1. LESSON 4–3
Congruent Triangles

2. Five-Minute Check (over Lesson 4–2) TEKS Then/Now New Vocabulary
Key Concept: Definition of Congruent Polygons
Example 1: Identify Corresponding Congruent Parts
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

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

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

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

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

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

8. One angle in an isosceles triangle has a measure of 80°
One angle in an isosceles triangle has a measure of 80°. What is the measure of one of the other two angles?
A. 35
B. 40
C. 50
D. 100
5-Minute Check 6

9. Mathematical Processes G.1(F), G.1(G)
Targeted TEKS
G.6(D) Verify theorems about the relationships in triangles, including proof of the Pythagorean Theorem, the sum of interior angles, base angles of isosceles triangles, midsegments,
and medians, and apply these relationships to solve problems.
Mathematical Processes
G.1(F), G.1(G)
TEKS

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

11. congruent polygons
corresponding parts
Vocabulary

12. Concept 1

13. Identify Corresponding Congruent Parts
Show that the polygons are congruent by identifying all of the congruent corresponding parts. Then write a congruence statement.
Angles:
Sides:
Answer: All corresponding parts of the two polygons are congruent. Therefore, ABCDE  RTPSQ.
Example 1

14. The support beams on the fence form congruent triangles
The support beams on the fence form congruent triangles. In the figure ΔABC  ΔDEF, which of the following congruence statements correctly identifies corresponding angles or sides? A. B. C. D.
Example 1

15. In the diagram, ΔITP  ΔNGO. Find the values of x and y.
Use Corresponding Parts of Congruent Triangles
In the diagram, ΔITP  ΔNGO. Find the values of x and y.
O  P CPCTC
mO = mP Definition of congruence
6y – 14 = 40 Substitution
Example 2

16. NG = IT Definition of congruence x – 2y = 7.5 Substitution
Use Corresponding Parts of Congruent Triangles
6y = 54 Add 14 to each side.
y = 9 Divide each side by 6.
CPCTC
NG = IT Definition of congruence
x – 2y = 7.5 Substitution
x – 2(9) = 7.5 y = 9
x – 18 = 7.5 Simplify.
x = 25.5 Add 18 to each side.
Answer: x = 25.5, y = 9
Example 2

17. In the diagram, ΔFHJ  ΔHFG. Find the values of x and y.
A. x = 4.5, y = 2.75
B. x = 2.75, y = 4.5
C. x = 1.8, y = 19
D. x = 4.5, y = 5.5
Example 2

18. Concept 2

19. ΔJIK  ΔJIH Congruent Triangles
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 IJK  IKJ and mIJK = 72, find mJIH.
ΔJIK  ΔJIH Congruent Triangles
mIJK + mIKJ + mJIK = 180 Triangle Angle-Sum Theorem
Example 3

20. mIJK + mIJK + mJIK = 180 Substitution
Use the Third Angles Theorem
mIJK + mIJK + mJIK = 180 Substitution
mJIK = 180 Substitution
144 + mJIK = 180 Simplify.
mJIK = 36 Subtract 144 from each side.
mJIH = 36 Third Angles Theorem
Answer: mJIH = 36
Example 3

21. TILES A drawing of a tile contains a series of triangles, rectangles, squares, and a circle. If ΔKLM  ΔNJL, KLM  KML, and mKML = 47.5, find mLNJ.
A. 85
B. 45
C. 47.5
D. 95
Example 3

22. Write a two-column proof.
Prove That Two Triangles are Congruent
Write a two-column proof.
Prove: ΔLMN  ΔPON
Example 4

23. 2. Vertical Angles Theorem
Prove That Two Triangles are Congruent
Proof:
Statements Reasons
1. Given 1.
2. LNM  PNO
2. Vertical Angles Theorem
3. M  O
3. Third Angles Theorem
4. ΔLMN  ΔPON
4. CPCTC
Example 4

24. Find the missing information in the following proof.
Prove: ΔQNP  ΔOPN
Proof:
Reasons
Statements 1.
1. Given 2.
2. Reflexive Property of Congruence
3. Q  O, NPQ  PNO
3. Given
4. _________________
4. QNP  ONP ?
5. Definition of Congruent Polygons
5. ΔQNP  ΔOPN
Example 4

25. B. Vertical Angles Theorem
A. CPCTC
B. Vertical Angles Theorem
C. Third Angles Theorem
D. Definition of Congruent Angles
Example 4

26. Concept 3

27. LESSON 4–3
Congruent Triangles