# Proving Corresponding Angles are Congruent

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In the diagram given, lines [math]k[/math] and [math]m[/math] are parallel, and line [math]t[/math] is a transversal. Let point [math]A[/math] be the intersection of lines [math]k[/math] and [math]t[/math], and point [math]B[/math] be the intersection of lines [math]m[/math] and [math]t[/math] (points not labeled in the diagram). Also, let [math]h[/math] be the line parallel to lines [math]k[/math] and [math]m[/math] which is equidistant from these two lines (not pictured).

A.

If one were to perform a translation by vector [math]vec{AB}[/math], what would this vector look like, in terms of separate horizontal and vertical translations? (Assume that [math]m ang 4 < m ang 3[/math] for this purpose.)

- Translate down by the distance between lines [math]k[/math] and [math]m[/math] and translate left by the horizontal distance between [math]A[/math] and [math]B[/math].
- Translate up by the distance between lines [math]k[/math] and [math]m[/math] and translate left by the horizontal distance between [math]A[/math] and [math]B[/math].
- Translate up by the distance between lines [math]k[/math] and [math]m[/math] and translate right by the horizontal distance between [math]A[/math] and [math]B[/math].
- Translate down by the distance between lines [math]k[/math] and [math]m[/math] and translate right by the horizontal distance between [math]A[/math] and [math]B[/math].

B.

If the upper half plane (everything above line [math]h[/math], not pictured) is translated by the vector [math]vec{AB}[/math], which of the following is true? There may be more than one correct answer.

- [math]A[/math] maps to [math]B[/math].
- [math]k[/math] maps to [math]m[/math].
- [math]m[/math] maps to a new line, but parallel to itself.
- [math]t[/math] maps to [math]t[/math].

C.

Which of the following reasons ensures that lines [math]k[/math] and [math]m[/math] are coincident after the translation by [math]vec{AB}[/math], given that line [math]k[/math] now passes through point [math]B?[/math]

- Rigid transformations don't change size or shape.
- No rotations or reflections were used in the transformation, thus the slope of the line cannot change.
- Point [math]A[/math] maps to point [math]B[/math], and point [math]A[/math] was on line [math]k[/math], thus these two points define a line, which is [math]k[/math].
- The Parallel Postulate and the fact that the slope of line [math]k[/math] remains unchanged by translations.

D.

What is the reason we can conclude, including the information from parts A - C, that [math]m ang 1 = m ang 5,[/math] (and therefore, that [math]ang1 ~= ang 5[/math] by the definition of congruent angles).

- Substitution Property of Equality.
- Congruent Supplements Theorem.
- Rigid transformations preserve angle measure.
- Given by the diagram.