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CBSE Class 9 Mathematics Question 6 of 13

Quadrilaterals — Question 6

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Question 6

ABCD is a parallelogram and AP and CQ are perpendiculars from vertices A and C on diagonal BD. Show that

(i) Δ APB ≅ Δ CQD

(ii) AP = CQ

ABCD is a parallelogram and AP and CQ are perpendiculars from vertices A and C on diagonal BD. NCERT Class 9 Mathematics CBSE Solutions.
Answer

Given :

ABCD is a parallelogram with AP ⊥ BD and CQ ⊥ BD.

From figure,

AB || DC and BD is transversal.

(i) In Δ APB and Δ CQD,

⇒ ∠APB = ∠CQD (Both equal to 90°)

⇒ AB = CD (Opposite sides of parallelogram are equal)

⇒ ∠ABP = ∠CDQ (Alternate interior angles are equal)

∴ Δ APB ≅ Δ CQD (By A.A.S. congruence rule)

Hence, proved that Δ APB ≅ Δ CQD.

(ii) Since,

Δ APB ≅ Δ CQD

We know that,

Corresponding parts of congruent triangles are equal.

⇒ AP = CQ (By C.P.C.T.)

Hence, proved that AP = CQ.