ABCD is a parallelogram and AP and CQ are perpendiculars from vertices A and C on diagonal BD (see Fig. 8.21). Show that (i) ΔAPB ≅ ΔCQD  (ii) AP = CQ

Solution:

Given: ABCD is a parallelogram and AP ⊥ DB, CQ ⊥ DB

(i) In ΔAPB and ΔCQD,

∠APB = ∠CQD (Each 90°)

AB = CD (Opposite sides of parallelogram ABCD)

∠ABP = ∠CDQ (Alternate interior angles as AB || CD)

∴ ΔAPB ≅ ΔCQD (By AAS congruency)

(ii) By using the result ΔAPB ≅ ΔCQD., we obtain AP = CQ (By CPCT)

☛ Check: NCERT Solutions Class 9 Maths Chapter 8

Video Solution:

ABCD is a parallelogram and AP and CQ are perpendiculars from vertices A and C on diagonal BD (see Fig. 8.21). Show that (i) ΔAPB ≅ ΔCQD (ii) AP = CQ

NCERT Maths Solutions Class 9 Chapter 8 Exercise 8.1 Question 10

Summary:

If ABCD is a parallelogram where AP and CQ are perpendiculars from vertices A and C on BD, then ΔAPB ≅ ΔCQD using AAS congruency and AP = CQ.

☛ Related Questions:

• 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.
• 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