Download geometry module 1 lesson 22 congruenece

GEOMETRY
MODULE 1 LESSON 22
CONGRUENECE CRITERIA FOR TRIANGLES – SAS
OPENING EXERCISE

Notebook check

Homework Due Dates
DISCUSSION
Side-Angle-Side Triangle Congruence Criteria (SAS): Given two triangles ∆𝐴𝐵𝐶 and ∆𝐴′𝐵′𝐶′
so that 𝐴𝐵 = 𝐴’𝐵’ (Side), 𝑚∠𝐴 = 𝑚∠𝐴′ (Angle), and 𝐴𝐶 = 𝐴’𝐶’ (Side). Then the triangles
are congruent.
Consider the two triangles to the right.
From earlier lessons, we know that basic rigid motions preserve segment and angle
measurements. From this, we can say corresponding sides and angles are congruent.
We will determine the transformations taking ∆𝐴′𝐵′𝐶′ back to ∆𝐴𝐵𝐶. First, use a
translation T to map a common vertex.

What two points determine the appropriate vector for our translation? 𝐴 𝑎𝑛𝑑 𝐴’

Considering the given triangles, can any other pair of points be used? Explain.
No, We use 𝐴 𝑎𝑛𝑑 𝐴’ because only these angles are given as congruent.
MOD1 L22
1

What rigid motion will take side 𝐴’𝐶’ to 𝐴𝐶? How can we be sure that vertex 𝐶” maps to 𝐶?
A rotation maps 𝐴’𝐶’ to 𝐴𝐶. We can be sure that vertex 𝐶” maps to 𝐶 because rotations
preserve segment lengths.

What is the last rigid motion mapping ∆𝐴′𝐵′𝐶′ to ∆𝐴𝐵𝐶?
A reflection maps ∆𝐴′𝐵′𝐶′ to ∆𝐴𝐵𝐶.
What transformations are needed to demonstrate congruence?
PRACTICE
1. Given: 𝑚∠𝐿𝑁𝑀 = 𝑚∠𝐿𝑁𝑂 and 𝑀𝑁 = 𝑂𝑁. Do ∆𝐿𝑁𝑀 and ∆𝐿𝑁𝑂 meet the SAS criteria?
STEP
JUSTIFICATION
𝑚∠𝐿𝑁𝑀 = 𝑚∠𝐿𝑁𝑂
Given
𝑀𝑁 = 𝑂𝑁
Given
LN =LN
Reflexive Property
∆𝐿𝑁𝑀 ≅ ∆𝐿𝑁𝑂
SAS
What rigid motion(s) would map ∆𝐿𝑁𝑀 onto ∆𝐿𝑁𝑂?
Reflection
MOD1 L22
2
̅̅̅̅ and 𝑅𝑇
̅̅̅̅ bisect each other.
2. Given: 𝑆𝑈
Do ∆𝑆𝑉𝑅 and ∆𝑈𝑉𝑇 meet the SAS criteria?
STEP
JUSTIFICATION
̅̅̅̅ and 𝑅𝑇
̅̅̅̅ bisect each other.
𝑆𝑈
Given
𝑆𝑉 = 𝑈𝑉
Definition of Segment Bisector
𝑅𝑉 = 𝑉𝑇
Definition of Segment Bisector
𝑚∠𝑆𝑉𝑅 = 𝑀∠𝑈𝑉𝑇
Vertical Angles
∆𝑆𝑉𝑅 ≅ ∆𝑈𝑉𝑇
SAS
What rigid motion maps ∆𝑆𝑉𝑅 onto ∆𝑈𝑉𝑇? 180° 𝑅𝑜𝑡𝑎𝑡𝑖𝑜𝑛
3. Given: 𝑚∠𝑉𝑋𝑌 = 𝑚∠𝑉𝑌𝑋. ∆𝑉𝑋𝑊 and ∆𝑉𝑌𝑍 do not
meet the SAS criteria. Why?
Not enough information is given to show that the triangles
meet SAS. We need information to show that 𝑊𝑌 ≅ 𝑋𝑍.
SUMMARY
Two triangles, ∆𝐴𝐵𝐶 and ∆𝐴′𝐵′𝐶′, meet the Side-Angle-Side criteria when 𝐴𝐵 = 𝐴’𝐵’ (Side),
𝑚∠𝐴 = 𝑚∠𝐴′ (Angle), and 𝐴𝐶 = 𝐴’𝐶’ (Side). The SAS criteria implies the existence of
congruence that maps one triangle onto the other.
EXIT TICKET
If two triangles satisfy the SAS criteria, what rigid motion(s) would map one onto the other
when…

The two triangles share a single common vertex. Rotation, Reflection

The two triangles share a common side. Reflection

The two triangles are distinct from each other. Translation, Rotation, Reflection
MOD1 L22
3