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**Properties of Isosceles**

Triangles

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**A triangle with at least two sides congruent is called an Isosceles Triangle.**

In this triangle, sides b and c are congruent. b c a

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**The angles B and C are called the Base Angles of the triangle. **

Angle A is called the Vertex Angle A Vertex Angle b c Base Angles B C a

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A perpendicular segment from angle A to side a is called an altitude of the triangle. The altitude bisects side a. A b c B C a

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**The two triangles created by the altitude are congruent triangles…why??**

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Side-side-side b c a A B C

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Because the altitude divides the triangle into 2 congruent triangles, by the property of CPCTC, angles B and C are congruent. b c A B C a

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Using this property, you can solve many different problems dealing with isosceles triangles…here are 3 examples. 1. The vertex angle B of isosceles triangle ABC is 120 degrees. Find the degree measure of each base angle.

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**2. In isosceles triangle RST, angle S is the vertex angle**

2. In isosceles triangle RST, angle S is the vertex angle. Base angles R and T both measure 64 degrees. Find the degree measure of the vertex angle S.

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3. The degree measure of a base angle of isosceles triangle XYZ exceeds three times the degrees measure of the vertex Y by 60. Find the degree measure of the vertex angle Y.

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**Example 1: The vertex angle B of isosceles triangle ABC is 120 degrees**

Example 1: The vertex angle B of isosceles triangle ABC is 120 degrees. Find the degree measure of each base angle. Solution: (1) Let x = the measure of each base angle. (2) Set up an equation and solve for x. x + x degrees = 180 degrees 2x = 180 2x = x = 60 x = 60/2 x = 30 Each base angle of triangle ABC measures 30 degrees.

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**Example 2: In isosceles triangle RST, angle S is the vertex angle**

Example 2: In isosceles triangle RST, angle S is the vertex angle. Base angles R and T both measure 64 degrees. Find the degree measure of the vertex angle S. Solution: (1) Let x = measure of vertex angle S. (2) Set up an equation and solve for x. 64 degrees + 64 degrees + x = 180 degrees x = 180 x = x = 52 The measure of vertex angle S in triangle RST is 52 degrees.

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Example 3: The degree measure of a base angle of isosceles triangle XYZ exceeds three times the degrees measure of the vertex Y by 60. Find the degree measure of the vertex angle Y. We need to make an equation out of this problem, so let's figure out what it's trying to tell us. First we read "The degree measure of a base angle", so let's start with X= Now we see "exceeds three times... Y... by 60", which means 3Y + 60. Our equation now: X = 3Y + 60

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Since we know that X = Z because it is an isosceles triangle, then we can solve for the measures of all the angles. X + Y + Z = 180 (3Y + 60) + Y+ (3Y + 60) = Y = 180 7Y = 60 Y = 60/7 Y = 8.57 degrees The vertex angle Y of triangle XYZ equals 8.57 degrees.

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**That’s all for now. Remember these facts about Isosceles Triangles:**

An Isosceles Triangle has two (or more) congruent sides. The Base Angles of an Isosceles Triangle are congruent. The Altitude from the vertex angle of an Isosceles Triangle bisects the opposite side.

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