Download GEOMETRY MODULE 1 LESSON 24

GEOMETRY
MODULE 1 LESSON 24
CONGRUENECE CRITERIA FOR TRIANGLES –ASA and SSS
OPENING EXERCISE
Use the provide 30° angle as one base angle of an isosceles triangle. Use a compass and straight
edge to construct an appropriate isosceles triangle around it. You may extend the base if you
wish.
1. Strike an arc A with the compass needle at the vertex. Note the intersection points.
2. On the opposite end of the base segment, strike an arc B the same width as arc A.
3. Open your compass to the width of the two intersection points from arc A (the original arc).
4. With this width, strike an arc with the compass needle on one of the intersection points from
arc A.
5. Strike the same arc from the intersection point of arc B.
6. Draw the segment to complete the isosceles triangle.
Does using a given angle measure guarantee that all the triangles constructed in class have
corresponding sides of equal lengths?
No, side lengths may vary.
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DISCUSSION
We introduce two new triangle congruence criteria.

Angle-Side-Angle Triangle Congruence Criteria (ASA): Given two triangles
∆𝐴𝐵𝐶 and ∆𝐴′𝐵′𝐶′. If 𝑚∠𝐶𝐴𝐵 = 𝑚∠𝐶′𝐴′𝐵′ (Angle), 𝐴𝐵 = 𝐴’𝐵’ (Side), and 𝑚∠𝐶𝐵𝐴 =
𝑚∠𝐶′𝐵′𝐴′ (Angle), then the triangles are congruent.

Side-Side-Side Triangle Congruence Criteria (SSS): Given two triangles
∆𝐴𝐵𝐶 and ∆𝐴′𝐵′𝐶′. If 𝐴𝐵 = 𝐴’𝐵’ (Side), 𝐴𝐶 = 𝐴’𝐶’ (Side), and 𝐵𝐶 = 𝐵’𝐶’ (Side), then
the triangles are congruent.
As we did with SAS, we can prove these two congruence criteria through basic rigid motions.
ASA
SSS
NOTE: For ASA, the two angles given/found must be connected to the given/found side.
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PRACTICE
Based on the information provided, determine whether a congruence exists between triangles. If
a congruence exists between triangles or if multiple congruencies exist, state the congruencies
and the criteria used to determine them.
1. Given: M is the midpoint of ̅̅̅̅
𝐻𝑃 and 𝑚∠𝐻 = 𝑚∠𝑃.
ASA
𝑚∠𝐻 = 𝑚∠𝑃 is given.
𝐻𝑀 = 𝑀𝑃 by definition of midpoint
𝑚∠𝐺𝑀𝐻 = 𝑚∠𝑅𝑀𝑃 by vertical angles
So, ∆𝐺𝑀𝐻 ≅ ∆𝑅𝑀𝑃
2. Given: Rectangle JKLM with diagonal ̅̅̅̅̅
𝐾𝑀
SSS/SAS/ASA
𝑚∠𝐽 = 𝑚∠𝐾 = 𝑚∠𝐿 = 𝑚∠𝑀 by definition of rectangle
𝐽𝑀 = 𝐾𝐿 and 𝐽𝐾 = 𝑀𝐿 by definition of rectangle
𝐾𝑀 = 𝐾𝑀 by Reflexive
So, ∆𝐽𝐾𝑀 ≅ ∆𝐿𝑀𝐾
3. Given: 𝑅𝑌 = 𝑅𝐵, 𝐴𝑅 = 𝑋𝑅
SAS
𝑚∠𝐴𝑅𝑌 = 𝑚∠𝑋𝑅𝐵 by vertical angles
∆𝑌𝑅𝐵 is an isosceles triangle.
𝑚∠𝐴𝐵𝑌 = 𝑚∠𝑋𝑌𝐵 Base angles of an isosceles triangles are
equal in measure.
So, ∆𝐴𝑅𝑌 ≅ ∆𝑋𝑅𝐵 and ∆𝐴𝐵𝑌 ≅ ∆𝑋𝑌𝐵
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4. Given: 𝐴𝐵 = 𝐴𝐶, 𝐵𝐷 = 4 𝐴𝐵, 𝐶𝐸 = 4 𝐴𝐶
SAS
∆𝐵𝐴𝐶 is an isosceles triangle.
𝐵𝐷 = 𝐶𝐸 by Transitive Property
𝐴𝐷 = 𝐴𝐸 by Transitive Property thru work below

𝐴𝐵 − 𝐵𝐷 = 𝐴𝐷 Partition property (segment subtraction)

𝐴𝐶 − 𝐶𝐸 = 𝐴𝐸 Partition property (segment subtraction)
So, ∆𝐴𝐵𝐸 ≅ ∆𝐴𝐶𝐷
5. Given: Circles with centers A and B intersect at C and D
Prove: ∠𝐶𝐴𝐵 ≅ ∠𝐷𝐴𝐵
What congruence criteria would be best here?
STEP
JUSTIFICATION
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Circles with centers A and B intersect at C and D
Given
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𝐴𝐶 = 𝐴𝐷
Radius of Circle
3
𝐵𝐶 = 𝐵𝐷
Radius of Circle
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𝐴𝐵 = 𝐴𝐵
Reflexive Property
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∆𝐴𝐶𝐵 ≅ ∆𝐴𝐷𝐵
SSS
Corresponding angles of congruent
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∠𝐶𝐴𝐵 ≅ ∠𝐷𝐴𝐵
triangles are congruent.
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SUMMARY

Angle-Side-Angle Triangle Congruence Criteria (ASA): Given two triangles
∆𝐴𝐵𝐶 and ∆𝐴′𝐵′𝐶′. If 𝑚∠𝐶𝐴𝐵 = 𝑚∠𝐶′𝐴′𝐵′ (Angle), 𝐴𝐵 = 𝐴’𝐵’ (Side), and 𝑚∠𝐶𝐵𝐴 =
𝑚∠𝐶′𝐵′𝐴′ (Angle), then the triangles are congruent.

Side-Side-Side Triangle Congruence Criteria (SSS): Given two triangles
∆𝐴𝐵𝐶 and ∆𝐴′𝐵′𝐶′. If 𝐴𝐵 = 𝐴’𝐵’ (Side), 𝐴𝐶 = 𝐴’𝐶’ (Side), and 𝐵𝐶 = 𝐵’𝐶’ (Side), then
the triangles are congruent.

The ASA and SSS criteria implies the existence of a congruence that maps one triangle onto
the other.
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