Question

If a straight line drawn from the vertex of a triangle, parallel to the base of triangle bisect the exterior vertical angle, then the triangle is:

• A. equilateral
• B. right-angled
• C. obtuse-angled
• D. isosceles

Answer 1

Answer: (A)

equilateral

In $\Delta ABC,AEistheanglebisectorof\angle DAC.⟹\angle DAE=\angle EAC$
$\angle DAE$ is the exterior angle for triangle ABC extending AB.
Given, $AE\parallel BC⟹\angle EAC=\angle ACB\left[alternateangles\right]⟹\angle DAE=\angle ABC\left[correspondingangles\right]\therefore \angle ABC=\angle ACB=x\left(asshowninfigure\right)$
Similarly, extending AC to F and CG being the angle bisector of $\angle BCF$
$\angle BCF$ is the exterior angle for triangle ABC extending along AC.
Given, $AB\parallel CG⟹\angle GCB=\angle CBA\left[alternateangles\right]⟹\angle FCG=\angle CAB\left[correspondingangles\right]\therefore \angle CBA=\angle BAC=x\left(asshowninfigure\right)$
Hence the triangle is equilateral.