In the figure, it is given that $RT = TS, \angle 1 = 2\angle 2$ and $\angle 4 = 2\angle 3$. Prove that: $\triangle RBT = \triangle SAT$.
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Given:

$RT = TS, \angle 1 = 2\angle 2$ and $\angle 4 = 2\angle 3$.

To do:

We have to prove that $\triangle RBT = \triangle SAT$.

Solution:

$\angle 1 = \angle 4$                (Vertically opposite angles are equal)

$\angle 1 = 2\angle 2$ and $\angle 4 = 2\angle 3$

This implies,

$2\angle 2 = 2\angle 3$

$\angle 2 = \angle 3$

$RT = ST$

This implies,

$\angle R = \angle S$                 (Angles opposite to equal sides are equal)

$\angle R - \angle 2 = \angle S - \angle 3$

$\angle TRB = \angle TSA$

In $\triangle RBT$ and $\triangle SAT$,

$\angle TRB = \angle SAT$

$RT = ST$

$\angle T = \angle T$

Therefore, by SAS axiom,

$\triangle RBT \cong \triangle SAT$

Hence proved.

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Updated on: 10-Oct-2022

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