The side EF, FD and DE of a triangle DEF are produced in order forming three exterior angles DFP, EDQ and FER respectively. Prove that ∠ DFP+∠ EDQ+∠ FER=360&#x00b0

Question

The side EF, FD and DE of a triangle DEF are produced in order forming three exterior angles DFP, EDQ and FER respectively. Prove that &#x2220; DFP+&#x2220; EDQ+&#x2220; FER=360&#x00b0

Philoid · Accepted Answer

Given: DEF is a triangle . The sides EF, FD and DE are produced to P, Q, and R respectively.
To prove: ∠ DFP + ∠ EDQ + ∠ FER = 360° Proof:
∠ DFP = ∠ D + ∠ E………………….(i) [Exterior angle is equal to the sum of the interior opposite angles]
∠ EDQ = ∠ E +∠ F…….…………(ii) [Exterior angle is equal to the sum of the interior opposite angles]
∠ FER = ∠ F + ∠ D ………………….(iii) [Exterior angle is equal to the sum of the interior opposite angles]
Adding (i), (ii) and (iii),
∠ DFP + ∠ EDQ + ∠ FER = (∠ D + ∠ E) + (∠ E + ∠ F) + (∠ F + ∠ D)
= 2(∠ D + ∠ E + ∠ F)
= 2 × 180° [Sum of the three angles of a triangle is 180° .]
= 360°
∴∠ DFP + ∠ EDQ + ∠ FER = 360°

The side EF, FD and DE of a triangle DEF are produced in order forming three exterior angles DFP, EDQ and FER respectively. Prove that∠ DFP+∠ EDQ+∠ FER=360°

The side EF, FD and DE of a triangle DEF are produced in order forming three exterior angles DFP, EDQ and FER respectively. Prove that
∠ DFP+∠ EDQ+∠ FER=360°