Question

line segment ab is the image of ab after a 180° rotation around point m. select all of the true statements about the figure. a angle amb and angle amb are vertical angles. b am is the same length as am. c am is the same length as bm. d ab is parallel to ab. e angle amb is the same measure as angle amb

Explanation:

Step1: Recall rotation properties

A 180 - degree rotation around a point $M$ maps a point $A$ to $A'$ and $B$ to $B'$ such that $M$ is the mid - point of $AA'$ and $BB'$.

Step2: Analyze option A

Vertical angles are formed by two intersecting lines. Since $A$, $M$, $A'$ are collinear and $B$, $M$, $B'$ are collinear, $\angle AMB$ and $\angle A'MB'$ are vertical angles. This is true.

Step3: Analyze option B

In a 180 - degree rotation around point $M$, the distance from a point to the center of rotation is the same as the distance from its image to the center of rotation. So, $AM = A'M$. This is true.

Step4: Analyze option C

There is no reason for $AM$ to be equal to $BM$ based on the rotation alone. This is false.

Step5: Analyze option D

$AB$ and $A'B'$ are not parallel. They intersect at point $M$. This is false.

Step6: Analyze option E

Since $A$, $M$, $A'$ are collinear, $\angle AMM'=\angle A'MM' = 180^{\circ}$ (where $M'$ is just a way to think about the line containing these points), and the measures of angles related to the rotation around $M$ imply $\angle AMM'$ and $\angle A'MM'$ are equal in measure. This is true.

Answer:

A. Angle $AMB$ and angle $A'MB'$ are vertical angles.
B. $AM$ is the same length as $A'M$.
E. Angle $AMM'$ is the same measure as angle $A'MM'$.