1.1.Determine whether the regular pentagon has reflection symmetry, rotation symmetry, both, or neither. If it has reflection symmetry, state the number of axes of symmetry. If it has rotation symmetry, state the angle of rotation. For each type of symmetry, explain how you can tell that the figure does or does not have the given symmetry.

2.2.PQRS is a trapezoid. U is the midpoint of , and T is the midpoint of .
a.)Find the length of . Show your work.
b.)If RS is 42, what is the length of ? Explain how you calculated your answer.

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1.1. A regular pentagon has both reflection symmetry and rotation symmetry.

Reflection Symmetry:
To determine if a regular pentagon has reflection symmetry, we can imagine drawing a line (axis of symmetry) through the center of the pentagon that divides it into two congruent halves. If the reflection of one half over the axis of symmetry perfectly coincides with the other half, then the pentagon has reflection symmetry.

A regular pentagon has 5 equal sides and 5 equal angles. By drawing lines from each vertex to the center, we can see that there are 5 lines of symmetry passing through the center that divide the pentagon into 2 congruent halves. Therefore, a regular pentagon has 5 axes of reflection symmetry.

Rotation Symmetry:
To determine if a regular pentagon has rotation symmetry, we need to check if it can be rotated by a certain angle and still have the same appearance as the original pentagon.

A regular pentagon can be rotated by 72 degrees (360 degrees divided by 5) about its center and still appear the same. Therefore, a regular pentagon has rotation symmetry with an angle of rotation of 72 degrees.

2.2.

a.) To find the length of ST, we need information about the trapezoid PQRS. Without any given values or specific geometric properties mentioned, we cannot determine the length of ST.

b.) If RS is 42, and U is the midpoint of PS while T is the midpoint of QR, then we can deduce that PT is half the length of QR. Similarly, US is half the length of PS. Therefore, if RS is 42, QR is 2 times RS (42 * 2 = 84), and PT is half of QR (84 / 2 = 42). Thus, the length of PT is 42.

1. To determine the symmetry of a regular pentagon, we need to analyze reflection symmetry and rotation symmetry separately.

a) Reflection Symmetry:
- A figure has reflection symmetry if it can be reflected along a line and still appear the same.
- To determine if the regular pentagon has reflection symmetry, we need to check if there is a line through which we can fold the pentagon so that both sides will match.
- In the case of a regular pentagon, it does have reflection symmetry.
- The number of axes of symmetry for a regular pentagon is 5.
- To determine this, draw lines from the center of the pentagon to each vertex. You will notice that there are 5 lines which divide the pentagon into 2 equal halves. Each of these lines is an axis of symmetry.

b) Rotation Symmetry:
- A figure has rotation symmetry if it can be rotated by a certain angle and still appear the same.
- To determine if the regular pentagon has rotation symmetry, we need to check if we can rotate the pentagon by any angle and have it match up perfectly with the original shape.
- In the case of a regular pentagon, it does have rotation symmetry.
- The angle of rotation for a regular pentagon is 72 degrees.
- To determine this, divide 360 degrees (a full revolution) by the number of sides of the pentagon, which is 5. The result is 72 degrees.

2. Now let's move on to the trapezoid PQRS.

a) To find the length of , you need to apply the concept of midpoint formula.
- Let A be the point where the trapezoid PQRS intersects with the line segment UT.
- Since U is the midpoint of , the distance from P to U is equal to the distance from U to R.
- Similarly, since T is the midpoint of , the distance from S to T is equal to the distance from T to Q.
- Thus, by using the midpoint formula, you can find the length of :

= sqrt((S-T)^2 + (T-R)^2)

b) If RS is 42, we can find the length of by using the same concept of midpoint formula as explained in part a.
- Since U is the midpoint of , the distance from P to U is equal to the distance from U to R.
- So, we have:

PU = UR = RS / 2

- Hence, the length of will be half of RS.
- Therefore, the length of is 42 / 2 = 21.