The missing measures of the angles are:
[tex]m\angle 1= 60^{\circ}\\\\m\angle 2= 39^{\circ}\\\\m\angle 3= 21^{\circ}\\\\m\angle 4 = 39^{\circ}\\\\m\angle 5 = 21^{\circ}[/tex]
Given:
[tex]m\angle Z = 138^{\circ}[/tex]
Note the following:
- An equilateral triangle has all its three angles equal to each other. Each angle = [tex]60^{\circ}[/tex].
- This implies that, since [tex]\triangle WXY[/tex] is equilateral, therefore, [tex]m\angle Y = m\angle XWY = m\angle XYW = 60^{\circ}[/tex]
- Base angles of an isosceles triangle are congruent to each other.
- This implies that, since [tex]\triangle WZY[/tex] is isosceles, therefore, [tex]m\angle 3 = \angle 5[/tex]
Applying the above stated, let's find the measure of each angle:
- Find [tex]m\angle 1[/tex]
[tex]m\angle 1 = 60^{\circ}[/tex] (an angle in an equilateral triangle equals 60 degrees)
- Find [tex]m\angle 2[/tex]
[tex]m\angle 2 = 60 - m \angle 3[/tex]
[tex]m\angle 2 = 60 -\frac{1}{2}(180 - 138)[/tex] [tex](Note: \frac{1}{2}(180 - 138) = 1 $ base $ angle $ of $ \triangle WZY)[/tex]
[tex]m\angle 2 = 60 -21\\\\m\angle 2 = 39^{\circ}[/tex]
- Find [tex]m\angle 3[/tex] and [tex]m\angle 5[/tex] (base angles of isosceles triangle WZY)
[tex]m\angle 3 = \frac{1}{2}(180 - 138) (1 $ base $ angle $ of $ \triangle WZY)[/tex]
[tex]m\angle 3 = \frac{1}{2}(42) \\\\m\angle3 = 21^{\circ}[/tex]
[tex]m\angle3 = m\angle 5[/tex] (base angles of isosceles triangle are congruent)
Therefore,
[tex]m\angle5 = 21^{\circ}[/tex]
- Find [tex]m\angle 4[/tex]
[tex]m\angle 4 = 60 - m\angle 5[/tex]
Substitute
[tex]m\angle 4 = 60 - 21\\\\m\angle 4 = 39^{\circ}[/tex]
The missing measures of the angles are:
[tex]m\angle 1= 60^{\circ}\\\\ m\angle 2= 39^{\circ}\\\\m\angle 3= 21^{\circ}\\\\m\angle 4 = 39^{\circ}\\\\m\angle 5 = 21^{\circ}[/tex]
Learn more here:
https://brainly.com/question/2944195
