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:

is given that ABCD is a square and P is a point inside it such that PB = PD 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.