Q. 64.0( 17 Votes )
Prove that, if the bisector of ∠BAC of ΔABC is perpendicular to side BC, then ΔABC is an isosceles triangle.
Answer :
Given: Bisector of ∠BAC of ΔABC is perpendicular to side BC
To Prove: ΔABC is an isosceles triangle.
Proof:
In ΔABD and ΔACD
Since, AD is the angle Bisector of ΔABC
∴ ∠BAD = ∠CAD
AD = AD ……….[Common Side]
∠ADB = ∠ADC ……[Both equal to 90°]
So, by ASA congruency test
ΔABD ≅ ΔACD
Therefore,
AB = AC ………………. corresponding sides of congruent triangles.
∠ABD = ∠ACD ……………… corresponding angles of congruent triangles.
∴ ∠ABC = ∠ACB
Since, AB = AC and ∠ABC = ∠ACB so, ΔABC is an isosceles triangle.
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