Question

Given \overline{RS}\perp\overline{AB}, \overline{CB}\perp\overline{AC}. Prove \triangle BSR\sim \triangle BCA

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asked 2021-08-14

Given: \(\displaystyle\overline{{{R}{S}}}\perp\overline{{{A}{B}}},\overline{{{C}{B}}}\perp\overline{{{A}{C}}}\)
Prove: \(\displaystyle\triangle{B}{S}{R}\sim\triangle{B}{C}{A}\)
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Answers (1)

2021-08-15

Now,
\(\displaystyle\overline{{{R}{S}}}\perp\overline{{{A}{B}}}\Rightarrow{m}\angle{R}{S}{B}={90}^{{\circ}}\)
And
\(\displaystyle\overline{{{C}{B}}}\perp\overline{{{A}{C}}}\Rightarrow{m}\angle{A}{C}{B}={90}^{{\circ}}\)
To prove: \(\displaystyle\triangle{B}{S}{R}\sim\triangle{B}{C}{A}\)
In \(\displaystyle\triangle{B}{S}{R}\ {\quad\text{and}\quad}\ \triangle{B}{C}{A}\)
\(\displaystyle{m}\angle{R}{S}{B}={m}\angle{A}{C}{B}={90}^{{\circ}}\)
\(\displaystyle{m}\angle{R}{B}{S}={m}\angle{A}{B}{C}\) = common angle
\(\displaystyle\therefore\triangle{B}{S}{R}\sim\triangle{B}{C}{A}\) by Angle−Angle similarity theorem
(Angle−Angle similarity theorem:If two angles of one triangle are equal to two angles of another triangle,then the triangles are similar)
Hence, proved.

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