Question

Q12. ABCD is a rhombus. Show that diagonal AC bisects ∠ A as well as ∠ C.

OR

In given figure if AB || CD, ∠ APQ = 50° and ∠ PRD = 127°, find x and y.






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Answer 1

Given :- ABCD is a rhombus. Show that diagonal AC bisects ∠ A as well as ∠ C.

SOLUTION :- (from first figure.)

ABCD is a rhombus .

So,

→ AD = CD (All sides of rhombus are equal.)

→ ∠DAC = ∠DCA . (angle opposite to equal sides are equal.)

Also,

→ AD || BC and transversal AC intersects them.

So,

→ ∠DAC = ∠BCA (Alternate interior angles.)

therefore,

→ ∠DCA = ∠BCA

Hence,

→ AC bisects ∠C .

Similarly,

→ AC bisects ∠A.

_____________

Given :- In given figure if AB || CD, ∠ APQ = 50° and ∠ PRD = 127°, find x and y. ?

Solution :- (from second figure.)

we have,

→ AB || CD

So,

→ ∠APQ = ∠PQR (Alternate interior Angles.)

given,

→ ∠ APQ = 50°

therefore,

→ ∠PQR = ∠x = 50° .

now,

→ ∠x +∠y = 127° (sum of two interior angles is equal to exterior angle.)

→ 50 ° + ∠y = 127°

→ ∠y = 127° - 50°

→ ∠y = 77° .

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