4.3
Congruent Triangles
CCSS
Content Standards
G.CO.7 Use the definition of congruence in terms of rigid
motions to show that two triangles are congruent if and
only if corresponding pairs of sides and corresponding
pairs of angles are congruent.
G.SRT.5 Use congruence and similarity criteria for
triangles to solve problems and to prove relationships in
geometric figures.
Mathematical Practices
6 Attend to precision.
3 Construct viable arguments and critique the
reasoning of others.
Then/Now
You identified and used congruent angles.
• Name and use corresponding parts of congruent
polygons.
• Prove triangles congruent using the definition of
congruence.
Concept 1
Identify Corresponding Congruent Parts
Example 1
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
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 2
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
Example 2
6y = 54
Add 14 to each side.
y= 9
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.
Example 2
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
Concept 2
Example 3
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 Angles Theorem
Use the Third
mIJK + mIJK + mJIK = 180
Substitution
72 + 72 + mJIK = 180
Substitution
144 + mJIK = 180
Answer:
mJIH = 36
Simplify.
mJIK = 36
Subtract 144 from
each side.
mJIH = 36
Third Angles
Theorem
Example 3
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
Example 4
Prove: ΔLMN  ΔPON
Example 4
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
Example 4
Find the missing information in the following proof.
Prove: ΔQNP  ΔOPN
Proof:
Statements
Reasons
3. Q  O, NPQ  PNO
4. QNP  ONP
1. Given
2. Reflexive Property of
Congruence
3. Given
4. _________________
?
5. ΔQNP  ΔOPN
5. Definition of Congruent Polygons
1.
2.
Example 4
A. CPCTC
B. Vertical Angles Theorem
C. Third Angles Theorem
D. Definition of Congruent
Angles