Question

complete the proof that ( mangle u + mangle y + mangle uvy = 180^circ ) without using the triangle angle sum theorem.
image of a geometric figure with points u, y, v, w, x and a table with statements and reasons

1. ( overline{uy} parallel overline{vx} )
2. ( angle u cong angle wvx )
3. ( angle y cong angle xvy )
4. ( mangle wvy = mangle wvx + mangle xvy )
5. ( mangle wvy = mangle u + mangle y )
6. ( mangle wvy + mangle uvy = 180^circ )
7. ( mangle u + mangle y + mangle uvy = 180^circ )

Explanation:

Step1: Identify Given Parallel Lines

We are given that \(\overline{UY} \parallel \overline{VX}\). This is a given statement, so the reason for statement 1 is "Given".

Step2: Corresponding Angles (∠U and ∠WVX)

Since \(\overline{UY} \parallel \overline{VX}\) and \(\overline{UV}\) is a transversal, \(\angle U \cong \angle WVX\) by the Corresponding Angles Postulate (if two parallel lines are cut by a transversal, corresponding angles are congruent).

Step3: Corresponding Angles (∠Y and ∠XVY)

Similarly, since \(\overline{UY} \parallel \overline{VX}\) and \(\overline{YV}\) is a transversal, \(\angle Y \cong \angle XVY\) by the Corresponding Angles Postulate.

Step4: Angle Addition Postulate (∠WVY)

The measure of \(\angle WVY\) is equal to the sum of the measures of \(\angle WVX\) and \(\angle XVY\) by the Angle Addition Postulate (the measure of an angle formed by two adjacent angles is the sum of the measures of the two angles).

Step5: Substitution (∠U and ∠Y)

We substitute \(\angle U\) for \(\angle WVX\) and \(\angle Y\) for \(\angle XVY\) (from steps 2 and 3) into the equation from step 4. So \(m\angle WVY = m\angle U + m\angle Y\) by Substitution Property.

Step6: Linear Pair (∠WVY and ∠UVY)

\(\angle WVY\) and \(\angle UVY\) form a linear pair, so their measures add up to \(180^\circ\) by the Linear Pair Postulate (if two angles form a linear pair, they are supplementary, i.e., their measures sum to \(180^\circ\)).

Step7: Substitution (Final Proof)

We substitute \(m\angle WVY\) with \(m\angle U + m\angle Y\) (from step 5) into the equation from step 6. So \(m\angle U + m\angle Y + m\angle UVY = 180^\circ\) by Substitution Property.

Answer:

The proof is completed by using the Corresponding Angles Postulate, Angle Addition Postulate, Linear Pair Postulate, and Substitution Property as explained in the steps above, resulting in \(m\angle U + m\angle Y + m\angle UVY = 180^\circ\).