Question

angle-angle similarity theorem
frank then needs to show that line a is parallel to line b. frank knows that corresponding angles of similar triangles are congruent. which of the following strategies should frank use in his proof?
show that since the x-axis is a transversal to lines a and b and the alternate interior angles ∠kjl and ∠nml are congruent, line a is parallel to line b.
show that since the x-axis is a transversal to lines a and b and the corresponding angles ∠kjl and ∠nml are congruent, line a is parallel to line b.
show that since the x-axis is a transversal to lines a and b and the consecutive angles ∠kjl and ∠nml are supplementary, line a is parallel to line b.

Explanation:

To prove two lines are parallel, one of the key theorems is that if a transversal intersects two lines and the corresponding angles are congruent, then the two lines are parallel. In this case, the \(x -\)axis is the transversal for lines \(a\) and \(b\). Since Frank knows that corresponding angles of similar triangles are congruent (so \(\angle KJL\) and \(\angle NML\) are congruent as corresponding angles), this fits the corresponding - angles - for - parallel - lines theorem.
The alternate - interior - angles theorem would require the angles to be on opposite sides of the transversal and between the two lines. The consecutive - angles theorem requires the sum of the angles to be \(180^{\circ}\) (supplementary).

Answer:

Show that since the \(x -\)axis is a transversal to lines \(a\) and \(b\) and the corresponding angles \(\angle KJL\) and \(\angle NML\) are congruent, line \(a\) is parallel to line \(b\).