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

2. Proving triangles congruent. 4.2 Identify congruent figures.
Target
Proving triangles congruent.
GOAL:
4.2 Identify congruent figures.

3. Vocabulary
congruent figures - figures the same shape and size;
two figures are congruent if and only if all corresponding angles and all corresponding sides are congruent.
CPCTC - Corresponding Parts of Congruent Triangles
are Congruent
Third Angles Theorem 4.3 –
If two angles of one triangle are congruent to two angles
of another triangle, then the third angles are also congruent.
R-S-T Theorem for Congruent Triangles 4.4 –
Congruence of triangles is reflexive, symmetric, and
transitive.

4. EXAMPLE 1
Identify congruent parts
To write a congruence statement for the triangles first identify all pairs of congruent corresponding parts.
SOLUTION
Corresponding angles
J T, ∠ K S, L R
Corresponding sides
JK TS, KL SR, LJ RT
congruence statement
The diagram indicates that
JKL TSR.

5. Use properties of congruent figures
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.
You know that ∠ F Q. FG
= QR
m F
= m Q 12
= 2x – 4 68 o
= (6y + x) 16
= 2x 68
= 6y + 8 8
= x 10
= y

6. 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 .

7. GUIDED PRACTICE
for Examples 1, 2, and 3
In the diagram at the below, 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. 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

10. 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

11. 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

12. 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°
By the definition of congruence, what additional information is needed to know that
NDC NSR.
ANSWER
DC RS and DN SN