Question

proove that x+y=a+b.this question is from P.K. garg 9th standard unit 4 geometry (lines and angles) group c 76th question.

Answer 1

CONCEPT USED :-

★ In this question the concept of Angle sum property of a quardilatetal and linear pair property has been used. So what is angle sum property of a quardilatetal ? So the angle sum property of a quardilatetal is , the sum of all interior angles of a quardilatetal is 360° . What is linear pair ? so the linear pair is a property in which the sum of all the angles lie in a straight line is 180°.

TO PROVE :-

• x + y = a + b.

PROOF :-

By linear pair we have,

$\implies \sf \: y + \angle PSR = 180^{ \circ}$

$\implies \sf \:\angle PSR =180^{ \circ} - y \: ...(1)$

Similarly,

$\implies \sf \: x + \angle PQR = 180^{ \circ}$

$\implies \sf \: \angle PQR = 180^{ \circ} - x \: ...(2)$

Now by angle sum property we have,

$\implies \sf \: \angle P + \angle PQR +\angle R + \angle PSR = 360^{ \circ}$

$\implies \sf \:b + 180^{ \circ} - y + a + 180^{ \circ} - y = 360$

$\implies \sf \:b + a + 360 - x - y = 360$

$\implies \sf \:a + b - x - y = 360 - 360$

$\implies \sf \:a + b - x - y = 0$

$\implies \underline{ \boxed{ \sf \:a + b = x + y}}$

Hence Proved !

Answer 2

Answer:

Question :-

• Prove that x + y = a + b

PROVE :-

• x + y = a + b.

PROOF :-

${:}\implies \sf \: y + \angle PSR = 180^{ \circ}$

${:}\implies \sf \:\angle PSR =180^{ \circ} - y \: ------(1)$

Similarly,

${:}\implies \sf \: x + \angle PQR = 180^{ \circ}$

${:}\implies \sf \: \angle PQR = 180^{ \circ} - x \: ------(2)$

Now by using angle sum property we get,

${:}\implies \sf \: \angle P + \angle PQR +\angle R + \angle PSR = 360^{ \circ}$

${:}\implies \sf \:b + 180^{ \circ} - y + a + 180^{ \circ} - y = 360^{ \circ}$

${:}\implies \sf \:b + a + 360^{ \circ} - x - y = 360^{ \circ}$

${:}\implies \sf \:a + b - x - y = 360^{ \circ} - 360^{ \circ}$

${:}\implies \sf \:a + b - x - y = 0$

${:}\implies \underline{ \boxed{ \sf \:a + b = x + y}}$

${\underline{\boxed{\large{\mathscr{Hence\: Proved}}}}}$