Question

6. In the following figure ray YW stands on a line XYZ. WYZ : WYX = 1:2. Ray YQ and ray YP are angle bisectors of WYZ and WYX respectively. Find PYQ.​

Answer 1

In the following figure ray YW stands on a line XYZ. WYZ : WYX = 1:2. Ray YQ and ray YP are angle bisectors of WYZ and WYX respectively.

∠ PYQ = 90°

From figure, it's clear that,

∠ XYZ = ∠ WYX + ∠ WYZ = 180°          (as XYZ is a straight line)

∠ WYX + ∠ WYZ = 180°

2x + x = 180°

3x = 180°

x = 60°

∴ ∠ WYZ = x = 60°

∴ ∠ WYX = 2x = 2 × 60° = 120°

We have,

∠ WYZ = ∠ WYQ + ∠ QYZ = 60°

∴ ∠ WYQ = ∠ QYZ = 60°/2 = 30°

∴ ∠ WYQ = 30° ...........(1)

∠ XYW = ∠ XYP + ∠ PYW = 120°

∴ ∠ XYP = ∠ PYW = 120°/2 = 60°

∴ ∠ PYW = 60° ...........(2)

Adding (1) and (2), we get,

∠ WYQ + ∠ PYW = 30° + 60°

⇒ ∠ WYQ + ∠ PYW  = ∠ PYQ

∴ ∠ PYQ = 90°

Answer 2

∠PYQ = 90°

Step-by-step explanation:

• Given data

        $\mathbf{\frac{\angle WYZ}{\angle WYX}=\frac{1}{2}}$

        Let

        ∠WYZ = k

        ∠WYX = 2k

• Where XZ is a straight line.

        ∠WYZ + ∠WYX = 180°

        k+ 2k =  180°

        Dividing by 2 on both side, we get

        $\mathbf{\frac{k}{2}+\frac{2k}{2}=90^{\circ}}$        ...1)

• It is given that ray YQ and ray YP are angle bisectors of ∠WYZ and ∠WYX

       So

       $\mathbf{\angle PYW=\frac{2k}{2}}$        ...2)

       $\mathbf{\angle WYQ=\frac{k}{2}}$         ...3)

• Where

       ∠PYQ= ∠WYQ +∠PYW     ...4)

• From equation 2) ,equation 3) and equation 4)

        $\mathbf{\angle PYQ=\frac{k}{2}+\frac{2k}{2}}$        ...5)

• From equation 1) and equation 5)

        ∠PYQ =90°