Congruent Triangles

1. SECTION 4-3
Congruent Triangles

2. ESSENTIAL QUESTIONS
How do you name and use corresponding parts of congruent
polygons?
How do you prove triangles congruent using the definition
of congruence?

3. VOCABULARY
1. Congruent:
2. Congruent Polygons:
3. Corresponding Parts:
Theorem 4.3 - Third Angles Theorem:

4. VOCABULARY
1. Congruent:Two figures with exactly the same size and shape
2. Congruent Polygons:
3. Corresponding Parts:
Theorem 4.3 - Third Angles Theorem:

5. VOCABULARY
1. Congruent:Two figures with exactly the same size and shape
2. Congruent Polygons: All parts of one polygon are
congruent to matching parts of another polygon
3. Corresponding Parts:
Theorem 4.3 - Third Angles Theorem:

6. VOCABULARY
1. Congruent:Two figures with exactly the same size and shape
2. Congruent Polygons: All parts of one polygon are
congruent to matching parts of another polygon
3. Corresponding Parts: The matching parts between two
polygons; Corresponding parts have the same position in
each polygon
Theorem 4.3 - Third Angles Theorem:

7. VOCABULARY
1. Congruent:Two figures with exactly the same size and shape
2. Congruent Polygons: All parts of one polygon are
congruent to matching parts of another polygon
3. Corresponding Parts: The matching parts between two
polygons; Corresponding parts have the same position in
each polygon
Theorem 4.3 - Third Angles Theorem: If two angles of one
triangle are congruent to two angles of a second triangle,
then the third angles must be congruent

8. CPCTC

9. CPCTC
The Corresponding Parts of
Congruent Triangles are Congruent

10. EXAMPLE 1
Show that the polygons are congruent by identifying all of
the congruent corresponding parts. Then write a
congruence statement.

11. EXAMPLE 1
Show that the polygons are congruent by identifying all of
the congruent corresponding parts. Then write a
congruence statement.
AB ≅ JI

12. EXAMPLE 1
Show that the polygons are congruent by identifying all of
the congruent corresponding parts. Then write a
congruence statement.
AB ≅ JI BC ≅ IH

13. EXAMPLE 1
Show that the polygons are congruent by identifying all of
the congruent corresponding parts. Then write a
congruence statement.
AB ≅ JI BC ≅ IH CD ≅ HG

14. EXAMPLE 1
Show that the polygons are congruent by identifying all of
the congruent corresponding parts. Then write a
congruence statement.
AB ≅ JI BC ≅ IH CD ≅ HG
DE ≅ GF

15. EXAMPLE 1
Show that the polygons are congruent by identifying all of
the congruent corresponding parts. Then write a
congruence statement.
AB ≅ JI BC ≅ IH CD ≅ HG
DE ≅ GF EA ≅ FJ

16. EXAMPLE 1
Show that the polygons are congruent by identifying all of
the congruent corresponding parts. Then write a
congruence statement.
AB ≅ JI BC ≅ IH CD ≅ HG
DE ≅ GF EA ≅ FJ
∠A ≅ ∠J

17. EXAMPLE 1
Show that the polygons are congruent by identifying all of
the congruent corresponding parts. Then write a
congruence statement.
AB ≅ JI BC ≅ IH CD ≅ HG
DE ≅ GF EA ≅ FJ
∠A ≅ ∠J ∠B ≅ ∠I

18. EXAMPLE 1
Show that the polygons are congruent by identifying all of
the congruent corresponding parts. Then write a
congruence statement.
AB ≅ JI BC ≅ IH CD ≅ HG
DE ≅ GF EA ≅ FJ
∠A ≅ ∠J ∠B ≅ ∠I ∠C ≅ ∠H

19. EXAMPLE 1
Show that the polygons are congruent by identifying all of
the congruent corresponding parts. Then write a
congruence statement.
AB ≅ JI BC ≅ IH CD ≅ HG
DE ≅ GF EA ≅ FJ
∠A ≅ ∠J ∠B ≅ ∠I ∠C ≅ ∠H
∠D ≅ ∠G

20. EXAMPLE 1
Show that the polygons are congruent by identifying all of
the congruent corresponding parts. Then write a
congruence statement.
AB ≅ JI BC ≅ IH CD ≅ HG
DE ≅ GF EA ≅ FJ
∠A ≅ ∠J ∠B ≅ ∠I ∠C ≅ ∠H
∠D ≅ ∠G ∠E ≅ ∠F

21. EXAMPLE 1
Show that the polygons are congruent by identifying all of
the congruent corresponding parts. Then write a
congruence statement.
AB ≅ JI BC ≅ IH CD ≅ HG
DE ≅ GF EA ≅ FJ
∠A ≅ ∠J ∠B ≅ ∠I ∠C ≅ ∠H
∠D ≅ ∠G ∠E ≅ ∠F
Since all corresponding parts are congruent, ABCDE ≅ JIHGF

22. EXAMPLE 2
In the diagram, ∆ITP ≅ ∆NGO. Find the values of x and y.

23. EXAMPLE 2
In the diagram, ∆ITP ≅ ∆NGO. Find the values of x and y.
∠P ≅ ∠O

24. EXAMPLE 2
In the diagram, ∆ITP ≅ ∆NGO. Find the values of x and y.
6y − 14 = 40
∠P ≅ ∠O

25. EXAMPLE 2
In the diagram, ∆ITP ≅ ∆NGO. Find the values of x and y.
6y − 14 = 40
∠P ≅ ∠O
6y = 54

26. EXAMPLE 2
In the diagram, ∆ITP ≅ ∆NGO. Find the values of x and y.
6y − 14 = 40
∠P ≅ ∠O
6y = 54
y = 9

27. EXAMPLE 2
In the diagram, ∆ITP ≅ ∆NGO. Find the values of x and y.
6y − 14 = 40
∠P ≅ ∠O
6y = 54
y = 9
IT ≅ NG

28. EXAMPLE 2
In the diagram, ∆ITP ≅ ∆NGO. Find the values of x and y.
6y − 14 = 40
∠P ≅ ∠O
6y = 54
y = 9
x − 2 y = 7.5
IT ≅ NG

29. EXAMPLE 2
In the diagram, ∆ITP ≅ ∆NGO. Find the values of x and y.
6y − 14 = 40
∠P ≅ ∠O
6y = 54
y = 9
x − 2 y = 7.5
IT ≅ NG
x − 2(9) = 7.5

30. EXAMPLE 2
In the diagram, ∆ITP ≅ ∆NGO. Find the values of x and y.
6y − 14 = 40
∠P ≅ ∠O
6y = 54
y = 9
x − 2 y = 7.5
IT ≅ NG
x − 2(9) = 7.5
x − 18 = 7.5

31. EXAMPLE 2
In the diagram, ∆ITP ≅ ∆NGO. Find the values of x and y.
6y − 14 = 40
∠P ≅ ∠O
6y = 54
y = 9
x − 2 y = 7.5
IT ≅ NG
x − 2(9) = 7.5
x − 18 = 7.5
x = 25.5

32. EXAMPLE 3
Write a two-column proof.
Given:
∠L ≅ ∠P, LM ≅ PO, LN ≅ PN, MN ≅ ON
Prove: △LMN ≅△PON

33. EXAMPLE 3
Write a two-column proof.
Given:
∠L ≅ ∠P, LM ≅ PO, LN ≅ PN, MN ≅ ON
Prove: △LMN ≅△PON
1. ∠L ≅ ∠P, LM ≅ PO, LN ≅ PN, MN ≅ ON

34. EXAMPLE 3
Write a two-column proof.
Given:
∠L ≅ ∠P, LM ≅ PO, LN ≅ PN, MN ≅ ON
Prove: △LMN ≅△PON
1. ∠L ≅ ∠P, LM ≅ PO, LN ≅ PN, MN ≅ ON 1. Given

35. EXAMPLE 3
Write a two-column proof.
Given:
∠L ≅ ∠P, LM ≅ PO, LN ≅ PN, MN ≅ ON
Prove: △LMN ≅△PON
1. ∠L ≅ ∠P, LM ≅ PO, LN ≅ PN, MN ≅ ON 1. Given
2. ∠LNM ≅ ∠PNO

36. EXAMPLE 3
Write a two-column proof.
Given:
∠L ≅ ∠P, LM ≅ PO, LN ≅ PN, MN ≅ ON
Prove: △LMN ≅△PON
1. ∠L ≅ ∠P, LM ≅ PO, LN ≅ PN, MN ≅ ON 1. Given
2. ∠LNM ≅ ∠PNO 2. Vertical Angles Thm.

37. EXAMPLE 3
Write a two-column proof.
Given:
∠L ≅ ∠P, LM ≅ PO, LN ≅ PN, MN ≅ ON
Prove: △LMN ≅△PON
1. ∠L ≅ ∠P, LM ≅ PO, LN ≅ PN, MN ≅ ON 1. Given
2. ∠LNM ≅ ∠PNO 2. Vertical Angles Thm.
3. ∠M ≅ ∠O

38. EXAMPLE 3
Write a two-column proof.
Given:
∠L ≅ ∠P, LM ≅ PO, LN ≅ PN, MN ≅ ON
Prove: △LMN ≅△PON
1. ∠L ≅ ∠P, LM ≅ PO, LN ≅ PN, MN ≅ ON 1. Given
2. ∠LNM ≅ ∠PNO 2. Vertical Angles Thm.
3. ∠M ≅ ∠O 3. Third Angle Theorem

39. EXAMPLE 3
Write a two-column proof.
Given:
∠L ≅ ∠P, LM ≅ PO, LN ≅ PN, MN ≅ ON
Prove: △LMN ≅△PON
1. ∠L ≅ ∠P, LM ≅ PO, LN ≅ PN, MN ≅ ON 1. Given
2. ∠LNM ≅ ∠PNO 2. Vertical Angles Thm.
3. ∠M ≅ ∠O 3. Third Angle Theorem
4.△LMN ≅△PON

40. EXAMPLE 3
Write a two-column proof.
Given:
∠L ≅ ∠P, LM ≅ PO, LN ≅ PN, MN ≅ ON
Prove: △LMN ≅△PON
1. ∠L ≅ ∠P, LM ≅ PO, LN ≅ PN, MN ≅ ON 1. Given
2. ∠LNM ≅ ∠PNO 2. Vertical Angles Thm.
3. ∠M ≅ ∠O 3. Third Angle Theorem
4.△LMN ≅△PON 4. Def. of congruent polygons

41. PROBLEM SET

42. PROBLEM SET
p. 257 #1-23 odd, 29, 36, 39, 49, 53, 55
“I've always tried to go a step past wherever people expected
me to end up.” - Beverly Sills