Section 6: Triangles - Part 1123
Section 6: Triangles β Part 1
Topic 1: Introduction to Triangles β Part 1 .................................................................................................... 125 Topic 2: Introduction to Triangles β Part 2 .................................................................................................... 127 Topic 3: Area and Perimeter in the Coordinate Plane β Part 1 ................................................................ 130 Topic 4:Area and Perimeter in the Coordinate Plane β Part 2 ................................................................ 132 Topic 5: Triangle Congruence β SSS and SAS β Part 1 ................................................................................ 134 Topic 6:Triangle Congruence β SSS and SAS β Part 2 ................................................................................ 136 Topic 7: Triangle Congruence β ASA and AAS β Part 1 ............................................................................. 139 Topic 8: Triangle Congruence β ASA and AAS β Part 2 ............................................................................. 142 Topic 9: Base Angle of Isosceles Triangles ................................................................................................... 144 Topic 10: Using the Definition of Triangle Congruence in Terms of Rigid Motions .................................. 147 Topic 11: Using Triangle Congruency to Find Missing Variables ............................................................... 149
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Section 6: Triangles - Part 1124
The following Mathematics Florida Standards will be covered in this section: G-CO.2.8 - Explain how the criteria for triangle congruence (ASA, SAS, SSS, and Hypotenuse-Leg) follow from the definition of congruence in terms of rigid motions. G-CO.3.10 - Prove theorems about triangles; use theorems about triangles to solve problems. G-GPE.2.5 - Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems. G-GPE.2.7 - Use coordinates to compute perimeters of polygons and areas of triangles and rectangles. G-SRT.2.4 - Prove theorems about triangles. G-SRT.2.5 - Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
124
Section 6: Triangles - Part 1125
Section 6: Triangles β Part 1 Section 6 β Topic 1
Introduction to Triangles β Part 1
We can classify triangles by their angles and their sides. Complete each section of the following table with the most appropriate answers.
Description Representation Name
One right angle
Three acute angles
One obtuse angle
All 60Β° angles
Two congruent sides
No congruent sides
Three congruent sides
Can a triangle be both acute and isosceles? Justify your reasoning. Can a triangle be both equiangular and obtuse? Justify your reasoning. Letβs Practice! 1. Consider the diagram below of an equilateral triangle.
How long is each side of the triangle? Justify your answer.
(4π₯π₯ + 1)ft
(6π₯π₯β 6)ft
(8π₯π₯β 13)ft
Section 6: Triangles - Part 1126
Try It! 2. Consider the triangle below.
a. If Ξπ·π·π·π·π·π· is an isosceles triangle with base π·π·π·π·, what is the value of π₯π₯? Justify your answer.
b. What is the length of each leg?
c. What is the length of the base? 3. How can you determine if a triangle on the coordinate
plane is a right triangle?
D
E
F
Letβs Practice! 4. Consider the figure below.
a. After connecting the points on the plane, Marcos claims that angle π΅π΅is a right angle. Is Marcos correct? Explain your reasoning.
b. How can you classify a triangle on the coordinate plane by its sides?
A
B
C
Section 6: Triangles - Part 1127
Try It! 5. Consider the figure below.
Connect the points on the plane and classify the resulting triangle. Use two different approaches to justify your answer.
A T
C
Section 6 β Topic 2 Introduction to Triangles β Part 2
What is the sum of the measures of the interior angles of a triangle? Formulate how you can prove the sum of measures, if possible.
Triangle Sum Theorem
The sum of the interior angles in a triangle is 180Β°.
B A
C
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Section 6: Triangles - Part 1128
Consider the following figure and complete the following proof.
Given: βπ΄π΄π΄π΄π΄π΄ and π΄π΄π΅π΅ is parallel to π΄π΄π΄π΄. Prove: ππβ 1 + ππβ 2 + ππβ 3 = 180Β°
Statements Reasons 1. π΄π΄π΄π΄π΄π΄ is a triangle. 1.
2. π΄π΄π΅π΅||π΄π΄π΄π΄ 2.
3.ππβ 1 + ππβ 5 = ππβ π΅π΅π΄π΄π΄π΄ 3.
4. ππβ π΅π΅π΄π΄π΄π΄ + ππβ 4 = 180Β° 4.
5. ππβ 1 + ππβ 5 + ππβ 4 = 180Β° 5.
6. β 2 β β 4; β 3 β β 5 6.
7. ππβ 2 = ππβ 4; ππβ 3 = ππβ 5 7.
8. ππβ 1 + ππβ 2 + ππβ 3 = 180Β° 8.
A
B
C
P
4 5 1
2 3
Letβs Practice! 1. Joan knows the measures of two of the interior angles in a
triangle. How could she find the third measure? Explain your reasoning.
Try It! 2. Consider the figure below.
Timothy was trying to find the measure of β πΎπΎ in the triangle above. His answer was 7Β°. He is confused as he cannot understand why ππβ πΎπΎ = 7Β°. Is Timothyβs answer correct? Justify your answer.
(3π₯π₯ + 9)Β°
(15π₯π₯)Β°
(8π₯π₯β 11)Β° J L
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Section 6: Triangles - Part 1129
BEAT THE TEST!
1. Triangle π·π·π·π·π·π· has vertices at π·π·(5, 8), π·π·(β 3, 10), and π·π·(β 3, 6).
Part A: Determine what type of triangle π·π·π·π·π·π· is and mark the most appropriate answer.
A Scalene B Isosceles C Equilateral D Right
Part B: If you move vertex π·π· four units to the left, will the classification of triangle π·π·π·π·π·π· change? If so, what type of triangle will it be? Justify your answer.
A.
2. Stephen is fencing in his triangular garden as shown by the diagram below. Part A: Write an expression for the measure of angle ππ. Part B: Stephen measured angle ππ as 90Β°. He measures
angle ππ as 38Β°. Did he measure correctly? Justify your answer.
ππΒ°
50Β°X
Y
Z
Section 6: Triangles - Part 1130
Section 6 β Topic 3 Area and Perimeter in the Coordinate Plane β Part 1
Consider the rectangle below.
Explain the differences between the perimeter and the area of the rectangle. Each of the smaller squares has a side that is one inch long. What is the perimeter of the rectangle above? What is the area of the rectangle above? What is the formula for finding the perimeter of any rectangle? What is the formula for finding the area of any rectangle?
Consider the parallelogram below.
What is the formula for finding the area of any parallelogram? Trace the parallelogram above on a separate piece of paper. Try cutting the parallelogram into two triangular pieces. Use your observations to write the formula to find the area of any triangle. What is the formula of the perimeter of a triangle?
β
Section 6: Triangles - Part 1131
Letβs Practice! 1. Find the area of the following figure.
10
8
Try It! 2. A triangular poster is twice as long as its height. A
rectangular banner is 3 inches longer than its width. Both the poster and the banner have areas of 648 square inches.
a. What is the height and the base of the poster? Justify
your answer.
b. What is the length and the width of the banner? Justify your answer.
Section 6: Triangles - Part 1132
Section 6 β Topic 4 Area and Perimeter in the Coordinate Plane β Part 2
How can we find area and perimeter when a figure is on the coordinate plane? Letβs Practice! 1. Consider the triangle π΅π΅π΅π΅π΅π΅ below.
a. Which side should be considered the base? Justify your answer.
b. Find the area and perimeter of the triangle.
A
T
M
B
π¦π¦
π₯π₯
Try It! 2. Consider the figure below.
Deenaβs mother is helping her sew a large flag for color guard. Each square on Deenaβs plan above represents a square foot.
a. Determine the amount of fabric Deena need in
square feet.
b. The flag will be sewn along the edges of the flag. How much ribbon will be needed to the nearest tenth of a foot?
O
D
G
π¦π¦
π₯π₯
Section 6: Triangles - Part 1133
BEAT THE TEST! 1. Consider the right triangle below.
If the perimeter is 40 units, find the value of π₯π₯ and the area of the triangle. The value of π₯π₯is . The area is square units.
5π₯π₯ β 3 2π₯π₯ + 7
π₯π₯ + 4
2. Dallas is putting down hardwood floors in his home. His living room is pentagonal. Each unit on the coordinate plane represents 5 feet. Find the area of flooring needed in square feet.
Which of the following is the total area of the living room? A 18ft^ B 24ft^ C 450ft^ D 600ft^
H
S U
O E
π¦π¦
π₯π₯
Section 6: Triangles - Part 1134
Section 6 β Topic 5 Triangle Congruence β SSS and SAS β Part 1
What information do we need in order to determine whether two different triangles are congruent? When we state triangle congruency, the order of the letters in the names of the triangles is extremely important.
How can this congruency be stated?
A
B
C
D
F E
Letβs Practice! 1. If βπ½π½π½π½π½π½ β βπΆπΆπΆπΆπΆπΆ, finish the following congruence statements
and mark the corresponding congruent sides and the corresponding congruent angles.
π½π½π½π½ β ______ β π½π½ β ______
______β πΆπΆπΆπΆ ______ β β πΆπΆ
π½π½π½π½ β ______ β π½π½ β ______
2. Complete the congruence statements for the triangles
below.
βπΆπΆππππ β β_______ β ππ β β ______
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R A G
N
J L
K T C
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Section 6: Triangles - Part 1135
Try It! 3. Letβs consider the same triangles whereβππππππ β βπΊπΊπΊπΊπΊπΊ.
a. Mark the corresponding congruent sides with hash
marks and the corresponding congruent angles with arcs.
b. To state that two triangles are congruent, we donβt need to know that all three sides and all three angles are congruent. Four postulates help us determine triangle congruency.
1. 2. 3. 4.
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A
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We can prove the following triangles are congruent by the SSS Congruence Postulate.
Write the congruency statement for the triangles above. Determine if Angle-Angle-Angle congruence exists and explain why it does or does not.
Side-Side-Side (SSS) Congruence Postulate If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent.
Y
X Z
U
T V
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Section 6: Triangles - Part 1136
We can prove the following triangles are congruent by the SAS Congruence Postulate.
Write the congruency statement for the triangles above. Determine if Side-Side-Angle congruence exists and explain why it does or does not.
Side-Angle-Side (SAS) Congruence Postulate 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.
A
C T
U
R L
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Section 6 β Topic 6 Triangle Congruence β SSS and SAS β Part 2
Letβs Practice! 1. What information is needed to prove the triangles below
are congruent using the SSS Congruence Postulate?
2. What information is needed to prove the triangles below
are congruent using the SAS Congruence Postulate?
G
A M
C
R L
C
D R
O
S K
Section 6: Triangles - Part 1137
3. What information is needed to prove the triangles below are congruent using the SSS Congruence Postulate?
C A
S E
Try It!
4. Consider βπΆπΆπΆπΆπΆπΆ and βπΆπΆπΆπΆπΆπΆ in the figure below.
Given: πΆπΆπΆπΆ β πΆπΆπΆπΆ and πΆπΆπΆπΆ β πΆπΆπΆπΆ
Prove: βπΆπΆπΆπΆπΆπΆ β βπΆπΆπΆπΆπΆπΆ
Based on the above figure and the information below, complete the following two-column proof.
Statements Reasons
1. πΆπΆπΆπΆ β πΆπΆπΆπΆ 1. Given
2. πΆπΆπΆπΆ β πΆπΆπΆπΆ 2. Given
3.
3. Reflexive Property of Congruence
4. βπΆπΆπΆπΆπΆπΆ β βπΆπΆπΆπΆπΆπΆ 4.
C
A
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Section 6: Triangles - Part 1138
5. Consider βπΊπΊπΊπΊπΊπΊ and βπ΄π΄πΊπΊπ΄π΄ in the diagram below.
Given: πΊπΊ is the midpoint of π΄π΄πΊπΊ and π΄π΄πΊπΊ. Prove: βπΊπΊπΊπΊπΊπΊ β βπ΄π΄πΊπΊπ΄π΄
Complete the following two-column proof.
Statements Reasons
1. πΊπΊ is the midpoint of π΄π΄πΊπΊ and π΄π΄πΊπΊ
1. Given
2. 2. Definition of Midpoint
3. 3. Definition of Midpoint
4. β πΊπΊπΊπΊπΊπΊ β β π΄π΄πΊπΊπ΄π΄ 4.
5. βπΊπΊπΊπΊπΊπΊ β βπ΄π΄πΊπΊπ΄π΄ 5.
πΊπΊR
E
A
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BEAT THE TEST!
1. Moshi is making a quilt using the pattern below and wants to be sure her triangles are congruent before cutting the fabric. She measures and finds that ππππ β ππππ and β ππ β β ππ.
Can Moshi determine if the triangles are congruent with the given information? If not, what other information would allow her to do so? Justify your answer.
N
A
V
J
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Section 6: Triangles - Part 1139
2. Iskra is a structural engineer, designing a tri-bearing truss for the roof of a new building. She must determine if the triangles below are congruent for the stability of the roof.
Given: πΉπΉπΉπΉ β πΉπΉπΉπΉ; πΉπΉπΉπΉ bisects β πΉπΉπΉπΉπΉπΉ Prove: βπΉπΉπΉπΉπΉπΉ β βπΉπΉπΉπΉπΉπΉ
Which of the reasons for statement 5 is correct?
Statements Reasons
1. πΉπΉπΉπΉ β πΉπΉπΉπΉ 1. Given
2. πΉπΉπΉπΉ bisects β πΉπΉπΉπΉπΉπΉ
2. Given
3.β πΉπΉπΉπΉπΉπΉ β β πΉπΉπΉπΉπΉπΉ 3. Definition of angle bisector
4. πΉπΉπΉπΉ β πΉπΉπΉπΉ 4. Reflective Property of Congruence
5. βπΉπΉπΉπΉπΉπΉ β βπΉπΉπΉπΉπΉπΉ 5.
A AAA B SAS C SSS D Canβt prove congruency
F
IM
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Section 6 β Topic 7 Triangle Congruence β ASA and AAS β Part 1
Consider the figures below.
In the above diagram, βππππππ β βπΉπΉπΉπΉπΉπΉ based on the ASA Congruence Postulate. Name the congruent sides and angles in these two triangles.
Angle-Side-Angle (ASA) Congruence Postulate 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.
E
D
πΉπΉ W
I
F
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Section 6: Triangles - Part 1140
Consider the figures below.
In the above diagram, βππππππ β βππππππ based on the AAS Congruence Postulate. Name the congruent sides and angles in these two triangles.
Angle-Angle-Side (AAS) Congruence Postulate If two angles and a non-included side of one triangle are congruent to two angles and a non-included side of a second triangle, then the two triangles are congruent.
N O
S M
T A
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Consider the triangles below.
Identify the postulate you could use to prove that the two triangles are congruent, given each additional congruence statement below.
Congruency Statement Postulate
π΅π΅π΅π΅ β ππππ
π΄π΄π΅π΅ β ππππ
π΄π΄π΅π΅ β ππππ
B
O
P
M
W
A
Section 6: Triangles - Part 1141
Consider the figure below.
Nadia would like to use the AAS Congruence Postulate to prove that βππππππ β βπππππΌπΌ. Would knowing that β ππ β β πΌπΌ be enough information for Nadia to use this postulate? If not, find the missing congruence statement.
S
I
T
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Letβs Practice!
1. Consider βππππππ and βππππππ in the diagram below.
Given: β ππ and β ππ are right angles; ππ is the midpoint of ππππ
Prove: βππππππ β βππππππ Complete the following two-column proof.
Statements Reasons
1. β ππ and β ππ are right angles 1. Given
2. β ππ β β ππ 2.
3.ππ is the midpoint of ππππ 3. Given
4. 4. Definition of midpoint
5. β ππππππ β β ππππππ 5.
6. βππππππ β βππππππ 6.
N I
P
A
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Section 6: Triangles - Part 1142
Section 6 β Topic 8 Triangle Congruence β ASA and AAS β Part 2
Letβs Practice!
1. Consider the figure to the right.
Given: ππππ β ππππ, πΏπΏππ β₯ ππππ,ππππ β₯ ππππ πΏπΏππ bisects β ππππππ,ππππ bisects β πΏπΏππππ. Prove: βπππππΏπΏ β βππππππ Complete the following two-column proof.
Statements Reasons
1. ππππ β ππππ 1. Given
2. πΏπΏππ β₯ ππππ,ππππ β₯ ππππ 2. Given
3. πΏπΏππ bisects β ππππππand ππππ bisects β πΏπΏππππ.
3. Given
4. 4. Definition of β₯ lines.
5. ππβ πΏπΏππππ = ππβ ππππππ = 45Β° 5.
6. 6. Vertical Angles
7. βπππππΏπΏ β βππππππ 7.
L
M
ππ
P
ππ
Try It!
2. Consider βππππππ and βπΈπΈππππ in the diagram below.
Given: ππππ β₯ πΈπΈππ ; πΈπΈππ β₯ ππππ Prove: βππππππ β βπΈπΈππππ Complete the following two-column proof.
Statements Reasons
1. ππππ β₯ πΈπΈππ 1. Given
2.
2. Alternate interior angles theorem
3. πΈπΈππ β₯ ππππ 3. Given
4. β πππππΈπΈ β β ππππππ 4.
5. 5. Reflexive property
6. βππππππ β βπΈπΈππππ 6.
O
PE
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Section 6: Triangles - Part 1143
Try It!
2. Consider the figures below.
How would you prove βπ΄π΄π΄π΄π΄π΄ β βπΊπΊπΊπΊπΊπΊ by applying ideas of transformations?
G
A
B C
F
E
π¦π¦
π₯π₯
BEAT THE TEST!
1. Consider the diagram below.
Given: ππππ β ππππ; ππππ β₯ ππππ Prove: βππππππ β βππππππ Select the most appropriate reason for #5.
Statements Reasons
1. ππππ β ππππ 1. Given
2. ππππ β₯ ππππ
2. Given
3. β ππππππ β β ππππππ
3. Alternate Interior Angles Theorem
4. β ππππππ β β ππππππ 4. Vertical angle theorem
5. βππππππ β βππππππ 5.
A AAS B ASA C SAS D SSS
E
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Section 6: Triangles - Part 1144
Section 6 β Topic 9 Base Angle of Isosceles Triangles
By definition, an _______________ _______________ is a triangle with two equal sides. Consider β³ ππππππ below.
Draw the angle bisector ππππ of β ππ, whereππ is the intersection of the bisector and ππππ. Use paragraph proofs to show that ππβ ππ = ππβ ππ in two ways: by using transformations and triangle congruence postulates.
Transformations Triangle Congruence Postulates
ππ
ππ
ππ
2. Consider the figure below.
Part A: What transformation(s) will prove βπ΅π΅π΅π΅π΅π΅ β βπππ΅π΅π΅π΅? Justify your answer.
Part B: If he knows that π΅π΅π΅π΅ is the angle bisector of β π΅π΅π΅π΅ππ, what additional information is needed to prove that βπ΅π΅π΅π΅π΅π΅ β βπππ΅π΅π΅π΅ using ASA?
A π΅π΅π΅π΅ β π΅π΅ππ B β π΅π΅π΅π΅π΅π΅ β β π΅π΅πππ΅π΅ C β π΅π΅π΅π΅π΅π΅ β β π΅π΅πππ΅π΅ D β π΅π΅π΅π΅π΅π΅ β β πππ΅π΅π΅π΅
B
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Section 6: Triangles - Part 1145
Letβs Practice! 1. Consider the diagram below.
For each of the following congruence statements, name the isosceles triangle and the pair of congruent angles for the triangle based on the diagram above. a. πΆπΆπΆπΆ β π΅π΅πΆπΆ b. πΉπΉπΉπΉ β πΊπΊπΉπΉ
c. π΄π΄π΄π΄ β πΈπΈπ΄π΄
d. πΈπΈπΉπΉ β πΉπΉπΉπΉ
e. πΉπΉπ΅π΅ β πΊπΊπ΅π΅
Base Angle Theorem and its Converse The Base Angle Theorem states that if two sides in a triangle are congruent, then the angles opposite to these sides are also congruent. The converse of this theorem is also true. If two angles of a triangle are congruent, then the sides opposite to these angles are congruent.
πΆπΆ
π΅π΅
π΄π΄
πΆπΆ
πΈπΈπΉπΉ
πΊπΊπ΄π΄
πΉπΉπΎπΎ
πΏπΏ
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2. Consider the figure below.
Given: β³ π π π π π π , π·π·π·π· is the angle bisector of β π π π·π·π π , and π π π π β₯ π·π·π·π·
Prove: π·π·π π = π·π·π π
Complete the following two-column proof.
Statements Reasons
1. π π π π β₯ π·π·π·π· 1. Given
2. β π·π·π·π·π π β β π π π π π·π·
2.
3.
3. Corresponding Angles Theorem
4. β π·π·π·π·π π β β π·π·π·π·π π
4.
5. 5. Transitive Property
6. 6. Definition of Congruence
7. π·π·π π = π·π·π π 7.
C
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D
Section 6: Triangles - Part 1146
BEAT THE TEST!
1. Consider the following β³ ππππππ, with ππβ ππππππ = ππβ ππππππ.
Waseem was asked to prove that ππππ = ππππ. His work is
shown below. There is at least one error in Waseemβs work. Describe and explain his error(s).
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Try It! 3. Consider the figure below.
Given: ππππ = ππππ and ππππ β₯ ππππ Prove: ππππ = ππππ
Complete the following two-column proof.
Statements Reasons
1. ππππ = ππππ 1. Given
2. β³ ππππππ is isosceles.
2.
3.
3. Base Angle Theorem
4. ππππ β₯ ππππ
4. Given
5. β ππππππ β β ππππππ and β ππππππ β β ππππππ 5.
6. 6. Transitive Property
7. 7. Definition of Congruence
8.ππππ = ππππ 8.
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Section 6: Triangles - Part 1147
Section 6 β Topic 10 Using the Definition of Triangle Congruence in Terms of
Rigid Motions How can rigid motion(s) be used to determine congruence?
Γ Rigid motions move figures to a new location without altering their ____________ or ____________, thus maintaining the conditions for the figures to be congruent.
By definition, two figures are ____________ if and only if there exists one, or more, rigid motions which will map one figure onto the other. Consider the diagram below.
Find a rigid motion that will map β³ ππππππ onto β³ π½π½π½π½π½π½.
Justify the use of the SSS Congruence Postulate to prove that β³ ππππππ β β³ π½π½π½π½π½π½.
ππ
ππ ππ
π½π½
π½π½
π½π½
π¦π¦
π₯π₯
Letβs Practice!
1. Consider the diagram below.
Find a rigid motion that will map β³ ππππππ onto β³ ππππππ.
ππ
ππ
ππ
ππ
ππ
ππ
π¦π¦
π₯π₯
Section 6: Triangles - Part 1148
3. Consider the diagram below.
Which of the following rigid motions will map β³ ππππππ onto β³ πΎπΎπΎπΎπΎπΎ? Select all that apply.
o A reflection over the π¦π¦-axis o A rotation of 270Β° clockwise about the origin o The translation (π₯π₯, π¦π¦) β (π₯π₯ + 5, π¦π¦ β 1) o A reflection over the line π¦π¦ = βπ₯π₯ followed by a rotation
of 180Β° clockwise about the origin o The translation (π₯π₯, π¦π¦) β (π₯π₯ β 5, π¦π¦ + 1) followed by a
rotation of 90Β° counterclockwise about vertex ππ
πΎπΎ
ππ
πΎπΎ
ππ
ππ
πΎπΎ
π₯π₯
π¦π¦
Try It! 2. Consider the diagram below.
a. Find a rigid motion that will map β³ π΄π΄π΄π΄π΄π΄ onto β³ πΆπΆπΆπΆπΆπΆ.
b. Suppose that ππβ π΄π΄π΄π΄π΄π΄ = 112.71Β°, π΄π΄π΄π΄ = 6.4β, and π΄π΄π΄π΄ = 7.28β. Justify the use of the SAS Congruence Postulate to prove that β³ π΄π΄π΄π΄π΄π΄ β β³ πΆπΆπΆπΆπΆπΆ.
c. Suppose that your friend suggests a translation as the rigid motion that maps β³ π΄π΄π΄π΄π΄π΄ onto β³ πΆπΆπΆπΆπΆπΆ. Is your friend correct? Justify your answer.
π΄π΄
π΄π΄
π΄π΄
πΆπΆ
πΆπΆ
πΆπΆ
ππ
Section 6: Triangles - Part 1149
Section 6 β Topic 11 Using Triangle Congruency to Find Missing Variables
Consider the figures below.
Find the value of π₯π₯ in order to prove that the two triangles are congruent by the SAS Congruence Postulate. Justify your work. Letβs Practice! 1. Consider the figures below.
Find the value of π¦π¦in order to prove that the two triangles are congruent using the ASA Congruence Postulate. Justify your work.
π₯π₯β 8 46Β° 7
(4π¦π¦ + 11)Β°
2π¦π¦ β 3
π¦π¦ + 5
BEAT THE TEST!
1. Consider the diagram below.
Find a combination of rigid motions that will map β³ πΆπΆπΆπΆπΆπΆ onto β³ πΆπΆππππ and determine if β³ πΆπΆπΆπΆπΆπΆ β β³ πΆπΆππππ.
π₯π₯
π¦π¦
πΆπΆ
πΆπΆ
πΆπΆ πΆπΆ
ππ ππ
Section 6: Triangles - Part 1150
BEAT THE TEST!
1. Consider the figure below.
Part A: If ππππ β ππππ and ππππ β₯ ππππ, which triangle congruency postulate can we use to determine βππππππ β βππππππ given the information on the figure? Select all that apply.
o AAS o ASA o SAS o SSS o SSA
Part B: What are the values of π₯π₯and π¦π¦?
Test Yourself! Practice Tool
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50Β°
5π₯π₯ + 6
21
4π¦π¦ β 4π₯π₯
20ππ
ππ
ππ
ππ
ππ
Try It!
2. Consider the figure below.
Find the values of π₯π₯ and π¦π¦ that prove the two triangles are congruent using the SSS Congruence Postulate.
3. Consider the figure below.
Find the values of π₯π₯and π¦π¦that prove the two triangles are congruent using the AAS Congruence Theorem. Justify your work.
π₯π₯ β 3
12 π¦π¦ β 4
2π₯π₯ β 7
(7π¦π¦ β 4)Β°80Β°
13π₯π₯ β 9
17