Question

Suppose ABC is an equiangular triangle.. Prove that it is equilateral. (You have seen earlier that an equilateral triangle is equiangular. Thus for triangles equiangularity is equivalent to equilaterality.)

Answer 1

Equiangular triangle is a triangle where all angles are equal

$\angle P=\angle Q=\angle R$

construction: Draw a triangle

$PQR$ and draw a perpendicular $PT$ where

$\angle PTQ=\angle PTR={90}^{\circ }$

$\therefore PT=PT$

From triangle $PQT$ and triangle $PTR$, we have

$PT=PT$ (same side)

$\angle PQT=\angle PRT$ (by definition)

$\angle PTQ=\angle PTR={90}^{\circ }$ (by construction)

$\therefore$ one side and two angles are equal.

Hence by $ASA$ congruency rule

$△PQT=△PTR$

$\therefore$ by defintion of congruency, we have that congruent triangles are triangles having corresponding sides and angles to

be equal

Hence $PQ=PR=QR$