Question

for each part below, use the figure to fill in the blank.
if necessary, you may learn what the markings on a figure indicate.
(a) find ( mangle x ).
( mangle x=square^{circ} )
(b) find ( mangle v ).
( mangle v=square^{circ} )

Explanation:

Step1: Analyze the first triangle (a)

The triangle in part (a) is an isosceles triangle (two sides are equal). The base - angles (the angles opposite the equal sides) are equal. Let the measure of each base - angle be \(x\). Using the angle - sum property of a triangle (\(m\angle X + m\angle Y+m\angle W=180^{\circ}\)). Given \(m\angle Y = 40^{\circ}\), and \(m\angle W=m\angle Y\) (since the sides opposite them are equal). So \(m\angle X+40^{\circ}+40^{\circ}=180^{\circ}\).

Step2: Solve for \(m\angle X\)

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Step3: Analyze the second triangle (b)

The triangle in part (b) is an equilateral triangle (all sides are equal). In an equilateral triangle, all angles are equal. Let \(m\angle V = m\angle U=m\angle T\). Using the angle - sum property of a triangle (\(m\angle V + m\angle U+m\angle T=180^{\circ}\)).

Step4: Solve for \(m\angle V\)

Since \(m\angle V = m\angle U=m\angle T\), then \(m\angle V=\frac{180^{\circ}}{3}=60^{\circ}\)

Answer:

\(m\angle X = 100^{\circ}\), \(m\angle V = 60^{\circ}\)