Question 145
In rectangle PAIR, find $∠ A R I, ∠ R M I$ and $∠ P M A$ .

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Solution

Given, $∠ R A I = 35^{\circ}$ .
$∴ ∠ P R A = 35^{\circ}$ [Corresponding angles as $P R ∥ A I$ and AR is transversal]
$\Rightarrow ∠ A R I = 90^{\circ} - ∠ P R A = 90^{\circ} - 35^{\circ} = 55^{\circ}$ .
$∵ R M = I M, ∠ M R I = ∠ M I R = 55^{\circ}$ .

In $Δ R M I,$
$∠ R M I + ∠ M R I + ∠ M I R = 180^{\circ}$ . [Angle sum property]
$\Rightarrow ∠ R M I = 180^{\circ} - 55^{\circ} - 55^{\circ} = 180^{\circ} - 110^{\circ} = 70^{\circ}$
Also, $∠ P M A = ∠ R M I$ [vertically opposite angles]
$\Rightarrow ∠ P M A = 70^{\circ}$

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Question 145 In rectangle PAIR, find ∠ARI,∠RMI and ∠PMA.

Question 145
In rectangle PAIR, find $∠ A R I, ∠ R M I$ and $∠ P M A$ .