Question

In the given figure, prove that

∠P +∠Q+∠R+∠S+∠T= 2 right angles.​

Answer 1

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 ∘

=2×90

=2Rightangles

Answer 2

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