Question

show that the bisector of the vertical angle of an isoceles triangle bisects the base at right angle

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Answer 1

$\textbf{\huge{ANSWER:}}$

Refer to the figure I have attached

$\sf{Given:}$

Triangle ABC is isosceles, in which, AB = AC

AD bisects Angle A

In Triangles ABD and ACD:

Angle B = Angle C (Angles opposite to equal sides of an isosceles triangle)

Angle BAD = Angle DAC (AD bisects Angle A)

AB = AC (Triangle ABC is isosceles, Given)

Thus, Triangle ABD is congruent to Triangle ACD by ASA criterion of congruence

Therefore,

=》 BD = DC (C.P.C.T.)

=》 Angle ADB = Angle ADC (C.P.C.T.)

=》Angle ADB + Angle ADC = 180° (Linear Pair, BC is a straigt line)

=》2 (Angle ADB) = 180°

=》 Angle ADB = $\frac{180}{2}$

=》 Angle ADB = 90°

Thus, AD bisects the base angle at right angle.

$\textbf{Hence Proved!}$

Hope it Helps!! :)