Question

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

Answer 1

Answer:

a. m∠Z =  62

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

c.  m∠W = 62

d. $m\widehat{WX}$ = 122°

Step-by-step 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°.