# Prove that If the Two Arms of an Angle Are Perpendicular to the Two Arms of Another Angle, Then the Angles Are Either Equal Or Supplementary - Mathematics

Sum

Prove that if the two arms of an angle are perpendicular to the two arms of another angle, then the angles are either equal or supplementary

#### Solution

Consider be angles AOB and ACB

Given OA  ⊥ AC, OB ⊥ BC

To prove: ∠AOB = ∠ACB (or)

∠AOB + ∠ACB = 180°

⇒∠A + ∠O + ∠B + ∠C = 360°

⇒ 180 + ∠O + ∠C = 360°

⇒ ∠O + ∠C = 360 -180 = 180°

Hence, ∠AOB + ∠ACB = 180°        ......(i )

Also,

∠B + ∠ACB = 180°              ......(i )

Also,

∠B + ∠ACB = 180°         ......(i )

Also,

∠B + ∠ACB = 180°

⇒ ∠ACB = 180° - 90°

⇒∠ACB = 90°        .....(ii)

From (i) and (ii)

∴∠ACB = ∠AOB = 90°

Hence, the angles are equal as well as supplementary

Concept: Pairs of Angles
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Chapter 10: Lines and Angles - Exercise 10.4 [Page 49]

#### APPEARS IN

RD Sharma Mathematics for Class 9
Chapter 10 Lines and Angles
Exercise 10.4 | Q 23 | Page 49

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