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Lesson Menu Five-Minute Check (over Lesson 4–2) 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 A.A B.B C.C D.D 5-Minute Check 1 A.115 B.105 C.75 D.65 Find m  1.

Over Lesson 4–2 A.A B.B C.C D.D 5-Minute Check 2 A.75 B.72 C.57 D.40 Find m  2.

Over Lesson 4–2 A.A B.B C.C D.D 5-Minute Check 3 A.75 B.72 C.57 D.40 Find m  3.

Over Lesson 4–2 A.A B.B C.C D.D 5-Minute Check 4 A.18 B.28 C.50 D.75 Find m  4.

Over Lesson 4–2 A.A B.B C.C D.D 5-Minute Check 5 A.70 B.90 C.122 D.140 Find m  5.

Over Lesson 4–2 A.A B.B C.C D.D 5-Minute Check 5 A.35 B.40 C.50 D.100 One angle in an isosceles triangle has a measure of 80°. What is the measure of one of the other two angles?

Then/Now You identified and used congruent angles. (Lesson 1–4) Name and use corresponding parts of congruent polygons. Prove triangles congruent using the definition of congruence.

Vocabulary congruent congruent polygons corresponding parts

Concept 1

Example 1 Identify 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:

A.A B.B C.C D.D Example 1 The support beams on the fence form congruent triangles. In the figure ΔABC  ΔDEF, which of the following congruence statements directly matches corresponding angles or sides? A. B. C. D.

Example 2 Use Corresponding Parts of Congruent Triangles  O  PCPCTC m  O=m  PDefinition of congruence 6y – 14=40Substitution In the diagram, ΔITP  ΔNGO. Find the values of x and y.

Example 2 Use Corresponding Parts of Congruent Triangles 6y=54Add 14 to each side. y=9Divide each side by 6. NG=ITDefinition of congruence x – 2y=7.5Substitution x – 2(9)=7.5y = 9 x – 18=7.5Simplify. x=25.5Add 18 to each side. CPCTC Answer: x = 25.5, y = 9

A.A B.B C.C D.D 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  J   K and m  J = 72, find m  JIH. m  KJI + m  IKJ + m  JIK=180Triangle Angle-Sum Theorem  H   K,  I   I and  J   JCPCTC ΔJIK  ΔJIH Congruent Triangles

Example 3 Use the Third Angles Theorem m  JIK=180Simplify. m  JIK=36 Subtract 144 from each side. Answer: m  JIH = m  JIK =180 Substitution m  JIH = 36 Third Angles Theorem

A.A B.B C.C D.D 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 m  KML = 47.5, find m  LNJ.

Example 4 Prove That Two Triangles are Congruent Write a two-column proof. Prove:ΔLMN  ΔPON

Example 4 Prove That Two Triangles are Congruent 2.  LNM   PNO 2. Vertical Angles Theorem Proof: StatementsReasons 3.  M   O 3. Third Angles Theorem 4.ΔLMN  ΔPON 4. CPCTC 1. Given 1.

Example 4 Find the missing information in the following proof. Prove:ΔQNP  ΔOPN Proof: ReasonsStatements 3.  Q   O,  NPQ   PNO 3. Given 5. Definition of Congruent Polygons 5. ΔQNP  ΔOPN 4. _________________ 4.  QNP   ONP ? Reflexive Property of Congruence Given

A.A B.B C.C D.D Example 4 A.CPCTC B.Vertical Angles Theorem C.Third Angle Theorem D.Definition of Congruent Angles

Concept 3

End of the Lesson