LESSON 4–3
Congruent Triangles
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
Over Lesson 4–2
Find m1.
A. 115
B. 105
C. 75
D. 65
Over Lesson 4–2
Find m2.
A. 75
B. 72
C. 57
D. 40
Over Lesson 4–2
Find m3.
A. 75
B. 72
C. 57
D. 40
Over Lesson 4–2
Find m4.
A. 18
B. 28
C. 50
D. 75
Over Lesson 4–2
Find m5.
A. 70
B. 90
C. 122
D. 140
Over Lesson 4–2
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
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)
You identified and used congruent angles.
• Name and use corresponding parts of
congruent polygons.
• Prove triangles congruent using the
definition of congruence.
• congruent polygons
• corresponding parts
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.
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.
Use Corresponding Parts of Congruent
Triangles
In the diagram, ΔITP  ΔNGO. Find the values of
x and y.
O  P
mO = mP
6y – 14 = 40
CPCTC
Definition of congruence
Substitution
Use Corresponding Parts of Congruent
Triangles
6y = 54
y= 9
Add 14 to each side.
Divide each side by 6.
CPCTC
NG = IT
x – 2y = 7.5
x – 2(9) = 7.5
x – 18 = 7.5
x = 25.5
Answer: x = 25.5, y = 9
Definition of congruence
Substitution
y=9
Simplify.
Add 18 to each side.
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
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
Use the Third Angles Theorem
mIJK + mIJK + mJIK = 180
Substitution
72 + 72 + 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
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
Prove That Two Triangles are Congruent
Write a two-column proof.
Prove: ΔLMN  ΔPON
Prove That Two Triangles are Congruent
Proof:
Statements
Reasons
1.
1. Given
2. LNM  PNO
2. Vertical Angles Theorem
3. M  O
3. Third Angles Theorem
4. ΔLMN  ΔPON
4. CPCTC
Find the missing information in the following proof.
Prove: ΔQNP  ΔOPN
Proof:
Statements
Reasons
1. Given
2. Reflexive Property of
Congruence
3. Q  O, NPQ  PNO 3. Given
4. _________________
4. QNP  ONP
?
1.
2.
5. ΔQNP  ΔOPN
5. Definition of Congruent Polygons
A. CPCTC
B. Vertical Angles Theorem
C. Third Angles Theorem
D. Definition of Congruent
Angles
LESSON 4–3
Congruent Triangles