In figure- if - 1- 2 and - NSQ - MTR- then prove that - PTS - PRQ

Given, Δ NSQ≅Δ MTR and ∠ 1=∠ 2 To prove Δ PTS∼Δ PRQ Proof since, Δ NSQ≅Δ MTR So, SQ = TR ….(i) Also, ∠ 1=∠ 2⇒ PT=PS ……(ii) [ since, sides opposite to equ ...

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Standard VIII

Mathematics

Construction of Parallel Lines

In figure, if...

Question

In figure, if $∠ 1 = ∠ 2$ and $Δ N S Q ≅ Δ M T R$ , then prove that $Δ P T S \sim Δ P R Q$

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Solution

Given, $Δ N S Q ≅ Δ M T R a n d ∠ 1 = ∠ 2$
To prove $Δ P T S \sim Δ P R Q$
Proof since, $Δ N S Q ≅ Δ M T R$
So, SQ = TR ….(i)
Also, $∠ 1 = ∠ 2 \Rightarrow P T = P S$ ……(ii)
[ since, sides opposite to equal angles are also equal]
From Eqs. (i) and (ii), $\frac{P S}{S Q} = \frac{P T}{T R}$
$\Rightarrow S T ∥ Q R$ [by convenes of basic proportionally theoreum]
$∴ ∠ 1 = ∠ P Q R$
And $∠ 2 = ∠ P R Q$
In $Δ P T S a n d Δ P R Q$ , [common angles]
$∠ P = ∠ P$
$∠ 1 = ∠ P Q R$
$∠ 2 = ∠ P R Q$
$∴ Δ P T S \sim Δ P R Q$ [by AAA similarity criterion]

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In figure, if ∠1=∠2 and ΔNSQ≅ΔMTR, then prove that ΔPTS∼ΔPRQ

In figure, if $∠ 1 = ∠ 2$ and $Δ N S Q ≅ Δ M T R$ , then prove that $Δ P T S \sim Δ P R Q$