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Author: nsharp1
No. Questions: 4
Created: Nov 3, 2018

# Proving Corresponding Angles are Congruent

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In the diagram given, lines $k$ and $m$ are parallel, and line $t$ is a transversal. Let point $A$ be the intersection of lines $k$ and $t$, and point $B$ be the intersection of lines $m$ and $t$ (points not labeled in the diagram). Also, let $h$ be the line parallel to lines $k$ and $m$ which is equidistant from these two lines (not pictured).
Grade 10 Points, Lines, and Planes CCSS: HSG-CO.C.9
A.
If one were to perform a translation by vector $vec{AB}$, what would this vector look like, in terms of separate horizontal and vertical translations? (Assume that $m ang 4 < m ang 3$ for this purpose.)
1. Translate down by the distance between lines $k$ and $m$ and translate left by the horizontal distance between $A$ and $B$.
2. Translate up by the distance between lines $k$ and $m$ and translate left by the horizontal distance between $A$ and $B$.
3. Translate up by the distance between lines $k$ and $m$ and translate right by the horizontal distance between $A$ and $B$.
4. Translate down by the distance between lines $k$ and $m$ and translate right by the horizontal distance between $A$ and $B$.
Grade 10 Points, Lines, and Planes CCSS: HSG-CO.C.9
B.
If the upper half plane (everything above line $h$, not pictured) is translated by the vector $vec{AB}$, which of the following is true? There may be more than one correct answer.
1. $A$ maps to $B$.
2. $k$ maps to $m$.
3. $m$ maps to a new line, but parallel to itself.
4. $t$ maps to $t$.
Grade 10 Points, Lines, and Planes CCSS: HSG-CO.C.9
C.
Which of the following reasons ensures that lines $k$ and $m$ are coincident after the translation by $vec{AB}$, given that line $k$ now passes through point $B?$
1. Rigid transformations don't change size or shape.
2. No rotations or reflections were used in the transformation, thus the slope of the line cannot change.
3. Point $A$ maps to point $B$, and point $A$ was on line $k$, thus these two points define a line, which is $k$.
4. The Parallel Postulate and the fact that the slope of line $k$ remains unchanged by translations.
Grade 10 Points, Lines, and Planes CCSS: HSG-CO.C.9
D.
What is the reason we can conclude, including the information from parts A - C, that $m ang 1 = m ang 5,$ (and therefore, that $ang1 ~= ang 5$ by the definition of congruent angles).
1. Substitution Property of Equality.
2. Congruent Supplements Theorem.
3. Rigid transformations preserve angle measure.
4. Given by the diagram.