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Geometry Section 4-4 1112

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Triangle Congruence -- SSS, SAS

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Geometry Section 4-4 1112

1. 1. Section 4-4 Proving Congruence: SSS, SAS
2. 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. 3. Vocabulary 1. Included Angle: Postulate 4.1 - Side-Side-Side (SSS) Congruence: Postulate 4.2 - Side-Angle-Side (SAS) Congruence:
4. 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. 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. 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. 7. Example 1 Prove the following. Given: QU ≅ AD, QD ≅ AU Prove: △QUD ≅△ADU
8. 8. Example 1 Prove the following. Given: QU ≅ AD, QD ≅ AU Prove: △QUD ≅△ADU 1. QU ≅ AD, QD ≅ AU
9. 9. Example 1 Prove the following. Given: QU ≅ AD, QD ≅ AU Prove: △QUD ≅△ADU 1. QU ≅ AD, QD ≅ AU 1. Given
10. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 27. Example 3 Prove the following. Prove: △FEG ≅△HIG Given: EI ≅ FH, G is the midpoint of EI and FH
28. 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. 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. 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. 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. 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. 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. 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. 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. 36. Problem Set
37. 37. Problem Set p. 266 #1-19 odd, 31, 39, 41 "Doubt whom you will, but never yourself." - Christine Bovee