This document provides examples and explanations for proving triangle congruence using the Side-Side-Side (SSS) and Side-Angle-Side (SAS) postulates. It begins with definitions of key vocabulary like included angle. Example 1 uses SSS to prove two triangles congruent by showing that corresponding sides are congruent. Example 2 has students graph triangles on a coordinate plane and determine if they are congruent. Example 3 uses the midpoint theorem and vertical angles theorem to prove triangles congruent via SAS. The document concludes with practice problems for students.

1. Section 4-4
Proving Congruence: SSS, SAS

2. Essential Questions
How do you use the SSS Postulate to test for triangle
congruence?
How do you use the SAS Postulate to test for triangle
congruence?

3. Vocabulary
1. Included Angle:
Postulate 4.1 - Side-Side-Side (SSS) Congruence:
Postulate 4.2 - Side-Angle-Side (SAS) Congruence:

4. Vocabulary
1. Included Angle: An angle formed by two adjacent sides of a
polygon
Postulate 4.1 - Side-Side-Side (SSS) Congruence:
Postulate 4.2 - Side-Angle-Side (SAS) Congruence:

5. Vocabulary
1. Included Angle: An angle formed by two adjacent sides of a
polygon
Postulate 4.1 - Side-Side-Side (SSS) Congruence: If three sides
of one triangle are congruent to three corresponding sides of
a second, then the triangles are congruent
Postulate 4.2 - Side-Angle-Side (SAS) Congruence:

6. Vocabulary
1. Included Angle: An angle formed by two adjacent sides of a
polygon
Postulate 4.1 - Side-Side-Side (SSS) Congruence: If three sides
of one triangle are congruent to three corresponding sides of
a second, then the triangles are congruent
Postulate 4.2 - Side-Angle-Side (SAS) Congruence: If two sides
and the included angle of one triangle are congruent to two
corresponding sides and included angle of a second triangle,
then the triangles are congruent

7. Example 1
Prove the following.
Given: QU ≅ AD, QD ≅ AU
Prove: △QUD ≅△ADU

8. Example 1
Prove the following.
Given: QU ≅ AD, QD ≅ AU
Prove: △QUD ≅△ADU
1. QU ≅ AD, QD ≅ AU

9. Example 1
Prove the following.
Given: QU ≅ AD, QD ≅ AU
Prove: △QUD ≅△ADU
1. QU ≅ AD, QD ≅ AU 1. Given

10. Example 1
Prove the following.
Given: QU ≅ AD, QD ≅ AU
Prove: △QUD ≅△ADU
1. QU ≅ AD, QD ≅ AU 1. Given
2. DU ≅ DU

11. Example 1
Prove the following.
Given: QU ≅ AD, QD ≅ AU
Prove: △QUD ≅△ADU
1. QU ≅ AD, QD ≅ AU 1. Given
2. DU ≅ DU 2. Reflexive

12. Example 1
Prove the following.
Given: QU ≅ AD, QD ≅ AU
Prove: △QUD ≅△ADU
1. QU ≅ AD, QD ≅ AU 1. Given
2. DU ≅ DU 2. Reflexive
3. DU ≅ UD

13. Example 1
Prove the following.
Given: QU ≅ AD, QD ≅ AU
Prove: △QUD ≅△ADU
1. QU ≅ AD, QD ≅ AU 1. Given
2. DU ≅ DU 2. Reflexive
3. DU ≅ UD 3. Symmetric

14. Example 1
Prove the following.
Given: QU ≅ AD, QD ≅ AU
Prove: △QUD ≅△ADU
1. QU ≅ AD, QD ≅ AU 1. Given
2. DU ≅ DU 2. Reflexive
3. DU ≅ UD 3. Symmetric
4. △QUD ≅△ADU

15. Example 1
Prove the following.
Given: QU ≅ AD, QD ≅ AU
Prove: △QUD ≅△ADU
1. QU ≅ AD, QD ≅ AU 1. Given
2. DU ≅ DU 2. Reflexive
3. DU ≅ UD 3. Symmetric
4. △QUD ≅△ADU 4. SSS

16. Example 2
∆DVW has vertices D(−5, −1), V(−1, −2), and W(−7, −4). ∆LPM
has vertices L(1, −5), P(2, −1), and M(4, −7). Graph both
triangles on the coordinate plane. Then, using your drawing,
determine whether the triangles are congruent or not,
providing a logical argument for your statement.

17. Example 2
∆DVW has vertices D(−5, −1), V(−1, −2), and W(−7, −4). ∆LPM
has vertices L(1, −5), P(2, −1), and M(4, −7). Graph both
triangles on the coordinate plane. Then, using your drawing,
determine whether the triangles are congruent or not,
providing a logical argument for your statement.
x
y

18. Example 2
∆DVW has vertices D(−5, −1), V(−1, −2), and W(−7, −4). ∆LPM
has vertices L(1, −5), P(2, −1), and M(4, −7). Graph both
triangles on the coordinate plane. Then, using your drawing,
determine whether the triangles are congruent or not,
providing a logical argument for your statement.
x
y
D

19. Example 2
∆DVW has vertices D(−5, −1), V(−1, −2), and W(−7, −4). ∆LPM
has vertices L(1, −5), P(2, −1), and M(4, −7). Graph both
triangles on the coordinate plane. Then, using your drawing,
determine whether the triangles are congruent or not,
providing a logical argument for your statement.
x
y
D
V

20. Example 2
∆DVW has vertices D(−5, −1), V(−1, −2), and W(−7, −4). ∆LPM
has vertices L(1, −5), P(2, −1), and M(4, −7). Graph both
triangles on the coordinate plane. Then, using your drawing,
determine whether the triangles are congruent or not,
providing a logical argument for your statement.
x
y
D
V
W

21. Example 2
∆DVW has vertices D(−5, −1), V(−1, −2), and W(−7, −4). ∆LPM
has vertices L(1, −5), P(2, −1), and M(4, −7). Graph both
triangles on the coordinate plane. Then, using your drawing,
determine whether the triangles are congruent or not,
providing a logical argument for your statement.
x
y
D
V
W

22. Example 2
∆DVW has vertices D(−5, −1), V(−1, −2), and W(−7, −4). ∆LPM
has vertices L(1, −5), P(2, −1), and M(4, −7). Graph both
triangles on the coordinate plane. Then, using your drawing,
determine whether the triangles are congruent or not,
providing a logical argument for your statement.
x
y
D
V
W
L

23. Example 2
∆DVW has vertices D(−5, −1), V(−1, −2), and W(−7, −4). ∆LPM
has vertices L(1, −5), P(2, −1), and M(4, −7). Graph both
triangles on the coordinate plane. Then, using your drawing,
determine whether the triangles are congruent or not,
providing a logical argument for your statement.
x
y
D
V
W
L
P

24. Example 2
∆DVW has vertices D(−5, −1), V(−1, −2), and W(−7, −4). ∆LPM
has vertices L(1, −5), P(2, −1), and M(4, −7). Graph both
triangles on the coordinate plane. Then, using your drawing,
determine whether the triangles are congruent or not,
providing a logical argument for your statement.
x
y
D
V
W
L
P
M

25. Example 2
∆DVW has vertices D(−5, −1), V(−1, −2), and W(−7, −4). ∆LPM
has vertices L(1, −5), P(2, −1), and M(4, −7). Graph both
triangles on the coordinate plane. Then, using your drawing,
determine whether the triangles are congruent or not,
providing a logical argument for your statement.
x
y
D
V
W
L
P
M

26. Example 2
∆DVW has vertices D(−5, −1), V(−1, −2), and W(−7, −4). ∆LPM
has vertices L(1, −5), P(2, −1), and M(4, −7). Graph both
triangles on the coordinate plane. Then, using your drawing,
determine whether the triangles are congruent or not,
providing a logical argument for your statement.
x
y
D
V
W
L
P
M
These are congruent. What
possible ways could we show this?

27. Example 3
Prove the following.
Prove: △FEG ≅△HIG
Given: EI ≅ FH, G is the midpoint of EI and FH

28. Example 3
Prove the following.
Prove: △FEG ≅△HIG
Given: EI ≅ FH, G is the midpoint of EI and FH
1. EI ≅ FH, G is the midpoint of EI and FH

29. Example 3
Prove the following.
Prove: △FEG ≅△HIG
Given: EI ≅ FH, G is the midpoint of EI and FH
1. EI ≅ FH, G is the midpoint of EI and FH 1. Given

30. Example 3
Prove the following.
Prove: △FEG ≅△HIG
Given: EI ≅ FH, G is the midpoint of EI and FH
1. EI ≅ FH, G is the midpoint of EI and FH 1. Given
FG ≅ HG, EG ≅ IG2.

31. Example 3
Prove the following.
Prove: △FEG ≅△HIG
Given: EI ≅ FH, G is the midpoint of EI and FH
1. EI ≅ FH, G is the midpoint of EI and FH 1. Given
FG ≅ HG, EG ≅ IG2. 2. Midpoint Thm.

32. Example 3
Prove the following.
Prove: △FEG ≅△HIG
Given: EI ≅ FH, G is the midpoint of EI and FH
1. EI ≅ FH, G is the midpoint of EI and FH 1. Given
FG ≅ HG, EG ≅ IG2. 2. Midpoint Thm.
∠FGE ≅ ∠HGI3.

33. Example 3
Prove the following.
Prove: △FEG ≅△HIG
Given: EI ≅ FH, G is the midpoint of EI and FH
1. EI ≅ FH, G is the midpoint of EI and FH 1. Given
FG ≅ HG, EG ≅ IG2. 2. Midpoint Thm.
∠FGE ≅ ∠HGI3. 3. Vertical Angles Thm.

34. Example 3
Prove the following.
Prove: △FEG ≅△HIG
Given: EI ≅ FH, G is the midpoint of EI and FH
1. EI ≅ FH, G is the midpoint of EI and FH 1. Given
FG ≅ HG, EG ≅ IG2. 2. Midpoint Thm.
∠FGE ≅ ∠HGI3. 3. Vertical Angles Thm.
4. △FEG ≅△HIG

35. Example 3
Prove the following.
Prove: △FEG ≅△HIG
Given: EI ≅ FH, G is the midpoint of EI and FH
1. EI ≅ FH, G is the midpoint of EI and FH 1. Given
FG ≅ HG, EG ≅ IG2. 2. Midpoint Thm.
∠FGE ≅ ∠HGI3. 3. Vertical Angles Thm.
4. △FEG ≅△HIG 4. SAS

36. Problem Set

37. Problem Set
p. 266 #1-19 odd, 31, 39, 41
"Doubt whom you will, but never yourself." - Christine Bovee