Question

Find each measurement. (The figure is not drawn to scale.)

Answer 1

a. m∠Z = 62

b.
`m\widehat{WZ}` = 118

c. m∠W = 62

d.
`m\widehat{WX}` = 122°

Explanation:

a. The given parameters are;

m∠X = 118

`\overline {WZ} \cong \overline {YZ}`

m∠Y = 120

m∠X + m∠Z = 180 Angles in opposite segment are supplementary

m∠Z = 180 - m∠X = 180 - 118 = 62

m∠Z = 62

b. Given
`\overline {WZ} \cong \overline {YZ}` line drawn from W to Y forms isosceles triangles WZY, with base angles ∠WYZ and ∠YWZ equal (Base angles of an isosceles triangle)

Therefore

∠WYZ + ∠YWZ + m∠Z = 180 (Angle sum theorem)

∠WYZ = ∠YWZ (Substitution property of equality)

∠WYZ + ∠YWZ + m∠Z = ∠WYZ + ∠WYZ + m∠Z =180

2×∠WYZ + 62 =180

2×∠WYZ = 180 -62 = 118°

∠WYZ = 118°/2 =59

∠WYZ = ∠YWZ = 59

`m\widehat{WZ}` subtends chord WZ at the center = ∠WYZ subtends chord WZ at the circumference

∴ 2×∠WYZ =
`m\widehat{WZ}`

`m\widehat{WZ}` = 2×59 = 118

`m\widehat{WZ}` = 118

c. m∠X + m∠Y + m∠Z + m∠W = 360 (Sum of angles in a quadrilateral)

m∠W = 360 - (m∠X + m∠Y + m∠Z) = 360 - (118 + 120 + 60) = 62

m∠W = 62

d.
`m\widehat{WZ}` +
`m\widehat{WX}` =
`m\widehat{XWZ}` (Angle addition postulate)

`m\widehat{XWZ}` = 2 × ∠Y (Angle subtended at the center = 2 × Angle subtended at the circumference

∴
`m\widehat{XWZ}` = 2 × 120 = 240

`m\widehat{WX}` =
`m\widehat{XWZ}` -
`m\widehat{WZ}`

`m\widehat{WX}` = 240 - 118 = 122°

`m\widehat{WX}` = 122°.