Lesson 4.3 Exploring Congruent Triangles

Definition of Congruent Triangles If ΔABC is congruent to ΔPQR, then there is a correspondence between their angles and sides such that corresponding angles are congruent and corresponding sides are congruent. The notation ΔABC  ΔPQR indicates the congruence and the correspondence, as shown below ΔABC  ΔPQR C B A P Q R Corresponding angles are: A   P  B   Q  C   R Corresponding sides are: AB  PQ BC  QR CA  RP

Congruent Triangles

Example 1 Naming Congruent Parts You and a friend have identical drafting triangles, as shown below. Name all congruent parts.

Example 1 Naming Congruent Parts Corresponding angles are:  D   R  E   S  F   T Corresponding sides are: DE  RS EF  ST FD  TR ∆DEF  ∆RST

Which of the following expresses the correct congruence statement for the figure below?

Classification of Triangles by Sides An equilateral triangle has 3 congruent sides An isosceles triangle has a least two congruent sides A scalene triangle has no sides congruent

Classification of Triangles by Angles An acute triangle has 3 acute angles. If these angles are all congruent, the triangle is also equiangular A right triangle has exactly one right angle An obtuse triangle has exactly one obtuse angle Obtuse

Vocabulary In ΔABC, each of the points A, B, and C is a vertex of the triangle The side BC is the side opposite  A Two sides that share a common vertex are adjacent sides

Vocabulary for right and isosceles triangles In a right triangle, the sides adjacent to the right angle are the legs of the triangle. The side opposite the right angle is the hypotenuse of the triangle An isosceles triangle can have 3 congruent sides. If is has only two, the two congruent sides are the legs of the triangle. The third side is the base of the triangle.

Example 2 Proving Triangles are Congruent The outside structure of the Bank of China is glass and aluminum and consists of more than 50 congruent triangles. Use the information given below to prove that ΔAEBΔDEC.

10.Def. of Congruent Triangles Statements 1. AB||CD 2.  EAB  EDC Reasons 1. Given 2. 2 lines ||  alt. int.  are  3.2 lines ||  alt. int.  are  4.Vertical angles are  5.Given 6.Given 7. Def. of midpoint 8. Given 9. Def. of Midpoint 10.Def. of Congruent Triangles Statements 1. AB||CD 2.  EAB  EDC 3. ABE  DCE 4. AEB  CED 5.ABCD 6.E is midpoint of AD 7. AE ED 8. E is midpoint of BC 9. BE EC 10.ΔAEBΔDEC s s

Find the values of x and y given that ∆MAS ≌ ∆NER.

Solution: Now we substitute 7 for x to solve for y:

Given: Prove:

Theorem 4.1 Properties of Congruent Triangles 1. Every triangle is congruent to itself 2. If ΔABCΔPQR, then ΔPQR  ΔABC 3. If ΔABCΔPQR and ΔPQRΔTUV, then ΔABCΔTUV