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# Common Core Standard HSG-CO.C.9 Questions

Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.

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Grade 10 Points, Lines, and Planes CCSS: HSG-CO.C.9

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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

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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

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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

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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.
Grade 10 Points, Lines, and Planes CCSS: HSG-CO.C.9

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What is the missing statement in step 3?
1. $l ~= l$
2. $bar{BP} ~= bar{BP$
3. $bar{AB} ~= bar{BC}$
4. $bar{AC} ~= bar{AC}$
Grade 10 Points, Lines, and Planes CCSS: HSG-CO.C.9

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Grade 10 Points, Lines, and Planes CCSS: HSG-CO.C.9

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What still needs to be done in order to complete the proof?
1. Consider other points on $l$, showing the same thing holds true for these other points.
2. Redraw the diagram, showing that the results are still true if $l$ is horizontal and $bar{AC}$ is vertical.
3. State that, if the point on $l$ lies on $bar{AC}$, that it is equidistant from points $A$ and $B$, due to $l$ bisecting $bar{AC}$ (from given information).
4. Nothing, it is complete.
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