1. 4.2 Apply Congruence and Triangles
You will identify congruent figures.
Essential Question: What are congruent figures?
Tell students they will learn how to answer this question by studying the definition of congruent figures

2. EXAMPLE 1
Identify congruent parts
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

3. EXAMPLE 2
Use properties of congruent figures
In the diagram, DEFG SPQR.
Find the value of x.
Find the value of y.
SOLUTION
You know that FG QR. FG
= QR 12
= 2x – 4 16
= 2x 8
= x

4. Use properties of congruent figures
EXAMPLE 2
Use properties of congruent figures
You know that ∠ F Q.
m F
= m Q 68 o
= (6y + x) 68
= 6y + 8 10
= y

5. EXAMPLE 3
Show that figures are congruent
PAINTING
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.
SOLUTION
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 .

6. EXAMPLE 3
Show that figures are congruent
Then, and 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.

7. GUIDED PRACTICE
for Examples 1, 2, and 3
In the diagram at the right, ABGH CDEF.
Identify all pairs of congruent corresponding parts.
SOLUTION
Corresponding sides:
AB CD, BG DE,
GH FE, HA FC
Corresponding angles:
A C, B D,
G E, H F.

8. GUIDED PRACTICE
for Examples 1, 2, and 3
In the diagram at the right, ABGH CDEF. 2.
Find the value of x and find m H.
SOLUTION
(a)
You know that H F
(4x+ 5)° = 105°
4x = 100
x = 25
(b)
You know that H F
m H m F =105°

9. GUIDED PRACTICE
for Examples 1, 2, and 3
In the diagram at the right, ABGH CDEF. 3.
Show that PTS RTQ.
SOLUTION
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.

10. EXAMPLE 4
Use the Third Angles Theorem
Find m BDC.
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° .
So, m ACD = m BDC = 105° by the definition of congruent angles.
ANSWER

11. EXAMPLE 5
Prove that triangles are congruent
Write a proof.
GIVEN
AD CB, DC AB
ACD CAB,
CAD ACB
PROVE
ACD CAB
Plan for Proof
AC AC.
Use the Reflexive Property to show that
Use the Third Angles Theorem to show that
B D

12. Prove that triangles are congruent
EXAMPLE 5
Prove that triangles are congruent
Plan in Action
STATEMENTS
REASONS
AD CB
, DC BA
Given
AC AC.
Reflexive Property of Congruence
ACD CAB,
CAD ACB
Given
B D
Third Angles Theorem
ACD CAB
Definition of

13. GUIDED PRACTICE
for Examples 4 and 5
DCN.
In the diagram, what is m
SOLUTION
CDN NSR, DNC SNR then the third angles are also congruent NRS DCN = 75°

14. GUIDED PRACTICE
for Examples 4 and 5
By the definition of congruence, what
additional information is needed to
know that
NDC NSR.
ANSWER
DC RS and DN SN

15. Daily Homework Quiz
In the diagram, ABC DEF. Complete the statement.
m A = ? 1.
60°
ANSWER 2.
FD ? CA
ANSWER 3.
EDF ?
BAC
ANSWER

16. WXZ YXZ; The diagram tell us that W Y and WZX YZX.
Daily Homework Quiz 4.
Write a congruence statement for the two small triangles. Explain your reasoning.
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 Segs. s
ANSWER

17. You will identify congruent figures.
Essential Question: What are congruent figures?
• Triangles can be proved congruent by showing that all 3 pairs of corresponding sides and all 3 pairs of corresponding angles
are congruent.
• If two angles of one triangle are congruent to two angles of another, then the third angles are congruent.
Congruent figures are figures that have exactly the same size and shape.