1. 4.24.2 Apply Congruence and Triangles
Bell Thinger
1. When are two angles congruent?
ANSWER when they have the same measure
ANSWER 45º
2. In ∆ABC, if m A = 64º and m B = 71º, what is m C?
ANSWER Transitive Property
3. What property of angle congruence is illustrated by
this statement? If A ≅ B and B ≅ C, then A ≅
C.

2. 4.2

3. 4.2Example 1
Write a congruence statement for the
triangles. Identify all pairs of congruent
corresponding parts.
SOLUTION
The diagram indicates that JKL ≅ TSR.
Corresponding angles ∠J ≅ ∠T, ∠K ≅ ∠S, ∠L ≅ ∠R
Corresponding sides JK ≅ TS, KL ≅ SR, LJ ≅ RT

4. 4.2Example 2
In the diagram, DEFG ≅ SPQR.
Find the value of x.a.
SOLUTION
FG = QR
12 = 2x – 4
16 = 2x
8 = x
You know that FG ≅ QR.a.

5. 4.2Example 2
In the diagram, DEFG ≅ SPQR.
SOLUTION
b. Find the value of y.
b. You know that ∠F ≅ ∠Q.
m∠F = m∠Q
68o
= (6y + x)
o
68 = 6y + 8
10 = y

6. 4.2Example 3
SOLUTION
If you divide the
wall into orange and blue
sections along JK , will the
sections of the wall be the
same size and shape?Explain.
PAINTING
From the diagram, ∠A ≅ ∠C and ∠D ≅ ∠B because all
right angles are congruent. Also, by the Lines
Perpendicular to a Transversal Theorem, AB || DC .

7. 4.2Example 3
Then, ∠1 ≅ ∠4 and ∠2 ≅ ∠3 by the Alternate Interior
Angles Theorem. So, all pairs of corresponding angles
are congruent.
The diagram shows AJ ≅ CK , KD ≅ JB , and DA ≅ BC . By
the Reflexive Property, JK ≅ KJ . All corresponding parts
are congruent, so AJKD ≅ CKJB.

8. 4.2Guided Practice
1. Identify all pairs of congruent
corresponding parts.
ANSWER
Corresponding sides: AB ≅ CD, BG ≅ DE,
GH ≅ FE, HA ≅ FC
Corresponding angles: ∠A ≅ ∠C, ∠B ≅ ∠D,
∠G ≅ ∠E, ∠H ≅ ∠F.
In the diagram at the right, ABGH ≅ CDEF.

9. 4.2
3. Show that PTS ≅ RTQ.
ANSWER
All of the corresponding parts of PTS are congruent
to those of RTQ by the indicated markings, the
Vertical Angle Theorem and the Alternate Interior
Angle theorem.
Guided Practice

10. 4.2
In the diagram at the right, ABGH ≅ CDEF.
2. Find the value of x and
find m∠H.
ANSWER 25, 105°
Guided Practice

11. 4.2

12. 4.2Example 4
Find m∠BDC.
So, m∠ACD = m∠BDC = 105° by the definition of
congruent angles.
ANSWER
SOLUTION
∠A ≅ ∠B and ∠ADC ≅ ∠BCD, so by the Third Angles
Theorem, ∠ACD ≅ ∠BDC. By the Triangle Sum
Theorem, m∠ACD = 180° – 45° – 30° = 105° .

13. 4.2Example 5
Plan for Proof
b. Use the Third Angles Theorem to show that ∠B ≅ ∠D.
Write a proof.
PROVE: ACD ≅ CAB
GIVEN: AD ≅ CB, DC ≅ BA,
∠ACD ≅ ∠CAB, ∠CAD ≅ ∠ACB
a. Use the Reflexive Property to show that AC ≅ AC.

14. 4.2Example 5
Plan in Action
1. Given
2. Reflexive Property of
Congruence
STATEMENTS REASONS
3. Given
4. Third Angles Theorem
1. AD ≅ CB , DC ≅ BA
2.a. AC ≅ AC.
3. ∠ACD ≅ ∠CAB,
∠CAD ≅ ∠ACB
4.b. ∠B ≅ ∠D
5. ACD ≅ CAB Definition of ≅ figures5.

15. 4.2
4. In the diagram, what is m∠DCN.
ANSWER 75°
Guided Practice

16. 4.2
By the definition of congruence, what
additional information is needed to
know that NDC ≅ NSR.
5.
Guided Practice
ANSWER
DC ≅ RS and DN ≅ SN

17. 4.2

18. 4.2Exit Slip
CAANSWER
3. EDF ≅ ?
In the diagram, ABC ≅ DEF.
Complete each statement.
BACANSWER
60°ANSWER
2. FD ≅ ?
m A = ?1.

19. 4.2
4. Write a congruence statement for the two small
triangles. Explain your reasoning.
Exit Slip
WXZ ≅ YXZ; The diagram tell us that
W ≅ Y and WZX ≅ YZX.
WXZ ≅ YXZ by the Third Thm.
From the diagram WX ≅ YX and
WZ ≅ YZ, and XZ ≅ XZ by Refl. Prop.
Of ≅.
ANSWER

20. 4.2
Homework
Pg 238-241
#5, 15, 16, 19, 20