Upcoming SlideShare
×

# Geometry Section 4-5 1112

2,160 views

Published on

Proving Congruence: ASA, AAS

Published in: Education, Technology
0 Likes
Statistics
Notes
• Full Name
Comment goes here.

Are you sure you want to Yes No
• Be the first to comment

• Be the first to like this

Views
Total views
2,160
On SlideShare
0
From Embeds
0
Number of Embeds
926
Actions
Shares
0
13
0
Likes
0
Embeds 0
No embeds

No notes for slide

### Geometry Section 4-5 1112

1. 1. SECTION 4-5 Proving Congruence: ASA, AAS
2. 2. ESSENTIAL QUESTIONS How do you use the ASA Postulate to test for triangle congruence? How do you use the AAS Postulate to test for triangle congruence?
3. 3. VOCABULARY 1. Included Side: Postulate 4.3 - Angle-Side-Angle (ASA) Congruence: Theorem 4.5 - Angle-Angle-Side (AAS) Congruence:
4. 4. VOCABULARY 1. Included Side: The side between two consecutive angles in a triangle Postulate 4.3 - Angle-Side-Angle (ASA) Congruence: Theorem 4.5 - Angle-Angle-Side (AAS) Congruence:
5. 5. VOCABULARY 1. Included Side: The side between two consecutive angles in a triangle Postulate 4.3 - Angle-Side-Angle (ASA) Congruence: If two angles and the included side of one triangle are congruent to two angles and included side of a second triangle, then the triangles are congruent Theorem 4.5 - Angle-Angle-Side (AAS) Congruence:
6. 6. VOCABULARY 1. Included Side: The side between two consecutive angles in a triangle Postulate 4.3 - Angle-Side-Angle (ASA) Congruence: If two angles and the included side of one triangle are congruent to two angles and included side of a second triangle, then the triangles are congruent Theorem 4.5 - Angle-Angle-Side (AAS) Congruence: If two angles and the nonincluded side of one triangle are congruent to the corresponding angles and nonincluded side of a second triangle, then the triangles are congruent
7. 7. EXAMPLE 1 Prove the following. Prove: △WRL ≅△EDL Given: L is the midpoint of WE, WR ! ED
8. 8. EXAMPLE 1 Prove the following. Prove: △WRL ≅△EDL Given: L is the midpoint of WE, WR ! ED 1. L is the midpoint of WE, WR ! ED
9. 9. EXAMPLE 1 Prove the following. Prove: △WRL ≅△EDL 1. Given Given: L is the midpoint of WE, WR ! ED 1. L is the midpoint of WE, WR ! ED
10. 10. EXAMPLE 1 Prove the following. Prove: △WRL ≅△EDL 1. Given Given: L is the midpoint of WE, WR ! ED 2. WL ≅ EL 1. L is the midpoint of WE, WR ! ED
11. 11. EXAMPLE 1 Prove the following. Prove: △WRL ≅△EDL 1. Given Given: L is the midpoint of WE, WR ! ED 2. WL ≅ EL 2. Midpoint Thm. 1. L is the midpoint of WE, WR ! ED
12. 12. EXAMPLE 1 Prove the following. Prove: △WRL ≅△EDL 1. Given Given: L is the midpoint of WE, WR ! ED 2. WL ≅ EL 2. Midpoint Thm. 3. ∠WLR ≅ ∠ELD 1. L is the midpoint of WE, WR ! ED
13. 13. EXAMPLE 1 Prove the following. Prove: △WRL ≅△EDL 1. Given Given: L is the midpoint of WE, WR ! ED 2. WL ≅ EL 2. Midpoint Thm. 3. ∠WLR ≅ ∠ELD 3. Vertical Angles Thm. 1. L is the midpoint of WE, WR ! ED
14. 14. EXAMPLE 1 Prove the following. Prove: △WRL ≅△EDL 1. Given Given: L is the midpoint of WE, WR ! ED 2. WL ≅ EL 2. Midpoint Thm. 3. ∠WLR ≅ ∠ELD 3. Vertical Angles Thm. 4. ∠LWR ≅ ∠LED 1. L is the midpoint of WE, WR ! ED
15. 15. EXAMPLE 1 Prove the following. Prove: △WRL ≅△EDL 1. Given Given: L is the midpoint of WE, WR ! ED 2. WL ≅ EL 2. Midpoint Thm. 3. ∠WLR ≅ ∠ELD 3. Vertical Angles Thm. 4. ∠LWR ≅ ∠LED 4. Alternate Interior Angles Thm 1. L is the midpoint of WE, WR ! ED
16. 16. EXAMPLE 1 Prove the following. Prove: △WRL ≅△EDL 1. Given Given: L is the midpoint of WE, WR ! ED 2. WL ≅ EL 2. Midpoint Thm. 3. ∠WLR ≅ ∠ELD 3. Vertical Angles Thm. 4. ∠LWR ≅ ∠LED 4. Alternate Interior Angles Thm 5. △WRL ≅△EDL 1. L is the midpoint of WE, WR ! ED
17. 17. EXAMPLE 1 Prove the following. Prove: △WRL ≅△EDL 1. Given Given: L is the midpoint of WE, WR ! ED 2. WL ≅ EL 2. Midpoint Thm. 3. ∠WLR ≅ ∠ELD 3. Vertical Angles Thm. 4. ∠LWR ≅ ∠LED 4. Alternate Interior Angles Thm 5. △WRL ≅△EDL 5. ASA 1. L is the midpoint of WE, WR ! ED
18. 18. EXAMPLE 2 Prove the following. Prove: LN ≅ MN Given: ∠NKL ≅ ∠NJM, KL ≅ JM
19. 19. EXAMPLE 2 Prove the following. Prove: LN ≅ MN Given: ∠NKL ≅ ∠NJM, KL ≅ JM 1. ∠NKL ≅ ∠NJM, KL ≅ JM
20. 20. EXAMPLE 2 Prove the following. Prove: LN ≅ MN Given: ∠NKL ≅ ∠NJM, KL ≅ JM 1. ∠NKL ≅ ∠NJM, KL ≅ JM 1. Given
21. 21. EXAMPLE 2 Prove the following. Prove: LN ≅ MN Given: ∠NKL ≅ ∠NJM, KL ≅ JM 1. ∠NKL ≅ ∠NJM, KL ≅ JM 1. Given 2. ∠N ≅ ∠N
22. 22. EXAMPLE 2 Prove the following. Prove: LN ≅ MN Given: ∠NKL ≅ ∠NJM, KL ≅ JM 1. ∠NKL ≅ ∠NJM, KL ≅ JM 1. Given 2. ∠N ≅ ∠N 2. Reﬂexive
23. 23. EXAMPLE 2 Prove the following. Prove: LN ≅ MN Given: ∠NKL ≅ ∠NJM, KL ≅ JM 1. ∠NKL ≅ ∠NJM, KL ≅ JM 1. Given 2. ∠N ≅ ∠N 2. Reﬂexive 3. △JNM ≅△KNL
24. 24. EXAMPLE 2 Prove the following. Prove: LN ≅ MN Given: ∠NKL ≅ ∠NJM, KL ≅ JM 1. ∠NKL ≅ ∠NJM, KL ≅ JM 1. Given 2. ∠N ≅ ∠N 2. Reﬂexive 3. △JNM ≅△KNL 3. AAS
25. 25. EXAMPLE 2 Prove the following. Prove: LN ≅ MN Given: ∠NKL ≅ ∠NJM, KL ≅ JM 1. ∠NKL ≅ ∠NJM, KL ≅ JM 1. Given 2. ∠N ≅ ∠N 2. Reﬂexive 3. △JNM ≅△KNL 3. AAS 4. LN ≅ MN
26. 26. EXAMPLE 2 Prove the following. Prove: LN ≅ MN Given: ∠NKL ≅ ∠NJM, KL ≅ JM 1. ∠NKL ≅ ∠NJM, KL ≅ JM 1. Given 2. ∠N ≅ ∠N 2. Reﬂexive 3. △JNM ≅△KNL 3. AAS 4. LN ≅ MN 4. Corresponding Parts of Congruent Triangles are Congruent (CPCTC)
27. 27. EXAMPLE 3 On a template design for a certain envelope, the top and bottom ﬂaps are isosceles triangles with congruent bases and base angles. If EV = 8 cm and the height of the isosceles triangle is 3 cm, ﬁnd PO.
28. 28. EXAMPLE 3 On a template design for a certain envelope, the top and bottom ﬂaps are isosceles triangles with congruent bases and base angles. If EV = 8 cm and the height of the isosceles triangle is 3 cm, ﬁnd PO. EV ≅ PL, so each segment has a measure of 8 cm. If an auxiliary line is drawn from point O perpendicular to PL, you will have a right triangle formed.
29. 29. EXAMPLE 3 On a template design for a certain envelope, the top and bottom ﬂaps are isosceles triangles with congruent bases and base angles. If EV = 8 cm and the height of the isosceles triangle is 3 cm, ﬁnd PO. EV ≅ PL, so each segment has a measure of 8 cm. If an auxiliary line is drawn from point O perpendicular to PL, you will have a right triangle formed. In the right triangle, we have one leg (the height) of 3 cm. The auxiliary line will bisect PL, as point O is equidistant from P and L.
30. 30. EXAMPLE 3
31. 31. EXAMPLE 3 a2 + b2 = c2
32. 32. EXAMPLE 3 a2 + b2 = c2 42 + 32 = (PO)2
33. 33. EXAMPLE 3 a2 + b2 = c2 42 + 32 = (PO)2 16 + 9 = (PO)2
34. 34. EXAMPLE 3 a2 + b2 = c2 42 + 32 = (PO)2 16 + 9 = (PO)2 25 = (PO)2
35. 35. EXAMPLE 3 a2 + b2 = c2 42 + 32 = (PO)2 16 + 9 = (PO)2 25 = (PO)2 25 = (PO)2
36. 36. EXAMPLE 3 a2 + b2 = c2 42 + 32 = (PO)2 16 + 9 = (PO)2 25 = (PO)2 25 = (PO)2 PO = 5 cm
37. 37. PROBLEM SET
38. 38. PROBLEM SET p. 276 #1-23 odd, 35 “There is only one you...Don’t you dare change just because you’re outnumbered.” - Charles Swindoll