Question

14. In the given figure, ABCD is a square and P is
a point inside it such that PB = PD. Prove that
CPA is a straight line.

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Answer 1

Answer:

It is given that ABCD is a square and P is a point inside it such that PB = PD expain below

Step-by-step explanation:

Considering △ APD and △ APB

We know that all the sides are equal in a square

So we get DA = AB

AP is common i.e. AP = AP

According to SSS congruence criterion

△ APD ≅ △ APB

We get ∠ APD = ∠ APB (c. p. c. t)…..

(1) Considering △ CPD and △ CPB

We know that all the sides are equal in a square

So we get CD = CB CP is common

i.e. CP = CP

According to SSS congruence criterion

△ CPD ≅ △ CPB

We get ∠ CPD = ∠ CPB (c. p. c. t)…..

(2) By adding both the equation

(1) and (2) ∠ APD + ∠ CPD = ∠ APB + ∠ CPB …….

(3) From the figure we know that the angles

surrounding the point P is 360o

So we get ∠ APD + ∠ CPD + ∠ APB + ∠ CPB = 360o

By grouping we get ∠ APB + ∠ CPB = 360o – (∠ APD + ∠ CPD) ……

(4) Now by substitution of

(4) in (3) ∠ APD + ∠ CPD = 360o – (∠ APD + ∠ CPD)

On further calculation 2 (∠ APD + ∠ CPD) = 360o

By division we get ∠ APD + ∠ CPD = 180o

Therefore, it is proved that CPA is a straight line.