Question

$\angle vwx \cong \angle vux$ and $\angle uvw \cong \angle uxw$. complete the proof that $\overline{vw} \parallel \overline{ux}$.

statement	reason
2 $\angle uvw \cong \angle uxw$
3 $m\angle vux + m\angle uvw + m\angle vwx + m\angle uxw = 360^\circ$
4 $m\angle vux + m\angle uvw + m\angle vux + m\angle uvw = 360^\circ$
5 $m\angle vux + m\angle uvw = 180^\circ$
6 $\overline{vw} \parallel \overline{ux}$

Explanation:

Step1: Identify Given Information

The problem states \( \angle VWX \cong \angle VUX \) and \( \angle UVW \cong \angle UXW \). These are given, so the reason for statements 1 and 2 is "Given".

Step2: Sum of Angles in a Quadrilateral

The sum of the interior angles of a quadrilateral is \( 360^\circ \). So for statement 3, the reason is "Sum of interior angles of a quadrilateral is \( 360^\circ \)".

Step3: Substitute Congruent Angles

Since \( \angle VWX \cong \angle VUX \) (so \( m\angle VWX = m\angle VUX \)) and \( \angle UVW \cong \angle UXW \) (so \( m\angle UXW = m\angle UVW \)), we substitute these into the equation from step 3. The reason for statement 4 is "Substitution (using \( \angle VWX \cong \angle VUX \) and \( \angle UVW \cong \angle UXW \))".

Step4: Simplify the Equation

From \( 2m\angle VUX + 2m\angle UVW = 360^\circ \), we divide both sides by 2 to get \( m\angle VUX + m\angle UVW = 180^\circ \). The reason for statement 5 is "Simplify (divide both sides by 2)".

Step5: Consecutive Interior Angles Theorem

If two lines are cut by a transversal and the consecutive interior angles are supplementary (sum to \( 180^\circ \)), then the lines are parallel. Here, \( \angle VUX \) and \( \angle UVW \) are consecutive interior angles for lines \( \overline{VW} \) and \( \overline{UX} \) cut by transversal \( \overline{UV} \). So the reason for statement 6 is "Consecutive Interior Angles Theorem (if consecutive interior angles are supplementary, lines are parallel)".

Answer:

1. Reason: Given
2. Reason: Given
3. Reason: Sum of interior angles of a quadrilateral is \( 360^\circ \)
4. Reason: Substitution (using \( \angle VWX \cong \angle VUX \) and \( \angle UVW \cong \angle UXW \))
5. Reason: Simplify (divide both sides by 2)
6. Reason: Consecutive Interior Angles Theorem (if consecutive interior angles are supplementary, lines are parallel)