Question

complete the proof that ( mangle u + mangle y = mangle wvy ) without using the triangle angle sum theorem.

1. ( overline{uy} parallel overline{vx} ) (given)
2. ( angle u cong angle wvx ) (reason: ______)
3. ( angle y cong angle xvy ) (reason: alternate exterior angles theorem / alternate interior angles theorem / converse of alternate exterior angles theorem / converse of alternate interior angles theorem / converse of corresponding angles theorem / ...)
4. ( mangle wvx + mangle xvy = mangle wvy ) (reason: ______)
5. ( mangle u + mangle y = mangle wvy ) (reason: ______)

Explanation:

Step1: Analyze Statement 2

We know that \(\overline{UY} \parallel \overline{VX}\) (given). \(\angle U\) and \(\angle WVX\) are alternate interior angles because they lie between the parallel lines \(\overline{UY}\) and \(\overline{VX}\) and on alternate sides of the transversal \(\overline{WV}\). So the reason for \(\angle U \cong \angle WVX\) is the Alternate Interior Angles Theorem.

Step2: Analyze Statement 3

For \(\angle Y \cong \angle XVY\), since \(\overline{UY} \parallel \overline{VX}\) and considering the transversal \(\overline{VY}\), \(\angle Y\) and \(\angle XVY\) are alternate interior angles. So the reason here is also the Alternate Interior Angles Theorem (though the dropdown for statement 3 is already filled, we confirm the logic).

Step3: Analyze Statement 4

\(\angle WVX\) and \(\angle XVY\) are adjacent angles that together form \(\angle WVY\), so by the Angle Addition Postulate, \(m\angle WVX + m\angle XVY = m\angle WVY\).

Step4: Analyze Statement 5

From statements 2 and 3, we have \(m\angle U = m\angle WVX\) and \(m\angle Y = m\angle XVY\) (since congruent angles have equal measures). Substituting these into the equation from statement 4 (using substitution property), we get \(m\angle U + m\angle Y = m\angle WVY\).

For the reason of statement 2:

Answer:

Alternate Interior Angles Theorem