Question

given ( mangle qvr = 49^circ )
prove: ( x = 15 )
diagram: horizontal line ( qvy ), line ( rvu ) intersecting at ( v ), vertical line ( vs ) above ( v ); ( angle uvt = (3x + 4)^circ )
complete the steps of the proof.
(selected: vertical angles theorem)
options: ( 3x + 4 = 15 ), ( 3x + 4 = 49 ), ( \text{measure of angle } qvr = 49 \text{ degrees} ), ( \text{measure of angle } uvt = (3x + 4) \text{ degrees} )

Explanation:

Step1: Apply Vertical Angles Theorem

By the Vertical Angles Theorem, \( \angle QVR \) and \( \angle UVT \) are vertical angles, so their measures are equal. Thus, \( m\angle QVR = m\angle UVT \).

Step2: Substitute Known Values

We know \( m\angle QVR = 49^\circ \) and \( m\angle UVT = (3x + 4)^\circ \). Substituting these into the equation from Step 1 gives \( 3x + 4 = 49 \).

Step3: Solve for \( x \)

Subtract 4 from both sides: \( 3x = 49 - 4 = 45 \). Then divide both sides by 3: \( x = \frac{45}{3} = 15 \).

Answer:

\( x = 15 \) is proven as shown in the steps above.