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Finding Unknown Angles

In figure, if...

Question

In figure, if $A B ∥ C D$ , then find the value of $∠ C H F$

188^{o}

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178^{o}

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118^{o}

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108^{o}

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Solution

The correct option is D $108^{o}$
$I n f i g u r e A B ∥ C D ∠ P R B = 2 x (G i v e n) A s l i n e s A B a n d P Q i n t e r s e c t a t R ∠ P R B = ∠ G R S ⟹ ∠ G R S = 2 x A l s o, A B ∥ C D ∠ G R S + ∠ H S R = (s u m o f i n t e r i o r a n g l e i s s u p p l e m e n t a r y ∠ H S R = 3 x (G i v e n) ⟹ 2 x + 3 x = 180 ⟹ 5 x = \frac{180}{5} ⟹ x = 36 ∠ E G A = 2 x (G i v e n) ∠ E G A + ∠ E G R = 180 ⟹ 2 x + ∠ E G R = 180 ⟹ ∠ E G R = 180 - 2 x A l s o ∠ G H S = ∠ E G R (C o r r o s p o n d i n g a n g l e s a r e e q u a l) ⟹ ∠ G H S = 180 - 2 x P u t t i n g v a l u e o f x = 36 ∠ G H S = 180 - 2 \times 36 = 180 - 72 = 108 ⟹ ∠ G H S = 108 N o w ∠ G H S = ∠ C H F (V e r t i c a l l y o p p o s i t e a n g l e s) H e n c e ∠ C H F = 108$

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