The sides AB and CD of a quadrilateral ABCD are extended to points P and Q respectively. Is ∠ADQ + ∠CBP = ∠A + ∠C? Give reasons.
Solution:
Given, ABCD is a quadrilateral.
The sides AB and CD are extended to points P and Q.
We have to determine if ∠ADQ + ∠CBP = ∠A + ∠C
On joining A and C,
By exterior angle property of a triangle,
If a side of a triangle is produced, then the exterior angle so formed is equal to the sum of the two interior opposite angles.
∠CBP = ∠BCA + ∠BAC ------------------- (1)
∠ADQ = ∠ACD + ∠DAC ------------------- (2)
Adding (1) and (2), we get
∠CBP + ∠ADQ = ∠BCA + ∠BAC + ∠ACD + ∠DAC
= (∠BCA + ∠ACD) + (∠BAC + ∠DAC)
From the figure,
∠BCA + ∠ACD = ∠C
∠BAC + ∠DAC = ∠A
Therefore, ∠CBP + ∠ADQ = ∠C + ∠A
✦ Try This: In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP=BQ. Show that AQ = CP.
☛ Also Check: NCERT Solutions for Class 8 Maths
NCERT Exemplar Class 8 Maths Chapter 5 Solved Problem 24
The sides AB and CD of a quadrilateral ABCD are extended to points P and Q respectively. Is ∠ADQ + ∠CBP = ∠A + ∠C? Give reasons.
Summary:
The sides AB and CD of a quadrilateral ABCD are extended to points P and Q respectively. It is shown that ∠ADQ + ∠CBP = ∠A + ∠C by exterior angle property of a triangle.
☛ Related Questions: