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

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:

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  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

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 m  IJK = 72, find m  JIH. m  IJK + m  IKJ + m  JIK=180Triangle Angle-Sum Theorem ΔJIK  ΔJIH Congruent Triangles

Example 3 Use the Third Angles Theorem m  IJK + m  IJK + m  JIK =180Substitution m  JIK =180Substitution m  JIK =180Simplify. m  JIK =36Subtract 144 from each side. m  JIH =36Third Angles Theorem Answer: m  JIH = 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 m  KML = 47.5, find m  LNJ.

Example 4 Prove That Two Triangles are Congruent 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

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