arianna wants to prove the vertical angles theorem. in the diagram, $\overleftrightarrow{rs}$ intersects $\overleftrightarrow{tu}$ at point $v$.
complete arianna’s proof that vertical angles $\angle tvr$ and $\angle svu$ are congruent.
| | statement | reason |
| 1 | $m\angle tvr + m\angle tvs = 180^\circ$ | angles forming a linear pair sum to $180^\circ$ |
| 2 | | |
| 3 | $m\angle tvr + m\angle tvs = m\angle tvs + m\angle svu$ | |
| 4 | | |
| 5 | $\angle tvr \cong \angle svu$ | |

Question

arianna wants to prove the vertical angles theorem. in the diagram, $\overleftrightarrow{rs}$ intersects $\overleftrightarrow{tu}$ at point $v$.
complete arianna’s proof that vertical angles $\angle tvr$ and $\angle svu$ are congruent.
| | statement | reason |
| 1 | $m\angle tvr + m\angle tvs = 180^\circ$ | angles forming a linear pair sum to $180^\circ$ |
| 2 |  |  |
| 3 | $m\angle tvr + m\angle tvs = m\angle tvs + m\angle svu$ |  |
| 4 |  |  |
| 5 | $\angle tvr \cong \angle svu$ |  |

Accepted Answer

# Explanation:
## Step1: Identify the linear pair for ∠SVU
We know that ∠TVS and ∠SVU form a linear pair (since $ \overleftrightarrow{RS} $ is a straight line and $ \overleftrightarrow{TU} $ intersects it at $ V $), so their measures sum to $ 180^\circ $. Thus, the statement for step 2 is $ m\angle TVS + m\angle SVU = 180^\circ $ and the reason is "Angles forming a linear pair sum to $ 180^\circ $".

## Step2: Justify step 3
Step 3 is a substitution. We substitute the expressions from step 1 ($ m\angle TVR + m\angle TVS = 180^\circ $) and step 2 ($ m\angle TVS + m\angle SVU = 180^\circ $) into each other. So the reason for step 3 is "Substitution Property of Equality" (since both left - hand sides equal $ 180^\circ $, they are equal to each other).

## Step3: Simplify step 3
To get step 4, we use the Subtraction Property of Equality. If $ m\angle TVR + m\angle TVS = m\angle TVS + m\angle SVU $, we can subtract $ m\angle TVS $ from both sides of the equation. So the statement for step 4 is $ m\angle TVR = m\angle SVU $ and the reason is "Subtraction Property of Equality" (subtracting $ m\angle TVS $ from both sides of the equation).

## Step4: Justify step 5
If the measures of two angles are equal, then the angles are congruent. So the reason for step 5 ($ \angle TVR\cong\angle SVU $) is "Definition of Congruent Angles" (angles with equal measures are congruent).

# Answer:
- Step 2:
  - Statement: $ m\angle TVS + m\angle SVU = 180^\circ $
  - Reason: Angles forming a linear pair sum to $ 180^\circ $
- Step 3:
  - Reason: Substitution Property of Equality
- Step 4:
  - Statement: $ m\angle TVR = m\angle SVU $
  - Reason: Subtraction Property of Equality
- Step 5:
  - Reason: Definition of Congruent Angles

	statement	reason
1	\\(m\angle tvr + m\angle tvs = 180^\circ\\)	angles forming a linear pair sum to \\(180^\circ\\)
2
3	\\(m\angle tvr + m\angle tvs = m\angle tvs + m\angle svu\\)
4
5	\\(\angle tvr \cong \angle svu\\)

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QUESTION IMAGE

Question

Explanation:

Step1: Identify the linear pair for ∠SVU

Step2: Justify step 3

Step3: Simplify step 3

Step4: Justify step 5

Answer: