1. Five-Minute Check (over Lesson 4–1) Mathematical Practices Then/Now
New Vocabulary
Key Concept: Definition of Congruent Polygons
Example 1: Proving Polygons Congruent
Example 2: Use Corresponding Parts of Congruent Triangles
Theorem 4.3 Third Angles Theorem
Example 3: Real-World Example: Use the Third Angles Theorem
Example 4: Prove that Two Triangles Are Congruent
Theorem 4.4 Properties of Triangle Congruence
Lesson Menu

2. Find m1.
A. 115
B. 105
C. 75
D. 65
5-Minute Check 1

3. Find m2.
A. 75
B. 72
C. 57
D. 40
5-Minute Check 2

4. Find m3.
A. 75
B. 72
C. 57
D. 40
5-Minute Check 3

5. Find m4.
A. 18
B. 28
C. 50
D. 75
5-Minute Check 4

6. Find m5.
A. 70
B. 90
C. 122
D. 140
5-Minute Check 5

7. Mathematical Practices
3 Construct viable arguments and critique the reasoning of others.
Content Standards
G.CO.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. MP

8. G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.).
G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. MP

9. You identified and used congruent angles.
Name and use corresponding parts of congruent polygons.
Prove triangles congruent using the definition of congruence.
Then/Now

10. principle of superposition
Congruent polygons
corresponding parts
principle of superposition
New Vocabulary

11. Key Concept

12. Proving Polygons Congruent
Show that the polygons are congruent by using rigid motions and by identifying all of the congruent corresponding parts. Then write a congruence statement.
Example 1

13. Proving Polygons Congruent
Does a translation one polygon to the other? Yes
Are the corresponding angles of the polygons congruent? Yes
List the congruent angle pairs.  A ≅  R,  B ≅  T,  C ≅  P,  D ≅
 S,  E ≅  Q
Are the corresponding sides of the polygons congruent? Yes
AB ≅ RT, BC ≅ TP, CD ≅ PS, DE ≅ SQ , EA ≅ QR
List the corresponding congruent sides.
List the corresponding congruent sides. Yes
Write the congruence statement. ABCDE ≅ RTPSQ
Example 1

14. Proving Polygons Congruent
Answer: A translation maps one polygon exactly onto the other.  A ≅  R,  B ≅  T,  C ≅  P,  D ≅  S,  E ≅  Q All corresponding parts of the two polygons are congruent. Therefore, ABCDE ≅ RTPSQ.
Example 1

15. In the diagram, △ITP ≅ △NGO. Find the values of x and y.
Use Corresponding Parts of Congruent Triangles
In the diagram, △ITP ≅ △NGO. Find the values of x and y.
Example 2

16. 6y = 54 Solve for y. Add 14 to both sides.
Use Corresponding Parts of Congruent Triangles
6y – 14 = 40  P ≅  O
6y = 54 Solve for y. Add 14 to both sides.
y = 9 Divide both sides by 6.
IT  NG
x – 2y = 7.5
x – 2(9) = 7.5 y = 9
x – 18 = 7.5 Solve for x by adding 18 to both sides.
x = 25.5
Answer: x = 25.5, y = 9
Example 2

17. Theorem

18. Use the Third Angles Theorem
ARCHITECTURE A drawing of a tower’s roof is composed of congruent triangles all converging at a point at the top. If J ≅ K and mJ = 72, find mJIH.
Real-World Example 3

19. 72 + 72 +  JIK = 180 The angles of a triangle add together to 180.
Use the Third Angles Theorem
m  J = m  K = 72  J ≅  K
 JIK = 180 The angles of a triangle add together to 180.
 JIK = 36  JIK ≅  JIH
 JIH = 36 Corresponding parts of congruent triangles are congruent.
Answer: mJIH = 36
Real-World Example 3

20. Write a two-column proof. Given: Prove:  LMN ≅  PON
Prove that Two Triangles Are Congruent
Write a two-column proof.
Given:
Prove:  LMN ≅  PON
Example 4

21. Prove that Two Triangles Are Congruent
 J ≅  K, LM ≅ PO , LN ≅ PN , MN ≅ ON, Write the given statements.
Why is  LNM ≅  PNO? Vertical Angles Theorem.
Why is  M ≅  O? Third Angles Theorem.
Why is ∆ LMN ≅ ∆ PON? Definition of congruent polygons
Example 4

22. Prove that Two Triangles Are Congruent
Answer:
Example 4

23. Theorem