Question

In the given figure, prove that ∆P + ∆Q + ∆R + ∆S + ∆T = 2 x right angles.​

Answer 1

Answer:

Sum of angles on a straight line=180

∘

⟹∠PAE=180

∘

−∠1

Similarly,∠PEA=180

∘

−∠5

In△PAE

∠PAE+∠PEA+∠APE=180

∘

⟹180−∠1+180−∠5+∠APE=180

∘

⟹∠APE=∠1+∠5−180

∘

=∠P

Similarly,∠BSC=∠2+∠3−180

∘

=∠S

∠DRC=∠3+∠4−180

∘

=∠R

∠DQE=∠4+∠∠5−180

∘

=∠Q

∠ATS=∠1+∠2−180

∘

=∠T

⟹∠P+∠Q+∠R+∠S+∠T

=∠1+∠2+∠3+∠4+∠2+∠3+∠4+∠5+∠1+∠5−(180×5)

=2(∠1+∠2+∠3+∠4+∠5)−900−(1)

From pentagon ABCDE,

Sum of angles of pentagon=((2×5)−4)×90

∘

(∵ Sum of angles=(2n−4)×90

∘

)

=6×90=540

∘

⟹∠P+∠Q+∠R+∠S+∠T=2(540

∘

)−900

∘

=1080

∘

−900

∘

=180