Question

2 figure mrjkgh is the image of figure af ekjb after being rotated 90° counter - clockwise around point k.
a. draw a segment in figure af ekjb to create a quadrilateral. draw the image of the segment when rotated 90° counter - clockwise around point k.
b. write a congruence statement for the quadrilateral you created in figure af ekjb and the image of the quadrilateral in figure mbjkgh.
3 triangle hef is the image of triangle fgh after a 180° rotation around point k.
select all statements that must be true.
a triangle fgh is congruent to triangle feh.
b triangle efh is congruent to triangle gfh.
c angle khe is congruent to angle kfg.
d angle ghk is congruent to angle khe.
e segment eh is congruent to segment fg.
f segment gh is congruent to segment ef.

Explanation:

Step1: Recall rotation - congruence property

A rotation is a rigid transformation. Rigid transformations preserve side - lengths and angle - measures, so the pre - image and the image are congruent.

Step2: Analyze triangle congruence for question 3

Since $\triangle HEF$ is the image of $\triangle FGH$ after a $180^{\circ}$ rotation around point $K$, $\triangle FGH\cong\triangle HEF$. Corresponding angles and corresponding sides of congruent triangles are congruent.

• For option A: $\triangle FGH$ is not congruent to $\triangle FEH$. The correct congruence is $\triangle FGH\cong\triangle HEF$.
• For option B: $\triangle EFH$ is not congruent to $\triangle GFH$.
• For option C: $\angle KHE$ and $\angle KFG$ are corresponding angles in the rotation of $\triangle FGH$ to $\triangle HEF$, so $\angle KHE\cong\angle KFG$.
• For option D: $\angle GHK$ and $\angle KHE$ are not corresponding angles in the rotation.
• For option E: $EH$ and $FG$ are corresponding sides in the rotation of $\triangle FGH$ to $\triangle HEF$, so $EH\cong FG$.
• For option F: $GH$ and $EF$ are corresponding sides in the rotation of $\triangle FGH$ to $\triangle HEF$, so $GH\cong EF$.

Step3: Answer part a of question 2

To create a quadrilateral in figure $AFEKB$, we can draw a segment, for example, draw a segment from $A$ to $K$. To find its image after a $90^{\circ}$ counter - clockwise rotation around point $K$, we use the rule for a $90^{\circ}$ counter - clockwise rotation of a point $(x,y)$ around a center $(a,b)$: $(x,y)\to-(y - b)+a,(x - a)+b$. In a geometric construction, we measure the distance and angle from $K$ to $A$, and then rotate that segment $90^{\circ}$ counter - clockwise around $K$.

Step4: Answer part b of question 2

Let the quadrilateral in $AFEKB$ be $AEKB$ (after drawing the segment $AK$). Its image in $MBJKG$ is $MBJK$. The congruence statement is $\text{Quadrilateral }AEKB\cong\text{Quadrilateral }MBJK$ because a $90^{\circ}$ counter - clockwise rotation is a rigid transformation that preserves side - lengths and angle - measures.

Answer:

For question 2a: Geometric construction of segment and its rotated image as described above.
For question 2b: $\text{Quadrilateral }AEKB\cong\text{Quadrilateral }MBJK$
For question 3: C. Angle $KHE$ is congruent to angle $KFG$, E. Segment $EH$ is congruent to segment $FG$, F. Segment $GH$ is congruent to segment $EF$