Question

1. lines ℓ and m are perpendicular. a point q has this property: rotating q 180 degrees using center p has the same effect as reflecting q over line m.

(from unit 1, lesson 11.)

a. give two possible locations of q.
b. do all points in the plane have this property?
(from unit 1, lesson 11.)

1. there is a sequence of rigid transformations that takes a to a, b to b, and c to c. the same sequence takes d to d. draw and label d:

(from unit 1, lesson 10.)

1. two distinct lines, ℓ and m, are each perpendicular to the same line n.

a. what is the measure of the angle where line ℓ meets line m?
b. what is the measure of the angle where line m meets line n?
(from unit 1, lesson 6.)

Explanation:

Step1: Analyze rotation and reflection properties for question 5a

Point $P$ and the intersection of $l$ and $m$ are two possible locations. When $Q = P$, rotating $Q$ 180 - degrees around $P$ results in the same point, and reflecting over $m$ (since $P$ lies on $m$) also results in the same point. When $Q$ is at the intersection of $l$ and $m$, rotating 180 - degrees around $P$ and reflecting over $m$ (which passes through the intersection) have the same effect.

Step2: Answer question 5b

No. Consider a point $Q$ that is not on the line passing through $P$ and perpendicular to $m$. A 180 - degree rotation around $P$ and a reflection over $m$ will result in different points.

Step3: Analyze question 6

Since it is a sequence of rigid - transformations (translations, rotations, reflections), we can use the fact that the relative position of $D$ with respect to $A$, $B$, and $C$ is the same as the relative position of $D'$ with respect to $A'$, $B'$, and $C'$. We can find $D'$ by using vector or coordinate - like relationships based on the transformation of the known points. But without specific coordinate information, we can use a geometric construction. Connect the points $A$, $B$, $C$, $D$ to form a polygon and its image $A'$, $B'$, $C'$. Then, based on the shape and orientation of the first polygon and its image, we can locate $D'$.

Step4: Analyze question 7a and 7b

If two lines $l$ and $m$ are perpendicular to the same line $n$, then by the definition of perpendicular lines, the measure of the angle where line $l$ meets line $n$ is $90^{\circ}$, and the measure of the angle where line $m$ meets line $n$ is also $90^{\circ}$.

Answer:

a. 5a: Point $P$, the intersection of $l$ and $m$
5b: No
6: Locate $D'$ based on the relative position of $D$ with respect to $A$, $B$, $C$ and the transformation of $A$ to $A'$, $B$ to $B'$, $C$ to $C'$ (geometric construction required)
7a: $90^{\circ}$
7b: $90^{\circ}$