Prove that the bisectors of the angles of a linear pair are at right angles?

Question

Answer

Sum of Angles formed by linear pair=180...If it is bisected...Both angles will equal to 90...

Answer

Let AB is a straight line.

And,OC bisect the line AB.

Then,

As it bisect the angle,

So ,angle AOC = angle BOC

Now,we know that ,sum of linear pair angle is 180 degree .

so,angle AOC + angle BOC = 180 degree.

2(angle AOC)= 180 degree.

angle AOC = 90 degree.

Answer

Proof that the Bisectors of Angles of a Linear Pair are at Right Angles

Definition of a Linear Pair
- A linear pair consists of two adjacent angles whose non-common sides form a straight line.

Definition of Bisectors of Angles
- The bisectors of angles are lines that divide an angle into two equal parts.

Proof
1. Let ∠AOC and ∠COB be a linear pair of angles, where ∠AOB = 180°.
2. Let the bisector of ∠AOB be line OD and the bisector of ∠COB be line OE.
3. Since OD bisects ∠AOB, we have ∠AOD = ∠DOB.
4. Similarly, since OE bisects ∠COB, we have ∠COE = ∠EOB.
5. Therefore, ∠AOD + ∠COE = ∠DOB + ∠EOB = 180° (angles on a straight line).
6. By transitive property, we have ∠AOD + ∠COE = 180°.
7. This implies that OD and OE form a straight line.
8. Thus, the angles formed by the bisectors OD and OE are right angles.
9. Hence, the bisectors of the angles of a linear pair are at right angles.

Conclusion
- The proof above shows that the bisectors of the angles of a linear pair are at right angles, as the bisectors form a straight line and the sum of the angles is 180°.

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