# In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ (see Fig. 8.20). Show that:

(i) Δ APD ≅ Δ CQB

(ii) AP = CQ

(iii) Δ AQB ≅ Δ CPD

(iv) AQ = CP

(v) APCQ is a parallelogram

**Solution:**

Given: ABCD is a parallelogram and DP = BQ

**(i)** In ΔAPD and ΔCQB,

∠ADP = ∠CBQ (Alternate interior angles for BC || AD)

AD = CB (Opposite sides of parallelogram ABCD)

DP = BQ (Given)

∴ ΔAPD ≅ ΔCQB (Using SAS congruence rule)

**(ii) **Since ΔAPD ≅ ΔCQB,

∴ AP = CQ (By CPCT)

**(iii)** In ΔAQB and ΔCPD,

AB = CD (Opposite sides of parallelogram ABCD)

∠ABQ = ∠CDP (Alternate interior angles for AB || CD)

BQ = DP (Given)

∴ ΔAQB ≅ ΔCPD (Using SAS congruence rule)

**(iv)** Since ΔAQB ≅ ΔCPD,

∴ AQ = CP (CPCT)

**(v)** From the result obtained in (ii) and (iv), AQ = CP and AP = CQ

Since opposite sides in quadrilateral APCQ are equal to each other, thus APCQ is a parallelogram.

**☛ Check: **NCERT Solutions Class 9 Maths Chapter 8

**Video Solution:**

## In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ (see Fig. 8.20). Show that: (i) ΔAPD ≅ ΔCQB (ii) AP = CQ (iii) ΔAQB ≅ ΔCPD (iv) AQ = CP (v) APCQ is a parallelogram

NCERT Maths Solutions Class 9 Chapter 8 Exercise 8.1 Question 9

**Summary:**

If in parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ, then ΔAPD ≅ ΔCQB by SAS congruence, AP = CQ, ΔAQB ≅ ΔCPD by SAS congruence, AQ = CP, and APCQ is a parallelogram.

**☛ Related Questions:**

- Show that if the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square.
- Diagonal AC of a parallelogram ABCD bisects ∠A (see the given figure). Show that i) it bisects ∠C also, ii) ABCD is a rhombus.
- ABCD is a rhombus. Show that diagonal AC bisects ∠A as well as ∠C and diagonal BD bisects ∠B as well as ∠D.
- ABCD is a rectangle in which diagonal AC bisects ∠A as well as ∠C. Show that:(i) ABCD is a square(ii) diagonal BD bisects ∠B as well as ∠D.

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