Prentice Hall Geometry

Chapter 4 Congruent Triangles > 4-5 Isosceles and Equilateral Triangles

The proof of the Isosceles Triangle Theorem requires an auxiliary line.

Proof

Proof of Theorem 4-3: Isosceles Triangle Theorem

Begin with isosceles cap delta x y z with X Y ¯ ≅ X Z ¯ . x y bar , approximately equal to . x z bar , . Draw X B ¯ , x b bar , comma the bisector of the vertex angle angle y x z .

Triangle XYZ has sides XY and XZ equal. A segment from Y meets YZ at B, forming angles 1, at YXB, and 2, at ZYB.

Given: X Y ¯ ≅ X Z ¯ , X B ¯ x y bar , approximately equal to . x z bar , comma . x b bar bisects angle y x z

Prove: angle y approximately equal to . angle z

Proof: X Y ¯ ≅ X Z ¯ x y bar , approximately equal to . x z bar is given. By the definition of angle bisector, ∠ 1 ≅ ∠ 2 . angle 1 , approximately equal to , angle 2 . By the Reflexive Property of Congruence, X B ¯ ≅ X B ¯ . x b bar , approximately equal to . x b bar , . So by the SAS Postulate, Δ X Y B ≅ Δ X Z B . ∠ Y ≅ ∠ Z cap delta x y b approximately equal to cap delta x z b . , angle y approximately equal to . angle z since corresponding parts of congruent triangles are congruent.

Take note Theorem 4-4: Converse of the Isosceles Triangle Theorem

Theorem
If two angles of a triangle are congruent, then the sides opposite those angles are congruent.

If …
angle eh approximately equal to angle b Triangle ABC has angles A and B equal.

Then …
A C ¯ ≅ B C ¯ eh c bar , approximately equal to . b c bar Triangle ABC has angles A and B equal, and sides AC and BC equal.
You will prove Theorem 4-4 in Exercise 23.

Problem 1 Using the Isosceles Triangle Theorems

Is A B ¯ eh b bar congruent to C B ¯ ? c b bar , question mark Explain.
Yes. Since ∠ C ≅ ∠ A , A B ¯ ≅ C B ¯ angle c approximately equal to angle eh comma . eh b bar , approximately equal to . c b bar by the Converse of the Isosceles Triangle Theorem.

Is angle eh congruent to angle d e eh question mark Explain.
Yes. Since A D ¯ ≅ E D ¯ , ∠ A ≅ ∠ D E A eh d bar , approximately equal to . e d bar , comma . angle eh approximately equal to , angle d e eh by the Isosceles Triangle Theorem.

Triangle ABC, with angles A and C equal, has segment DE extending from AB to AC, forming triangle ADE with sides AD and ED equal.

✓ Got It?

1. Is angle w v s congruent to angle s question mark Is T R ¯ t r bar congruent to T S ¯ ? t s bar , question mark Explain.
2. Reasoning Can you conclude that cap delta r u v is isosceles? Explain.

Triangle RST has segments from U on RT and W on ST meeting at V on RS. Angles R and WVS are equal, and sides WV and WS are equal.

An isosceles triangle has a certain type of symmetry about a line through its vertex angle. The theorems in this lesson suggest this symmetry, which you will study in greater detail in Lesson 9-4.

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Proof

Take note Theorem 4-4: Converse of the Isosceles Triangle Theorem

Problem 1 Using the Isosceles Triangle Theorems

Think

✓ Got It?

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