Isosceles Triangle Theorems
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An isosceles triangle is a triangle
with two congruent sides.


Theorem:

If two sides of a triangle are congruent, the angles opposite them are congruent.

 

Theorem:
(converse)

If two angles of a triangle are congruent, the sides opposite them are congruent. 

 

More Info:

When the altitude is drawn in an isosceles triangle, two congruent triangles are formed, proven by Hypotenuse-Leg.

(The congruent legs of the isosceles triangle become the congruent hypotenuses and the altitude becomes a shared leg.)


These congruent triangles make it possible, by use of CPCTC, to conclude that the following statements are true regarding an isosceles triangle:

1.  The altitude to the base of an isosceles triangle bisects the vertex angle.


 

 

2.  The altitude to the base of an isosceles triangle bisects the base.
 

 

 

Examples:

1.

Find x.

 

Solution:

If two angles of a triangle are congruent, the sides opposite them are congruent.
Set: 
6x - 8 = 4x + 2
              2x = 10
                x = 5

Note:  The side labeled 2x + 2 is a distracter and is not used in finding x.

2.

Find the measures of angles 1, 2, 3, 4.
    

Solution:

If two sides of a triangle are congruent, the angles opposite them are congruent.
So m<1 = m<2       and       m<3 = 40 degrees.
180 - 50 = 130                   180 - (40 + 40) = 100
m<1 = 65 degrees               m <4 = 100 degrees
m<2 = 65 degrees 
              

3.


Solution:
 
Statements Reasons
1.  1. Given
2. 2. An isosceles triangle has two congruent sides.
3. 3. If two sides of a triangle are congruent, the angles opposite them are congruent.
4. <1 supp <2
<3 supp <4
4. If two angles form a linear pair, they are supplementary.
5. 5. Supplements of the same angle, or congruent angles, are congruent.
6. 6. SAS  If two sides and the included angle of one triangle are congruent to the corresponding parts of a second triangle, the triangles are congruent.