In the given figure, QTSR is a parallelogram. PQR is a straight line and QU = UT. Which of the following statements are correct?

△ P Q U ≅ △ S T U

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Q U = U S

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P Q = Q R

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P U = U S

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Solution

The correct options are
A $△ P Q U ≅ △ S T U$
C $P Q = Q R$
D $P U = U S$
Given, QTSR is a parallelogram
$\Rightarrow$ QR = TS and QT = RS

Consider $△ P Q U and △ S T U$

$∠ P U Q = ∠ S U T$ ..... (vertically opposite angles)

$Q U = U T$ .... (given)

$∠ P Q U = ∠ S T U$ ....... (alternate interior angles)

$\Rightarrow △ P Q U ≅ △ S T U$ .....( ASA criterion)

PU = SU ...... (corresponding parts of congruent triangles)

PQ = TS ...... (corresponding parts of congruent triangles)

TS=QR .........(opposite sides of a parallelogram)

$\Rightarrow P Q = T S = Q R$

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In the given figure PQ|| RS, TR|| QU and $∠ P T R = 42^{\circ}$ . Find $∠ Q U R .$

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