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