1. 4-3 Congruent Triangles Warm Up Lesson Presentation Lesson Quiz
Holt Geometry
Warm Up
Lesson Presentation
Lesson Quiz

2. Do Now
1. Name all sides and angles of ∆FGH.
2. What is true about K and L? Why?
3. What does it mean for two segments to be congruent?

3. Objectives TSW use properties of congruent triangles.
TSW prove triangles congruent by using the definition of congruence.

4. Vocabulary
corresponding angles
corresponding sides
congruent polygons

5. Geometric figures are congruent if they are the same size and shape
Geometric figures are congruent if they are the same size and shape. Corresponding angles and corresponding sides are in the same position in polygons with an equal number of sides.
Two polygons are congruent polygons if and only if their corresponding sides are congruent. Thus triangles that are the same size and shape are congruent.

7. For example, P and Q are consecutive vertices.
Two vertices that are the endpoints of a side are called consecutive vertices.
For example, P and Q are consecutive vertices.
Helpful Hint

8. To name a polygon, write the vertices in consecutive order
To name a polygon, write the vertices in consecutive order. For example, you can name polygon PQRS as QRSP or SRQP, but not as PRQS.
In a congruence statement, the order of the vertices indicates the corresponding parts.

9. When you write a statement such as ABC  DEF, you are also stating which parts are congruent.
Helpful Hint

10. Example 1: Naming Congruent Corresponding Parts
Given: ∆PQR  ∆STW
Identify all pairs of corresponding congruent parts.

11. Example 2
If polygon LMNP  polygon EFGH, identify all pairs of corresponding congruent parts.

12. Example 3: Using Corresponding Parts of Congruent Triangles
Given: ∆ABC  ∆DBC.
Find the value of x.

13. Example 3: Using Corresponding Parts of Congruent Triangles
Given: ∆ABC  ∆DBC.
Find mDBC.

14. Example 4
Given: ∆ABC  ∆DEF
Find the value of x.

15. Example 5
Given: ∆ABC  ∆DEF
Find mF.

16. Example 6: Proving Triangles Congruent
Given: YWX and YWZ are right angles.
YW bisects XYZ. W is the midpoint of XZ. XY  YZ.
Prove: ∆XYW  ∆ZYW

17. Statements
Reasons
1. YWX and YWZ are rt. s.
1. Given
2. YW bisects XYZ
2. Given
3. W is mdpt. of XZ
3. Given 5. 4. 4. 5. 6. 6. 7. 7 8. 8. 9. 9.
10. ∆XYW  ∆ZYW
10.

18. Example 7
Given: AD bisects BE.
BE bisects AD.
AB  DE, A  D
Prove: ∆ABC  ∆DEC

19. Statements
Reasons 1.
1. Given 2.
2. Given 3.
3. Given 4.
4. Given 5. 5. 6. 6. 7. 7 8. 8.

20. Example 8: Engineering Application
The diagonal bars across a gate give it support. Since the angle measures and the lengths of the corresponding sides are the same, the triangles are congruent.
Given: PR and QT bisect each other.
PQS  RTS, QP  RT
Prove: ∆QPS  ∆TRS

21. Statements
Reasons 1.
1. Given 2.
2. Given 3.
3. Given 4. 4. 5. 5. 6. 6. 7. 7 8. 8. 9. 9.

22. Example 9
Use the diagram to prove the following.
Given: MK bisects JL. JL bisects MK. JK  ML. JK || ML.
Prove: ∆JKN  ∆LMN

23. Statements
Reasons 1.
1. Given 2.
2. Given 3.
3. Given 4. 4. 5. 5. 6. 6. 7. 7 8. 8.

24. 10. ∆ABC  ∆JKL and AB = 2x + 12.
JK = 4x – 50. Find x and AB.

25. 11. Given that polygon MNOP  polygon QRST, identify the congruent corresponding part. NO  ____ T  ____

26. 12. Given: C is the midpoint of BD and AE.
A  E, AB  ED
Prove: ∆ABC  ∆EDC

27. Lesson Quiz
1. ∆ABC  ∆JKL and AB = 2x JK = 4x – 50. Find x and AB.
Given that polygon MNOP  polygon QRST, identify the congruent corresponding part.
2. NO  ____ T  ____
4. Given: C is the midpoint of BD and AE.
A  E, AB  ED
Prove: ∆ABC  ∆EDC

28. Lesson Quiz 4. 7. 6. Third s Thm. 6. 5. 5. ACB  ECD 4.
4. AC  EC; BC  DC 3. 2. 1.
Reasons
Statements

29. Lesson Quiz
1. ∆ABC  ∆JKL and AB = 2x JK = 4x – 50. Find x and AB.
Given that polygon MNOP  polygon QRST, identify the congruent corresponding part.
2. NO  ____ T  ____
4. Given: C is the midpoint of BD and AE.
A  E, AB  ED
Prove: ∆ABC  ∆EDC
31, 74 RS P

30. Lesson Quiz 4. 7. Def. of  ∆s 7. ABC  EDC 6. Third s Thm.
6. B  D
5. Vert. s Thm.
5. ACB  ECD
4. Given
4. AB  ED
3. Def. of mdpt.
3. AC  EC; BC  DC
2. Given
2. C is mdpt. of BD and AE
1. Given
1. A  E
Reasons
Statements