Section 4.3 Congruent Triangles. If two geometric figures have exactly the same shape and size, they are congruent. In two congruent polygons, all of.

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Presentation transcript:

Section 4.3 Congruent Triangles

If two geometric figures have exactly the same shape and size, they are congruent. In two congruent polygons, all of the parts of one polygon are congruent to the corresponding parts or matching parts of the other polygon. These corresponding parts include corresponding angles and corresponding sides.

Concept 1 Other congruence statements for the triangles above exist. Valid congruence statements for congruent polygons list corresponding vertices in the same order.

Example 1: a) Show that the polygons are congruent by identifying all of the congruent corresponding parts. Then write a congruence statement. Angles:  A   R,  B   T,  C   P,  D   S,  E   Q All corresponding parts of the two polygons are congruent. Therefore, ABCDE  RTPSQ.

b) 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)

The phrase “if and only if” in the congruent polygon definition means that both the conditional and converse are true. So, if two polygons are congruent, then their corresponding parts are congruent. For triangles we say Corresponding Parts of Congruent Triangles are Congruent or CPCTC.

Example 2: a) In the diagram, ΔITP  ΔNGO. Find the values of x and y.  O   PCPCTC m  O = m  PDefinition of congruence 6y – 14= 40Substitution 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) = 7.5y = 9 x – 18 =7.5Simplify x = 25.5 Add 18 to each side CPCTC x = 25.5, y = 9

b) In the diagram, ΔFHJ  ΔHFG. Find the values of x and y.  GFH   HFG so: 6x + 8 = 35, solve for x. x = 4.5

Concept 2

Example 3: 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 = 180 ° Triangle Angle-Sum Theorem ΔJIK  ΔJIH Congruent Triangles 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

Example 4: Write a two-column proof. Prove: ΔLMN  ΔPON Given:  L   P StatementsReasons L  PL  P Given  LNM   PNO M  OM  O ΔLMN  ΔPON Vertical Angles Theorem Third Angles Theorem CPCTC

Concept 3