Question

Prove that angles opposite to equal sides of triangles are equal.

Answer 1

Hint: To prove such type of questions, we will use isosceles theorem which states that if there are two triangles which have one side common and one side and one angle is equal to each other then their opposite angle will also be equal.

Complete step-by-step answer:
Let ABC is a given triangle with, AB=AC
To prove: angle opposite to AB = angle opposite to AC
Construction: Draw AD perpendicular to BC
$\therefore \mathrm{\angle }ADB=\mathrm{\angle }ADC={90}^0$
Proof:
Consider $\mathrm{\Delta }ABD$and $\mathrm{\Delta }ACD$
$AD$ is common
$$\begin{gathered} AB=AC \\ \therefore \mathrm{\angle }ADB=\mathrm{\angle }ADC={90}^0 \end{gathered}$$
Hence, $\mathrm{\angle }ABD=\mathrm{\angle }ACD$
$\mathrm{\angle }ABC=\mathrm{\angle }ACB$
$\mathrm{\angle }B=\mathrm{\angle }C$
Hence, angle opposite to AB = angle opposite to AC.
This is known as isosceles triangle theorem.

Note: The isosceles triangle theorem states that if two sides of a triangle are congruent, then angles opposite those sides are congruent and if angles opposite those sides are congruent, then two sides of a triangle are equal. To solve these questions you must be familiar with all the theorems and congruence rules such as AAA, SAS, AAS, SSS. When two triangles are congruent means that their area, angle and sides are equal.