congruent and an angle that is not between the sides congruent. Draw and label a picture of two triangle with these properties.

4. Is that enough to say the triangles are congruent? What do we call this situation? Answer with complete sentences. ______________________________ ______________________________ ______________________________ ______________________________ ______________________________

Picture

Picture

Objective: Students will summarize and describe the discoveries they’ve made about triangle

congruence theorems.

IMP Lesson: Bringing It All Together on Triangle Congruence 2

5. Two triangles have two sides and the included angle congruent. Draw and label a picture of two triangle with these properties.

6. Is that enough to say the triangles are congruent? What do we call this situation? Answer with complete sentences. ______________________________ ______________________________ ______________________________ ______________________________ ______________________________

7. Two triangles have two angles and the included side congruent. Draw and label a picture of two triangle with these properties.

8. Is that enough to say the triangles are congruent? What do we call this situation? Answer with complete sentences. ______________________________ ______________________________ ______________________________ ______________________________ ______________________________

9. Fill out the following t-chart by placing each set of three triangle parts on the left if they are enough to prove two triangles are congruent, and on the right if they are not. SSS ASS (or SSA) SAS ASA

Picture

Picture

1 Proving More Congruence Theorems

Proving More Congruence Theorems

A mathematical proof is an argument that begins with known facts, proceeds from there

through a series of logical deductions, and ends with the thing you’re trying to prove.

A paragraph proof allows you to explain your reasoning. In a paragraph proof, the statements

and their justifications are written together in a logical order in paragraph form.

Using the triangle congruence theorems that we’ve already proved are true and other facts we

know about triangles, prove the Angle – Angle – Side is a congruence theorem.

1. The sum of all the interior angles of a triangle is _________________.

2. So, < 𝐴+ < 𝐵+ < 𝐶 = ________ and < 𝑋+ < 𝑌+< 𝑍 = _________.

3. Since < 𝐴 = < 𝑋 𝑎𝑛𝑑 < 𝐶 = _______, then < 𝐵 =___________. That is, if two angles of

one triangle are equal to two angles of another triangle, then the third angles are also

_____________________. Since 𝐵𝐶̅̅ ̅̅ ≅ 𝑌𝑍̅̅̅̅ the triangles are congruent by A – S – A. So, the

given Angle – Angle – Side congruence implies A – S – A so the triangles are congruent and A – A

- S is a congruence theorem.

A

B C Z

X

Y

Given: < 𝐴 ≅< 𝑋, < 𝐶 ≅< 𝑌,

𝑎𝑛𝑑 𝐵𝐶̅̅ ̅̅ ≅ 𝑌𝑍̅̅̅̅ .

Prove: ∆𝐴𝐵𝐶 ≅ ∆𝑋𝑍𝑌

The Angle-Angle-Side Congruence Theorem states that if two angles and any side of one triangle are congruent to two angles and any side of a second triangle, then the two triangles are congruent.

Objective: Students will practice paragraph proofs and prove other congruence theorem not easily

discovered through construction.

2 Proving More Congruence Theorems

Prove that in right triangle, Hypotenuse – Leg is a congruence theorem.

_______________________________________

_______________________________________

_______________________________________

_______________________________________

_______________________________________

_______________________________________

________________________________________

________________________________________

______________________________________________________________________________

______________________________________________________________________________

4

4

5

5

The Hypotenuse - Leg Congruence Theorem states that if ________________________________ of one triangle are congruent to _____________________________________ of a second triangle, then the two triangles are congruent.

What facts are we given about the two

triangles?

Given: _______________________

_____________________________

What else do we know about right triangles that

could give us MORE facts?

_____________________________

_____________________________

_____________________________

What are we trying to prove?

Prove: __________________________________

What theorem do we already know is a

congruence theorem that we could apply to

these given facts?

_______________________________________

A

B

C

M

N

O

IMP Activity: Formal Language 1

Using Formal Language

Part A:

Yesterday, we constructed 6 triangles given 3 side lengths. Sally was absent yesterday, again.

Sally says, “I constructed the two triangles below. They are not congruent because they don’t

even look alike.”

1. Explain to Sally how you know the triangles are congruent.

__________________________________________________________________________________

__________________________________________________________________________________

__________________________________________________________________________________

2. There are four ways that we have learned so far to show that two triangles are congruent.

The four ways are given below. How can you use each of the following to prove two triangles

are congruent?

a) Through Rigid Motions _____________________________________________________

________________________________________________________________________

b) 𝑆𝑆𝑆 ≅ 𝑇ℎ𝑒𝑜𝑟𝑒𝑚 _________________________________________________________

________________________________________________________________________

c) 𝑆𝐴𝑆 ≅ 𝑇ℎ𝑒𝑜𝑟𝑒𝑚_________________________________________________________

________________________________________________________________________

d) 𝐴𝑆𝐴 ≅ 𝑇ℎ𝑒𝑜𝑟𝑒𝑚______________________________________________________________

______________________________________________________________________________

Objective: Students will use paragraph proofs to prove two triangles are congruent.

IMP Activity: Formal Language 2

Part B:

Definition of Proof – evidence sufficient to establish a thing as true

Mathematical Proof - an argument that begins with known facts, proceeds from there through

a series of logical deductions, and ends with the statement you’re trying to prove.

2. What was it that you were trying to prove to Sally? ______________________

___________________________________________________________________

3. What facts did you know about the triangles that could be used as evidence?

___________________________________________________________________

___________________________________________________________________

A paragraph proof is one way a proof is often written. The advantage of a paragraph proof is

that you have the chance to explain your reasoning in your own words. In a paragraph proof,

the statements and their justifications are written together in a logical order in a paragraph

form. There is always a diagram and a statement of the given and prove sections before the

proof.

Given:

Prove:

From the Given, I know _______________________________________________.

Therefore, triangle _____ and _____ must be congruent because _____________

___________________________________________________________________

IMP Activity: Formal Language 3

Part C: 4. Prove that the two triangles are congruent, using a Triangle Congruence Theorem.

Given:

Prove:

From the Given, I know _______________________________________________.

Therefore, __________________________________________________________

___________________________________________________________________

5. Prove that the two triangles are congruent using Rigid Motions.

Given:

Prove:

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

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Name___________________________________

Period____Date________________Right Triangle Congruence

State if the two triangles are congruent. If they are, state how you know.

1) 2)

3) 4)

5) 6)

7) 8)

9) 10)

-1-

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State what additional information is required in order to know that the triangles are congruent for the

reason given.

11) HL

D

E

F VW

X

12) LL

A

BC

V

WX

13) LL

LK

M

H

14) HA

L M

N B C

D

15) LA

C

B

DI

J

16) HA

DE

C

UV

17) HL

C

D

E

HI

J

18) LA

D F

E

TV

-2-

Directions: Check which congruence postulate you would use to prove that the two triangles are congruent.

1.

2.

3.

4.

5.

Practice. Fill in the missing reasons

6. Given: YLF FRY, RFY LFY

Prove: FRY FLY

Statement Reason

1. YLF FRY

2. RFY LFY

3. FYFY

4. FRY FLY

7. Given: TRLT , ILT ETR, IT || ER

Prove: LIT TER

Statement Reason

1. TRLT

2. ILT ETR

3. IT || ER

4. LTI ERT

5. LIT TER

8. Given: DABC

AC bisects BCD

Prove: ABC CDA

Statement Reason

1. DABC

2. AC bisects BCD

3. DCABCA

4. ACAC

5. CDAABC

Practice. Write a 2-column proof for the following problems.

9.

Multiple Choice Practice

10. Which condition does not prove that two triangles are congruent?

(1) (2) (3) (4)

11. In the diagram of and below, , , and .

Which method can be used to prove ?

(1) SSS (2) SAS (3) ASA (4) HL

12. In the accompanying diagram of triangles BAT and FLU, and .

Which statement is needed to prove ?

(1) (2) (3) (4)

13. Complete the partial proof below for the accompanying diagram by providing reasons for

steps 3, 6, 8, and 9.

Given: , , , ,

Prove:

Statements Reasons

1 1 Given

2 , 2 Given

3 and are right angles. 3

4 4 All right angles are congruent.

5 5 Given

6 6

7 7 Given

8 8

IMP Practice: Who is Right? 1

Who’s right?

Amy says: I have drawn a triangle. It has one angle of 70° and another angle of 50°. Ben, you draw a triangle with the same two properties. Ben says: My triangle will be congruent to yours because all triangles that have those two

properties must be congruent.

Is Ben correct? Explain your answer with words and drawings.

Sara says: I have drawn a right triangle. It has one angle of 60° and one side is 10 cm long. Sanjay, you draw a right triangle with the same properties. Sanjay says: My triangle might be congruent to yours, but there are several non-congruent

triangles that have those properties. Is Sanjay correct? Explain your answer with words and drawings.

Objective: Students decide if given information is sufficient to describe a single triangle or if

several non-congruent triangles might have the same properties.

IMP Practice: Who is Right? 2

Burt says: I have drawn a right triangle. One side is 4 cm long and another side is 6 cm long. It has an angle that measures 120° between those two sides. Max, you draw a right triangle with the same properties. Max says: My triangle will be congruent to yours because all triangles that have those

properties must be congruent. Is Max correct? Explain your answer with words and drawings.

TASK D: Create your own discussion between to students. Determine who you want to be right

or wrong. When you’re finished, have a friend determine who they think is right.

IMP Toolkit: Triangle Congruence x 1

Triangle Congruence Toolkit

Congruence

Figures are called congruent if one can be mapped onto the other using rigid transformations. The three rigid transformations that define congruence are: ____________________ ____________________ ____________________ Corresponding or matching parts of congruent figures have the same measurements. This is a defining characteristic of congruent figures. When making a congruence statement, corresponding parts should match. For example: ∆𝐴𝐵𝐶 ≅ ∆𝑋𝑌𝑍 means that <A and <X are congruent, corresponding parts, 𝐴𝐶̅̅ ̅̅ and 𝑋𝑍̅̅ ̅̅ are congruent, corresponding parts, etc There are also three triangle congruence theorems which give us other ways to show triangles are congruent. These are: _________ which means ____________________________________________________________________ _________ which means ____________________________________________________________________ _________ which means ____________________________________________________________________

SSS

A compass and ruler can be used to draw a triangle given the three side measurements. ___________ ___________ ___________

My steps are:

_________________________

_________________________

_________________________

_________________________

_________________________

_________________________

Name:____________________________________Date:_________________Period:___________

IMP Toolkit: Triangle Congruence x 2

SAS

A ruler and protractor can be used to draw a triangle given two side measurements and the included angle. ___________ ___________ ___________

My steps are:

_________________________

_________________________

_________________________

_________________________

_________________________

_________________________

ASA

A ruler and protractor can be used to draw a triangle given two angle measurements and the included side. ___________ ___________ ___________ My steps are:

_________________________

_________________________

_________________________

_________________________

_________________________

_________________________

Triangle Inequality

The Triangle Inequality Theorem helps me know if three measurements could make a triangle. It says:

Name: ________________________________

UNIT 7: Triangle Congruence Test Prep

Describe/define the following:

1. reflection –

2. translation –

3. rotation –

4. dilation –

In # 5 – 8, identify the type of transformation represented by each diagram.

5. 6.

7. 8.

A

C B

Original

A

C B

Image

4

5

1 2

3 Image

1

2

3 4

5

Original

A B

D C

Original

D C

A B

Image

Original

A

D

E B

C

Image

A

C

B E

D

In # 9 - 11, sketch 2 triangles (without tools / ruler ok) that are congruent according to each of the

theorems stated. The second triangle should be oriented differently than the first triangle (i.e. rotate

and/or reflect) The ruler is to be used as a straight edge, not a measuring tool.

9. Side – Side – Side Theorem

10. Angle – Side – Angle Theorem

11. Side – Angle – Side Theorem

12. Sketch 2 triangles that are not congruent even though they have 2 pairs of congruent sides and 1 pair of

non-included angles congruent (angles are not between the 2 sides).

For # 13 – 16, use a compass, ruler and protractor, as needed, to construct the following. Label the angle

measures and side lengths given on the figures.

13. A triangle with a 50 angle formed by 2 sides that are both 5cm long.

14. A triangle with sides 4cm, 5cm and 8cm.

15. A triangle with angles of 25 40and and an included side of 9cm.

16. A triangle with sides 4 cm, 2 cm and 8cm.

17. Give an example of 3 side lengths that could form a triangle. Explain.

18. Give an example of 3 side lengths that could not form a triangle. Explain.

For # 19 – 20, without tools (ruler ok), draw the following and follow the directions.

19. Draw two scalene congruent right triangles. The second should be oriented differently than the first.

Label all vertices. Use ABC and XYZ for the vertices. Write a congruence statement for

the triangles. Identify all congruent pairs of sides and angles. Determine a sequence of transformations

that could prove the triangles congruent.

20. Draw two scalene congruent right triangles that share one side. Label the vertices ABC & ACD, not

necessarily in that order. Write a congruence statement for the triangles. Identify all congruent pairs of

sides and angles. Determine a sequence of transformations that could prove the triangles congruent.

21. Draw 2 triangles to match the statement: ΔXYZ = ΔWZY. (Hint: the triangles share vertices Y and Z)

Identify all congruent pairs of sides and angles. Determine a sequence of transformations that could

prove the triangles congruent.

For # 22 – 26, follow the directions given below.

22. Draw the reflection of AB about line y. Label it ' 'A B .

23. Draw the reflection of AB about line x. Label it '' ''A B .

24. Draw the reflection of ' 'A B about line x. Label it ''' '''A B .

25. The length of AB is __________ units.

26. Is the figure formed a:

b) Parallelogram? Show proof.

c) Rectangle? Show proof.

d) Square? Why?

e) Rhombus? Why?

For # 27 – 32, determine if the triangles are congruent according to the given information.

If they are congruent, give the reason. If there is not enough information, write not enough information.

27. 28.

5

5

29. 30.

31. 32.

6 6

10 10

9 4

11

9 4

11

33. Use patty paper to determine if the two triangles are congruent.

a.

b.

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