10. Prove that if the line bisecting the vertical angle of a triangle is perpendicular to the base, the triangle is isosceles.
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Answer:
Given: A AABC in which the bisector of the
vertical angle <BAC bisects the base BC, i.e., BD
= CD
To Prove: AABC is isosceles
Construction: Produce AD to E such that AD =
DE. Join EC.
Proof: In AADB and AEDC,
BD = CD | Given
AD = ED | By construction
ZADB = LEDC
Vertically opposite angles
:: AADB = AEDC
SAS congruence rule AB = EC ... (1) | CPCT
and ZBAD = <CED | CPCT
But <BAD = <CAD | Given
:: ZCAD = ZCED
AC = CE...(2)
Sides opposite to equal angles of a triangle are
equal
From (1) and (2),
AB = AC
:: AABC is isosceles.
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