Lesson 7.2

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Isosceles  Triangles  Fundamentals

• There  are  two  congruent  sides  in  an  isosceles  triangle  and  two  congruent  angles.  

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Vertex Angle

Base

Base Angles

Legs

Equilateral  Triangle  Fundamentals

• All  of  the  sides  and  angles  are  congruent.  • The  angles  in  an  equilateral  triangle  always  equal  60o.  

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�  If  two  sides  of  a  triangle  are  congruent,  then  the  base  angles  are  also  congruent.  

� Base  angles  are  the  angles  at  the  ends  of  the  2  congruent  segments.  

�  So,  in  the  diagram  angles  B  and  C  are  congruent.  

Base angle Base angle

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Isosceles  Triangle  Theorem  (AKA  Base  Angles  Theorem)  

• If  two  angles  in  a  triangle  are  congruent,  then  the  triangle  is  an  isosceles  triangle.  

• That  means  that  the  2  sides  of  the  triangles  are  also  congruent.  

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Converse  to  the  Isosceles  Triangle  Theorem  

(AKA  Base  Angles  Theorem)  

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Example  1:  Find  the  value  of  x.

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This is an isosceles triangle, so the 2 sides are congruent…

1.  

2.  3.  

Example  2:  Find  the  value  of  x  and  y.

5x + 5 = 35 5x = 30 x = 6

x + x + 102 = 180 2x + 102 = 180

2x = 78 x = 39 y = 39

55 + 55 + y = 180 110 + y = 180

y = 70 x + 7 = 55

x = 48

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Example  3:  Find  the  value  of  x.

x + x + 3x = 180 5x = 180

x = 36

x°  

x  

∠𝑇=3𝑥 ∠𝑇=3(36)

∠𝑇=108

Theorem

Converse

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Equilateral  Triangle  Theorem  Equiangular  Triangle  Theorem  

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C Isosceles Triangle Theorem

AC ≅ BC

Substitution

Transitive

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AC

Converse of the Isosceles Triangle Theorem

AB ≅ BC

∠A

Example 3A: Using Properties of Equilateral Triangles

Find the value of x.

∆LKM is equilateral.

(2x + 32)° = 60° The measure of each ∠ of an equiangular ∆ is 60°.

2x = 28 Subtract 32 both sides.

x = 14 Divide both sides by 2.

Equilateral ∆ à equiangular ∆

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Example 3B: Using Properties of Equilateral Triangles

Find the value of y. ∆NPO is equiangular. Equiangular ∆ à equilateral ∆

5y – 6 = 4y + 12 Definition of equilateral ∆.

y = 18 Subtract 4y and add 6 to both sides.

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