Question

given: ( mangle qvr = 49^circ ) prove: ( x = 15 ) complete the steps of the proof. definition of congruence definition of vertical angles vertical angles theorem vertical angles form a line

Explanation:

Step1: Identify Vertical Angles

$\angle QVR$ and $\angle TVU$ are vertical angles. By the vertical angles theorem, vertical angles are congruent, so $m\angle QVR = m\angle TVU$.

Step2: Set Up Equation

Given $m\angle QVR = 49^\circ$ and $m\angle TVU=(3x + 4)^\circ$, we set them equal: $3x + 4=49$.

Step3: Solve for x

Subtract 4 from both sides: $3x=49 - 4=45$. Divide by 3: $x=\frac{45}{3}=15$.

Answer:

To prove \( x = 15 \):

1. Vertical Angles Theorem: \( \angle QVR \) and \( \angle TVU \) are vertical angles, so \( m\angle QVR = m\angle TVU \).
2. Substitute \( m\angle QVR = 49^\circ \) and \( m\angle TVU=(3x + 4)^\circ \): \( 3x + 4 = 49 \).
3. Solve: \( 3x = 45 \implies x = 15 \).

(For the dropdown, the correct choice is "vertical angles theorem" to justify \( m\angle QVR = m\angle TVU \).)