NCERT Class 9 Maths Solutions: Lines and Angles

Question:

Adjacent angles on a straight line add up to 180 degrees (Linear Pair Axiom). An angle bisector divides an angle into two equal parts.


First, let’s draw the figure. We have an angle ∠XYZ = 64°. The line segment XY is extended to a point P, forming a straight line XZP. This means that ∠XYZ and ∠ZYP are adjacent angles on a straight line XZP.

Step 1: Identify and calculate ∠ZYP.
Since XZP is a straight line, the sum of angles on this line is 180°. Therefore, ∠XYZ + ∠ZYP = 180°.
We are given ∠XYZ = 64°.
So, 64° + ∠ZYP = 180°.
Subtracting 64° from both sides, we get ∠ZYP = 180° – 64° = 116°.

Step 2: Understand the role of the angle bisector YQ.
Ray YQ bisects ∠ZYP. This means that YQ divides ∠ZYP into two equal angles: ∠ZYQ and ∠QYP.
So, ∠ZYQ = ∠QYP = ∠ZYP / 2.

Step 3: Calculate ∠XYQ.
We need to find ∠XYQ. From the figure, we can see that ∠XYQ is the sum of ∠XYZ and ∠ZYQ.
We know ∠XYZ = 64°.
We calculated ∠ZYP = 116°.
Since YQ bisects ∠ZYP, ∠ZYQ = 116° / 2 = 58°.
Therefore, ∠XYQ = ∠XYZ + ∠ZYQ = 64° + 58° = 122°.

Step 4: Calculate reflex ∠QYP.
A reflex angle is an angle greater than 180° and less than 360°.
The complete angle around point Y is 360°.
We have already calculated ∠QYP = 58°.
The reflex ∠QYP is the angle formed by going the other way around from YQ to YP.
So, reflex ∠QYP = 360° – ∠QYP.
reflex ∠QYP = 360° – 58° = 302°.

Summary of Results:
∠XYQ = 122°
reflex ∠QYP = 302°