Please prove?

enter image source here

AB is the diameter of the circle.
DE is perpendicular on AB.
Arc AC=arc DB
To prove angle ABC=angle BDE

1 Answer
Apr 14, 2018

enter image source here

Given: AB is the diameter, arc AC= arcDB, DE_|_AB

RTP: angleABC=angleBDE

Construction: . A,D are joined.

Now in DeltaABD and DeltaBED

angle ADB=angle BED=90^@ as angle ADB is semicircular and DE_|_AB is drawn. angle DBE=angle ABD common

Hence remaining color(red)(angle BAD=angle BDE......[1])

So DeltaABD and DeltaBED are SIMILAR

Again in DeltaABC and DeltaABD

AC=BD as it is given arc AC= arc DB

angleACB and angleABD =90^ @ as they are semicircular angles.
and AB hypotenuse is common.

So DeltaABC~= DeltaABD

Hence color(red)(angle BAD=angle ABC......[2])

Combining [1] and [2] we get color(magenta)(angleABC=angleBDE)