Question

prove that the bisector of the vertical angle of an isosecles triangles bisects the base at right angles

Answer 1

Hi there!

Consider PQR is an  isosceles triangle such that PQ = PR and Pl is the bisector of ∠ P.

To prove : ∠PLQ = ∠PLR = 90°

and QL = LX

In ΔPLQ and ΔPLR

PQ = PR (Given)

PL = PL (Common)

∠QPL = ∠RPL ( PL is the bisector of ∠P)

ΔPLQ = ΔPLR ( SAS congruence criterion)

QL = LR (by c.p.c.t)

and ∠PLQ + ∠PLR = 180° (Linear pair)

2∠PLQ = 180°

∠PLQ = 180° / 2 = 90° 

∴ ∠PLQ = ∠PLR = 90°

Thus, ∠PLQ = ∠PLR = 90° and QL = LR.

Hence,
The bisector of the verticle angle an isosceles triangle bisects the base at right angle.

Answer 2

This two triangles are congruent by ASA congruence
So, BCA = DCA. (CPCT)

But they are linear pair, so,
BCA + DCA = 180°
2BCA = 180°
BCA = 90°

BCA = DCA = 90°
Hence, proved

Hope this helps!!!!
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