Question

in the figure
determine P+Q+R+S+T

Answer 1

∠P+∠Q+∠R+∠S+∠T =180°

Step-by-step explanation:

1. We know the sum of interior angle of pentagonal= (5-2)×180°=540°

  Means

  ∠U+∠V+∠W+∠X+∠Y =540°              ...1)

2. Apply sum of triangle angle property in following triangle

   In ΔPRX

   ∠P+∠R+∠X =180°                             ...2)

   In ΔQSY

   ∠Q+∠S+∠Y =180°                             ...3)

   In ΔRTU

   ∠R+∠T+∠U =180°                             ...4)

   In ΔSPV

   ∠S+∠P+∠V =180°                             ...5)

   In ΔQTW

   ∠Q+∠T+∠W =180°                            ...6)

3. Adding equation from equation 2) to equation 6)

   2∠P+2∠Q+2∠R+2∠S+2∠T+∠U+∠V+∠W+∠X+∠Y =180°×5

  2∠P+2∠Q+2∠R+2∠S+2∠T+∠U+∠V+∠W+∠X+∠Y =900°       ...7)

   

   From equation 1) and equation 7)

   2∠P+2∠Q+2∠R+2∠S+2∠T+540° =900°

   So

   2∠P+2∠Q+2∠R+2∠S+2∠T =900°-540°

   

  ⇒2∠P+2∠Q+2∠R+2∠S+2∠T =360°

      So

  ⇒2(∠P+∠Q+∠R+∠S+∠T ) =360°

  ⇒∠P+∠Q+∠R+∠S+∠T =180°