Geometry ~ Chapter 4 Congruent Triangles & Chapter 6 Proportions and Similarity Learning Targets: The student explains proofs or reasoning related to theorems about triangles. (Ch 4) o Theorems include, but are not limited to: measures of interior angles of a triangle sum to 180 degrees; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. The student will understand proportions and similarity among polygons and triangles. (Ch 6) Section Required Assignments Additional Assignments Sec 4-1 Classifying Triangles (Formative Assessments) ⎕ Chapter 4 Triple Entry Journal ⎕ Sec 4-1 #’s 13, 15, 17, 23, 25, 27, 29, 33, 35, 38, 39 (11 questions) ⎕ F.1 Classify triangles (iXL) ⎕ Sec 4-1 #’s1-12 ⎕ Pg 760 Lesson 4-1 #’s 1-9 Sec 4-2 Angles of Triangles (Formative Assessments) ⎕ Sec 4-2 #’s 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 40, (16 questions) ⎕ F.2 Triangle Angle-Sum Theorem (iXL) ⎕ K.8 Congruency in isosceles and equilateral triangles (iXL) ⎕ Sec 4-2 #’s 1-10 ⎕ Pg 761 Lesson 4-2 #’s 1-10 Sec 4-3 Congruent Triangles (Formative Assessments) ⎕ Sec 4-3 #’s 9, 11, 13, 15, 17, 23, 25, 27, 29, 30, 31, 32, 34 (13 questions) ⎕ Sec 4-3 #’s 1-8 ⎕ Pg 761 Lesson 4-3 #’s 1-5 Quiz Sec 4-1 to 4-3 (Summative Assessment) Must receive a grade of > 70 Things you should know: How to classify a triangle by angles (acute, obtuse or right) How to classify a triangle by sides (scalene, isosceles or equilateral) What is an equiangular triangle? Using the distance formula to help classify triangles. How many degrees do the angles in a triangle add up to be? Which two angles add up to equal an exterior angle of a triangle? How do you label congruent triangles? How to you write two triangles are congruent? Can you label 6 congruent parts of two triangles? Can you name the 3 congruence transformations?
Transcript
Geometry ~ Chapter 4 Congruent Triangles &
Chapter 6 Proportions and Similarity
Learning Targets:
The student explains proofs or reasoning related to theorems about triangles. (Ch 4) o Theorems include, but are not limited to: measures of interior angles of a triangle sum to 180
degrees; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.
The student will understand proportions and similarity among polygons and triangles. (Ch 6)
What parts are proportional in the Triangle Proportionality Theorem? (Sec 6-4)
What parts are congruent in the Triangle Midsegment Theorem? (Sec 6-4)
What if you have three or more parallel lines that intersect two transversals, which parts are proportional? (Sec 6-4)
If two triangles are similar, how do their perimeters compare? (Sec 6-5)
What is a median of a triangle? (Sec 6-5)
When you have an angle bisector in a triangle, which sides are proportional? (Sec 6-5)
Review
(Formative Assessment)
⎕ Chapter 4 Study Guide and Review #’s 1-8 vocabulary
#’s 9-11 Classifying Triangles
#’s 12-14 Angles of Triangles
#’s 15-17 Congruent Triangles
#’s 18-19 Proving Congruence – SSS, SAS
#’s20-21 Proving Congruence – ASA, AAS
#’s 22-25 Isosceles Triangles ⎕ Chapter 6 Study Guide and Review
#’s 1-9 True/False
#’s 10-17 Proportions
#’s 18-21 Similar Polygons
#’s 22-26 Similar Triangles
#’s 27-34 Parallel Lines and Proportional Parts
#’s 35-38 Parts of Similar Triangles
⎕ Chapter 4 Practice Test #’s 1-18
⎕ Chapter 6 Practice Test
#’s 1-19
(Summative Assessment)
Must receive a grade > 70
⎕ Chapter 4/6 Project
(Summative Assessment)
Must receive a grade > 70
⎕ Chapter 4/6 TEST
Geometry ~ Chapter 4 Triple Entry Journal
Learning Targets:
The student explains proofs or reasoning related to theorems about triangles. (Ch 4) o Theorems include, but are not limited to: measures of interior angles of a triangle sum to 180
degrees; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.
The student will understand proportions and similarity among polygons and triangles. (Ch 6)
Word Definition Picture and/or Example acute triangle
(pg 178)
base angles (pg 216)
congruent triangles (pg 192)
coordinate proof
(pg 222)
equiangular triangle (pg 178)
equilateral triangle (pg 179)
exterior angle (pg 186)
included angle
(pg 201)
included side (pg 207)
isosceles triangle (pg 179)
obtuse triangle (pg 178)
remote interior angles
(pg 186)
right triangle (pg 178)
scalene triangle (pg 179)
vertex angle (pg 216)
cross products (pg 283)
extremes (pg 283)
means (pg 283)
midsegment (pg 308)
proportion (pg 283)
ratio (pg 282)
scale factor (pg 290)
similar polygons (pg 289)
Geometry ~ Chapter 4/6 Notes
Learning Targets:
The student explains proofs or reasoning related to theorems about triangles. (Ch 4) o Theorems include, but are not limited to: measures of interior angles of a triangle sum to 180
degrees; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.
The student will understand proportions and similarity among polygons and triangles. (Ch 6)
Sec 4-1 Classifying Triangles
Key Points Notes
Classifying Triangles by Sides
Classifying Triangles by Angles
Examples of Classifying Triangles
Classify each by their sides AND by their angles.
Sec 4-1 Classifying Triangles (continued……)
Key Points Notes
Solving using triangle
knowledge.
Coordinate Geometry
Find ‘d’ and the measure of EACH side of equilateral triangle KLM if KL = d + 2, LM = 12 – d, and KM = 4d – 13. Find the measures of the sides of Triangle RST. Classify the triangle by sides. R = (-1, -3) S = (4, 4) T = (8, -1)
Sec 4-2 Angles of Triangles
Key Points Notes
Angle Sum Theorem
Example of Using the Angle Sum
Theorem
Sec 4-2 Angles of Triangles
Key Points Notes
Third Angle Theorem
Exterior Angle Theorem
If two angles of one triangle are congruent to two angles of a second triangle, then the third angles of the triangles are congruent.
Example of using the Exterior Angle Theorem
Sec 4-3 Congruent Triangles
Key Points Notes
Definition of Congruent Triangles (CPCTC)
Congruence Transformations
Two triangles are congruent if and only if their corresponding parts are congruent. C orresponding P arts in C ongruent T riangles are C ongruent
Based on the two triangles above, what are the 6 congruent parts?
Sec 4-3 Congruent Triangles (continued…….)
Key Points Notes
Make a
congruence statement.
Identify the congruence
transformation.
Prove these triangles are
congruent or not.
Sec 4-4 Proving Congruence – SSS, SAS
Key Points Notes
Example of SSS and SAS
Example of a proof involving
SSS or SAS
SAS
Given: C is the midpoint of AE C is the midpoint of BD Prove:
Sec 4-5 Proving Congruence – ASA, AAS
Key Points Notes
Example of ASA, AAS and HL
Answer as SSS, SAS, ASA, AAS, HL
or none
Sec 4-5 Proving Congruence – ASA, AAS
Key Points Notes
Proofs involving SSS, SAS, ASA,
AAS, or HL
Given: Prove:
Given: , Prove:
Sec 4-6 Isosceles Triangles
Key Points Notes
Isosceles Triangle
Theorem
Corollaries
If two sides of a triangle are congruent, then the angles opposite those sides are congruent.
4.3 A triangle is equilateral if and only if it is equiangular. 4.4 Each angle of an equilateral triangle measures 60 degrees.
Sec 4-6 Isosceles Triangles (continued……)
Key Points Notes
Proof with
isosceles triangles.
Examples with isosceles triangles.
Given: is isosceles with vertex ‘A’ Prove:
If , , 12 , what is the measure of <ABX?
Sec 6-1 Proportions
Key Points Notes
What is a ratio?
What is a proportion?
(2 ratios that are equal to each
other.)
Example of a ratio.
Example of a proportion.
Solve the proportion.
5 girls : 15 boys (this is a ratio)
This can also be written as
The total number of students who participate in sports programs at Central High School is 520. The total number of students in the school is 1850. Find the athlete-to-student ratio to the nearest tenth. A boxcar on a train has a length of 40 feet and a width of 9 feet. A scale model is made with a length of 16 inches. Find the width of the model. 4 5
3
26
6
Sec 6-2 Similar Polygons
Key Points Notes
Similar polygons
vs Congruent polygons
Similarity statements
Similar polygons Congruent polygons
(same shape….different sizes) (same shape & same size)
Which triangle(s) are similar to Triangle ABC? Write a similarity
statement.
Sec 6-2 Similar Polygons (continued…….)
Key Points Notes
Examples of
Similar Polygons
Determine whether the figures are similar.
Justify your answer.
a) Write a similarity statement. Then find x, y, and UV.
b) Find the scale factor of polygon ABCDE to polygon RSTUV.
Sec 6-3 Similar Triangles
Key Points Notes
Angle-Angle
Similarity Theorem
Side-Side-Side Similarity Theorem
Side-Angle-Side
Similarity theorem
Example
Sec 6-3 Similar Triangles
Key Points Notes
Examples
Sec 6-4 Parallel Lines and Proportional Parts
Key Points Notes
Triangle
Proportionality Theorem
Triangle Midsegment
Theorem
Example of the Triangle
Proportionality Theorem
If a line is parallel to one side of a triangle and intersects the other two
sides in two distinct points, then it separates these sides into segments
of proportional lengths
If D and E are midpoints of and , and DE = ½ AB.
Sec 6-4 Parallel Lines and Proportional Parts (continued……)
Key Points Notes
Example of the
Triangle Midsegment
Theorem
*Things you need
to remember
Example
A and B are midpoints of and and . Find x.
Midpoint formula: (Average the x’s, Average the y’s)
Distance formula: ( (
Triangle ABC has vertices A(-2, 2), B(2, 4), and C(4, -4). is a
midsegment of Triangle ABC.
a) Find the coordinates of D and E (midpoints of and ).
b) Verify that .
c) Verify that DE = ½ BC
Sec 6-4 Parallel Lines and Proportional Parts (continued……)
Key Points Notes
More Examples
Sec 6-5 Parts of Similar Triangles
Key Points Notes
Proportional Perimeters Theorem
Special Segments of Similar Triangles
If two triangles are similar, then the perimeters are proportional to the
measures of corresponding sides.
a) Find the length of the missing sides.
b) Find the perimeter of the large triangle.
c) Find the perimeter of the smaller triangle.
d) What is the ratio of a side on the large triangle to the corresponding
side in the smaller triangle?
e) What is ratio of the large perimeter to smaller perimeter?
a) If you have similar triangles then the ALTITUDES will also be
proportional.
b) If you have similar triangles then the ANGLE BISECTORS will
also be proportional.
c) If you have similar triangles then the MEDIANS will also be
proportional.
Sec 6-5 Parts of Similar Triangles (continued…….)
Key Points Notes
Examples
Angle Bisector Theorem
If Triangle ABC is similar to Triangle XYZ, AC = 32, AB = 16, BC =
16 5, and XY = 24, find the perimeter of Triangle XYZ.
and BC = 1/3 NO. Find the ratio of the length of an
altitude of to the length of an altitude of .
An angle bisector in a triangle separates the opposite side into segments
that have the same ratio as the other two sides.
Geometry Chapter 4/6 Study Guide Sec 4-1 Classifying Triangles
⎕ Classify Triangles by Angles ⎕ Classify Triangles by Sides ⎕ Know how to set up equations based on which sides are equal
Sec 4-2 Angles of Triangles ⎕ The 3 angles in a triangle ALWAYS add to 180 degrees. ⎕ If two angles of one triangle are congruent to two angles of a second triangle, then the third angles are congruent. ⎕ What is the Exterior Angle Theorem? ⎕ The acute angles of a RIGHT triangle are complementary. ⎕ There can be at most one right or obtuse angle in a triangle.
Sec 4-3 Congruent Triangles ⎕ Can you name the 6 congruent parts of 2 congruent triangles? ⎕ What does CPCTC stand for? ⎕ What is the Distance Formula?
Sec 4-4 & 4-5 Proving Congruence – SSS, SAS, ASA, AAS, HL ⎕ Know how to identify markings as SSS, SAS, ASA, AAS, HL ⎕ Know how to do triangle proofs using SSS, SAS, ASA, AAS, HL ⎕ Remind yourself of the ‘Reflexive Property’ for when triangles share a side or angle. This will need to be in your proof. ⎕ Be aware of what you are proving.
If you are proving two triangles are congruent, your last step in your proof will be SSS, SAS, ASA, AAS, HL.
If you are proving two sides or two angles of congruent triangles are congruent, then AFTER you prove the triangles are congruent with SSS, SAS, ASA, AAS, HL you will then prove the parts are congruent with CPCTC.
Sec 4-6 Isosceles Triangles ⎕ In an Isosceles Triangle the opposite angles of the equal sides are equal and vice versa. ⎕ In an equilateral triangle all the sides are equal and all the angles are 60 degrees.
Sec 6-1 Proportions ⎕ Can you set up proportions? Be sure you are consistent when you set up fractions. ⎕ Can you cross multiply?
Sec 6-2 Similar Polygons ⎕ What is the difference between congruent and similar?
⎕ Can you write a similarity statement? ⎕ Can you prove a shape is similar? ⎕ Given two similar triangles, can you find the scale factor?
Sec 6-3 Similar Triangles ⎕ What are the three Triangle Similarity Theorems? ⎕ How do you prove two triangles are similar by AA, SSS, SAS?
Sec 6-4 Parallel Lines and Proportional Parts
⎕ Triangle Proportionality Theorem & Converse of the Triangle Proportionality Theorem
⎕ Triangle Midsegment Theorem
Sec 6-5 Parts of Similar Triangles ⎕ Proportional Perimeters Theorem ⎕ Theorem 6.8 – If two triangles are similar then their ALTITUDES are
proportional. ⎕ Theorem 6.9 – If two triangles are similar then their ANGLE BISECOTRS are proportional. ⎕ Theorem 6.10 – If two triangles are similar then their MEDIANS are proportional. ⎕ Angle Bisector Theorem
If D and E are midpoints of and
respectively, and DE =
½ BC
is another
proportion that
works as well.
CSI Geometry: Triangles
Geometry ~ Chapter 4/6 Project ~ Scoring Rubric
NO CALCULATOR!!
Learning Targets
Triangle Classification
Interior and Exterior Angles
Triangle Congruence
The Centers of a Triangle
Scene Earned Points
Possible Points
What is being scored? Done?
Scene 1
3 pts 3 pts
State the rules of sides in a triangle.
Showed your work to prove which distance is NOT possible for the flight from Haiti back to Cuba.
Scene 2
3 pts
3 pts
Labeled EACH triangle with which statement ruled out that it was his Mom’s.
Correctly identified ‘Which Dorsal Fin is My Mom’s’.
Scene 3
3 pts 3 pts 3 pts 3 pts 3 pts 3 pts
Identified Captain Jack Sparrow correctly.
Identified Dread Pirate Roberts correctly.
Identified Long John Silver correctly.
Identified Morgan Adams correctly.
Identified Captain Hook correctly.
Correctly determined How many pirates were wrong.
Scene 4
3 pts 3 pts 3 pts 3 pts 3 pts 3 pts 3 pts
Labeled ALL angles in the 1st triangle correctly.
Labeled ALL angles in the 2nd triangle correctly.
Labeled ALL angles in the 3rd triangle correctly.
Labeled ALL angles in the 4th triangle correctly.
Labeled ALL angles in the 5th triangle correctly.
Labeled ALL angles in the 6th triangle correctly.
Calculated letter O correctly.
Scene 5
3 pts 3 pts 3 pts 3 pts 3 pts
Labeled ALL angles in the 1st triangle correctly.
Labeled ALL angles in the 2nd triangle correctly.
Labeled ALL angles in the 3rd triangle correctly.
Identified A, B and D clearly.
Decided on the variable of the largest angle and it measure correctly.
Scene 6
3 pts
We did not cover this so the answers are the following: Circumcenter 5 = S, Incenter 3 = D, Centroid 4 = R, Orthocenter 1 = A, The surviving ship is B = 9.
Cryptic Puzzle Solver Text
Message
3 pts 3 pts 3 pts
Plugged numbers in correctly.
Showed work and came out with the correct answer.
Explained ‘Who did it’?
Case Summary
5 pts
5 pts
5 pts
7 pts
Prepare a typed case in which you need to demonstrate your understanding of each of the following topics:
Triangle Classification (Find one from this project and one from your book. Explain HOW you did them both.)
Interior and Exterior Angles (Find one from this project and one from your book. Explain HOW you did them both.)
Triangle Congruence (Find one from this project and one from your book. Explain HOW you did them both.)
What vocabulary words and definitions were necessary throughout this project.
