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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
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