Lesson 4.3Lesson 4.3Exploring Congruent TrianglesExploring 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 PQBC QRCA RP

Congruent Triangles

Example 1Naming Congruent

Parts• You and a friend have identical drafting

triangles, as shown below. Name all congruent parts.

Example 1Naming Congruent

Parts

Corresponding angles are: D R E S F T

Corresponding sides are:DE RSEF STFD 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

Vocabularyfor 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 2Proving 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 ΔAEBΔDEC.

• Statements

1. AB||CD2. EAB EDC3. ABE DCE4. AEB CED5.ABCD6.E is midpoint of AD7. AE ED8. E is midpoint of BC9. BE EC10.ΔAEBΔDEC

• Reasons 1. Given2. 2 lines || alt. int. are 3.2 lines || alt. int. are 4.Vertical angles are 5.Given 6.Given7. Def. of midpoint8. Given9. Def. of Midpoint10.Def. of Congruent Triangles

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 4.1Properties of Congruent

Triangles

• 1. Every triangle is congruent to itself

• 2. If ΔABCΔPQR, then ΔPQR ΔABC

• 3. If ΔABCΔPQR and ΔPQRΔTUV, then ΔABCΔTUV