Question

rstu is a parallelogram. if ( mangle srv = 48^{circ} ) and ( mangle svr = 54^{circ} ), explain how you can find the measure of ( angle tuv ). show all steps of your work, and refer to any properties of triangles, parallelograms, or triangle congruency theorems as necessary to justify your response.

Explanation:

Step1: Find the measure of ∠RSV in △SRV

In △SRV, use the triangle - angle sum theorem (the sum of the interior angles of a triangle is \(180^{\circ}\)).
Let \(m\angle RSV=x\). Then \(x + 48^{\circ}+54^{\circ}=180^{\circ}\).

$$x=180^{\circ}-(48^{\circ} + 54^{\circ})=180^{\circ}-102^{\circ}=78^{\circ}$$

Step2: Use the properties of a parallelogram

Since \(RSTU\) is a parallelogram, \(SR\parallel TU\) and \(RU\parallel ST\), and the diagonals bisect each other (\(SV = VU\) and \(RV=VT\)). Also, \(SR\parallel TU\) implies that \(\angle RSV=\angle TUV\) (alternate - interior angles, because \(SR\parallel TU\) and \(SU\) is a transversal).

Answer:

\(m\angle TUV = 78^{\circ}\)