The sides BA and DC of quadrilateral ABCD are produced as shown in the figure Prove that x+y=a+b - Maths - Quadrilaterals

Question

The sides BA and DC of quadrilateral ABCD are produced as
shown in the figure Prove that x+y=a+b

Varun.Rawat · Accepted Answer

Solution :

Join AC
∠BCQ = x ; ∠PAD = y ; ∠ADC =a ; ∠ABC = bSince ∠ACQ is an exterior angle of a △ADC, then∠ACQ = ∠ADC + ∠DAC                  exterior angle theorem⇒∠ACB + ∠BCQ  = ∠ADC + ∠DAC⇒∠ACB + x = a + ∠DAC         
1Since ∠CAP is an exterior angle of a △ABC, then∠CAP = ∠ACB + ∠ABC                  exterior angle theorem⇒∠PAD +  ∠DAC = ∠ACB + b⇒y + ∠DAC = ∠ACB + b        
2adding 1 and 2, we get∠ACB + x  + y + ∠DAC  =  a + ∠DAC + ∠ACB + b  ⇒x + y = a + b

The sides BA and DC of quadrilateral ABCD are produced as shown in the figure.Prove that x+y=a+b