In the figure, if x + y = w + z, then prove that AOB is a line.

Understand the Problem

The question asks to prove that angle AOB is a straight line, given the relationship between the angles x, y, z, and w. It involves understanding properties of angles around a point and the definition of adjacent angles.

Answer

Angle $ AOB $ is a straight line.

Steps to Solve

State the Given Information

We are given that ( x + y = w + z ). We will also use the property of angles around a point.

Use the Angle Sum Property

Since the angles around point O sum up to ( 360^\circ ), we can express this as: $$ x + y + z + w = 360^\circ $$

Substitute for ( w + z )

Using the given information ( x + y = w + z ), we can replace ( w + z ) in the angle sum equation: $$ x + y + (x + y) = 360^\circ $$

Simplify the Equation

This simplifies to: $$ 2(x + y) = 360^\circ $$

Dividing both sides by 2: $$ x + y = 180^\circ $$

Conclude that ( AOB ) is a Straight Line

Since ( x + y = 180^\circ ), angles ( AOB ) are adjacent angles that sum to ( 180^\circ ). Therefore, angle ( AOB ) is a straight line.

Angle ( AOB ) is a straight line.

More Information

This proof uses the property that the sum of angles around a point is ( 360^\circ ) and the fact that adjacent angles can add up to form a straight line.

Tips

Misunderstanding Angle Relationships: Not recognizing that adjacent angles can sum to ( 180^\circ ).
Incorrectly Using the Sum of Angles: Forgetting that the angles around a point total ( 360^\circ ).

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