Question

Q5. The bisectors of the angles E and F of an isosceles triangle ∆ DEF with DE = DF (3m) intersect each other at O. EO is produced to meet DF at a point M. Prove that ∠ MOF = ∠DEF.

Answer 1

$\underline{\blue{ Given :}}$

$\triangle DEF \: is \: an \: isosceles \: triangle.\\DE = DF = 3 \:cm ,\\and \: angle \: bisectors \: of \\\angle E \:and \:\angle F \: intersect \: each \:other\\at \:O. \\EO \: is \: produced \: to \:meet \:DF \:at\:M.$

$\underline{\blue{ To \: prove :}}$

$\red{ \angle {MOF } = \angle {DEF}}$

$\underline{\blue{ Proof :}}$

$In \: \triangle {DEF} \: DE = DF$

$\implies \angle {DEF} = \angle {DFE} = 2x$

$\blue{ ( Angles \: opposite \:to \: equal \: angles) }$

$\implies \frac{\angle {DEF}}{2} = \frac{\angle {DFE}}{2} = \frac{2x}{2} = x\: --(1)$

$In \: figure \:(2),$

$In \:\triangle OEF ,$

$\red{\angle {MOF}} = \angle {OEF} + \angle {OFE}$

$\blue{( External \:angle \: at \: \angle {MOF}}\\\blue{ sum \: of \: interior \: opposite \: angles )}$

$= x + x \\= 2x \\\green {= \angle {DEF} }\: [From \:(1) ]$

$Hence \:Proved$

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