Question

prove: ( x = 15 )
complete the steps of the proof.

1. ( mangle qvr = 49 ) \t\t\t1. given
2. ( mangle uvt = 3x + 4 ) \t\t\t2. given
3. ( angle qvr ) and ( angle uvt ) are vertical angles. \t\t\t3. definition of vertical angles
4. ( angle uvt cong angle qvr ) \t\t\t4.
5. ( mangle uvt = mangle qvr ) \t\t\t5. definition of congruence
6. \t\t\t6. substitution property
7. ( 3x = 45 ) \t\t\t7. subtraction property
8. ( x = 15 ) \t\t\t8. division property

Explanation:

Step1: Identify the reason for step 4

Vertical angles are congruent by the Vertical Angles Theorem. So the reason for $\angle UVT \cong \angle QVR$ is the Vertical Angles Theorem.

Step2: Confirm step 6

We know from step 2 that $m\angle UVT = 3x + 4$ and from step 1 that $m\angle QVR = 49$, and from step 5 that $m\angle UVT = m\angle QVR$. By substitution property, we substitute the measures: $3x + 4 = 49$.

Answer:

1. Vertical Angles Theorem
2. $3x + 4 = 49$