Question

if △wxy is equilateral and △wzy is isosceles, find each missing measure:

Explanation:

Step1: Recall triangle - angle sum properties

The sum of interior angles of a triangle is 180°. For an equilateral triangle, each interior angle is 60°. For an isosceles triangle, two angles are equal.

Step2: Analyze \(\triangle WXY\)

Since \(\triangle WXY\) is equilateral, \(\angle WXY=\angle XYW=\angle YWX = 60^{\circ}\).

Step3: Analyze \(\triangle WZY\)

In \(\triangle WZY\), \(\angle WZY = 138^{\circ}\). Let \(\angle ZWY=\angle ZYW=x\) (because \(\triangle WZY\) is isosceles). Using the angle - sum property of a triangle (\(\angle WZY+\angle ZWY+\angle ZYW = 180^{\circ}\)), we have \(138 + x+x=180\).

Step4: Solve for \(x\) in \(\triangle WZY\)

Combining like terms gives \(138 + 2x=180\). Subtract 138 from both sides: \(2x=180 - 138=42\). Then divide both sides by 2: \(x = 21^{\circ}\).

Step5: Find the other angles

\(\angle1=\angle XYW-\angle ZYW=60 - 21=39^{\circ}\). \(\angle2 = 60^{\circ}\) (from equilateral \(\triangle WXY\)). \(\angle3=\angle ZWY = 21^{\circ}\). \(\angle4=\angle ZYW = 21^{\circ}\). \(\angle5=\angle XYW-\angle ZYW=39^{\circ}\).

Answer:

\(\angle1 = 39^{\circ}\), \(\angle2 = 60^{\circ}\), \(\angle3 = 21^{\circ}\), \(\angle4 = 21^{\circ}\), \(\angle5 = 39^{\circ}\)