Question

triangle uvw is shown with ( mangle wuv = 36^{circ} ). the measure of ( angle uvw ) is ( (5h - 64)^{circ} ), and the measure of ( angle awb ) is ( (5h - 16)^{circ} ). determine the value of ( h ). ( h = 20 ) ( h = 22.4 ) ( h = 48 ) ( h = 55.2 )

Explanation:

Step1: Use the exterior - angle theorem

The exterior - angle theorem states that an exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. In \(\triangle UVW\), \(\angle AWB\) is an exterior angle. So, \(m\angle AWB=m\angle WUV + m\angle UVW\).
Given \(m\angle WUV = 36^{\circ}\), \(m\angle UVW=(5h - 64)^{\circ}\), and \(m\angle AWB=(5h - 16)^{\circ}\).
Substitute the values into the equation: \((5h-16)=36+(5h - 64)\).

Step2: Simplify the right - hand side of the equation

Simplify \(36+(5h - 64)\):

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So the equation becomes \(5h-16 = 5h-28\) (This is wrong. Let's start again. The correct formula is \(m\angle AWB=m\angle WUV + m\angle UVW\), so \((5h - 16)=36+(5h - 64)\) is wrong. The correct one is \(m\angle AWB=m\angle WUV+m\angle UVW\), i.e., \(5h-16=36+(5h - 64)\) (no, another way. Since \(\angle AWB\) is an exterior angle of \(\triangle UVW\) with respect to \(\angle WUV\) and \(\angle UVW\). The correct equation is \(m\angle AWB=m\angle WUV + m\angle UVW\). So \(5h-16=36+(5h - 64)\) (wrong). Wait, no, \(\angle AWB\) and \(\angle VWU\) are vertical angles. \(\angle VWU = 5h-16\). And in \(\triangle UVW\), by the angle - sum property of a triangle \(m\angle WUV+m\angle UVW+m\angle VWU = 180^{\circ}\). So \(36+(5h - 64)+(5h - 16)=180\).

Step3: Combine like terms

Combine the terms:

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Step4: Solve for \(h\)

Add \(44\) to both sides of the equation \(10h-44 = 180\):
\(10h=180 + 44\), so \(10h=224\).
Divide both sides by \(10\): \(h=\frac{224}{10}=22.4\).

Answer:

\(h = 22.4\)