4.3 Congruent Triangles

CCSS

Content StandardsG.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 Practices6 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

Example 1Identify Corresponding Congruent Parts

Show that the polygons are congruent by identifying all of the congruent corresponding parts. Then write a congruence statement.

Answer: All corresponding parts of the two polygons are congruent. Therefore, ABCDE RTPSQ.

Sides:

Angles:

Example 1The 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 2Use Corresponding Parts of Congruent Triangles

O P CPCTC

mO = mP Definition of congruence

6y – 14 = 40 Substitution

In the diagram, ΔITP ΔNGO. Find the values of x and y.

Example 2Use Corresponding Parts of Congruent Triangles

6y = 54 Add 14 to each side.

y = 9 Divide each side by 6.

NG = IT Definition of congruence

x – 2y = 7.5 Substitution

x – 2(9) = 7.5 y = 9

x – 18 = 7.5 Simplify.

x = 25.5 Add 18 to each side.

CPCTC

Answer: x = 25.5, y = 9

Example 2

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

In the diagram, ΔFHJ ΔHFG. Find the values of x and y.

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 mIJK = 72, find mJIH.

mIJK + mIKJ + mJIK = 180 Triangle Angle-SumTheorem

ΔJIK ΔJIH Congruent Triangles

Example 3Use the Third Angles Theorem

mIJK + mIJK + mJIK = 180 Substitution

72 + 72 + mJIK = 180 Substitution

144 + mJIK = 180 Simplify.

mJIK = 36 Subtract 144 fromeach side.

mJIH = 36 Third Angles Theorem

Answer: mJIH = 36

Example 3

A. 85

B. 45

C. 47.5

D. 95

TILES A drawing of a tile contains a series of triangles, rectangles, squares, and a circle. If ΔKLM ΔNJL, KLM KML, and mKML = 47.5, find mLNJ.

Example 4Prove That Two Triangles are Congruent

Prove: ΔLMN ΔPON

Example 4Prove That Two Triangles are Congruent

2. LNM PNO 2. Vertical Angles Theorem

Proof:

Statements Reasons

3. M O

3. Third Angles Theorem

4. ΔLMN ΔPON

4. CPCTC

1. Given1.

Example 4Find the missing information in the following proof.

Prove: ΔQNP ΔOPNProof:

ReasonsStatements

3. Q O, NPQ PNO 3. Given

5. Definition of Congruent Polygons5. ΔQNP ΔOPN

4. _________________4. QNP ONP ?

2. 2. Reflexive Property ofCongruence

1. 1. Given

Example 4A. CPCTC

B. Vertical Angles Theorem

C. Third Angles Theorem

D. Definition of Congruent Angles