Base angles theorem 

The base angles theorem states that if the sides of a triangle are congruent (Isosceles triangle)then the angles opposite these sides are congruent.

Start with the following isosceles triangle. The two equal sides are shown with one red mark and the angles opposites to these sides are also shown in red.

Base angle


The strategy is to draw the perpendicular bisector from vertex C to segment AB

Then use SAS postulate to show that the two triangles formed are congruent

If the two triangles are congruent, then corresponding angles to will be congruent

Draw the perpendicular bisector from C

Perpendicular bisector
Since angle C is bisected,

angle (x) = angle (y)

Segment AC = segment BC ( This one was given)

Segment CF = segment CF (Common side is the same for both triangle ACF and triangle BCF)

Triangles ACF and triangle BCF are then congruent by SAS or side-angle-side

In other words, by

AC-angle(x)-CF

and

BC-angle(y)-CF

Since triange ACF and triangle BCF are congruent, angle A = angle B


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