This Grade VII worksheet helps students practice identifying angle pairs and calculating unknown values in geometric figures.

Problem Analysis and Solution

The image contains a worksheet on "Lines and Angles" with four problems. Let's solve each problem step by step.

---

Problem 1: Identify the following pairs of angles as complementary, supplementary, or equal.

#### Definitions:
- Complementary angles: Two angles whose measures add up to \(90^\circ\).
- Supplementary angles: Two angles whose measures add up to \(180^\circ\).
- Equal angles: Two angles with the same measure.

#### Given pairs:
a. \(15^\circ\) and \(75^\circ\)
b. \(133^\circ\) and \(133^\circ\)
c. \(90^\circ\) and \(90^\circ\)
d. \(45^\circ\) and \(5^\circ\)

#### Solutions:
1. Pair (a): \(15^\circ\) and \(75^\circ\)
- Sum: \(15^\circ + 75^\circ = 90^\circ\)
- These are complementary angles.

2. Pair (b): \(133^\circ\) and \(133^\circ\)
- Both angles have the same measure.
- These are equal angles.

3. Pair (c): \(90^\circ\) and \(90^\circ\)
- Both angles have the same measure.
- These are equal angles.

4. Pair (d): \(45^\circ\) and \(5^\circ\)
- Sum: \(45^\circ + 5^\circ = 50^\circ\)
- These are neither complementary nor supplementary. They are simply two angles.

#### Final Answer for Problem 1:
\[
\boxed{
\text{a. Complementary, b. Equal, c. Equal, d. Neither}
}
\]

---

Problem 2: Find the measure of \(x\) in the following figures.

#### Figure (a):
- The figure shows two intersecting lines forming vertical angles.
- One angle is given as \(42^\circ\), and the other angle is labeled \(x\).
- Vertical angles are equal.

\[
x = 42^\circ
\]

#### Figure (b):
- The figure shows a straight line with two angles labeled \(x\) and \(x + 105^\circ\).
- The sum of angles on a straight line is \(180^\circ\).

\[
x + (x + 105^\circ) = 180^\circ
\]
\[
2x + 105^\circ = 180^\circ
\]
\[
2x = 75^\circ
\]
\[
x = 37.5^\circ
\]

#### Figure (c):
- The figure shows a circle divided into three angles: \(90^\circ\), \(\frac{x}{2}\), and \(x\).
- The sum of angles around a point is \(360^\circ\).

\[
90^\circ + \frac{x}{2} + x = 360^\circ
\]
\[
90^\circ + \frac{x}{2} + \frac{2x}{2} = 360^\circ
\]
\[
90^\circ + \frac{3x}{2} = 360^\circ
\]
\[
\frac{3x}{2} = 270^\circ
\]
\[
3x = 540^\circ
\]
\[
x = 180^\circ
\]

#### Final Answers for Problem 2:
\[
\boxed{
\text{a. } 42^\circ, \text{ b. } 37.5^\circ, \text{ c. } 180^\circ
}
\]

---

Problem 3: An angle is \(30^\circ\) less than two times its supplement. Find the angles.

#### Definitions:
- The supplement of an angle \(y\) is \(180^\circ - y\).

#### Let the angle be \(y\).
- Its supplement is \(180^\circ - y\).
- According to the problem, the angle \(y\) is \(30^\circ\) less than two times its supplement.

\[
y = 2(180^\circ - y) - 30^\circ
\]
\[
y = 360^\circ - 2y - 30^\circ
\]
\[
y = 330^\circ - 2y
\]
\[
y + 2y = 330^\circ
\]
\[
3y = 330^\circ
\]
\[
y = 110^\circ
\]

#### The supplement of \(y\) is:
\[
180^\circ - y = 180^\circ - 110^\circ = 70^\circ
\]

#### Final Answer for Problem 3:
\[
\boxed{110^\circ \text{ and } 70^\circ}
\]

---

Problem 4: Find the value of \(x\) from the given figure.

#### Figure:
- The figure shows two angles that are vertically opposite.
- One angle is labeled \(x^\circ + 15^\circ\), and the other angle is not explicitly labeled but is equal to the first angle (since they are vertically opposite).

#### Since vertically opposite angles are equal:
\[
x^\circ + 15^\circ = x^\circ
\]

This equation suggests there might be a missing detail or another relationship in the figure. However, based on the information provided, we can only conclude that the angles are equal.

If additional context or another relationship is intended, please provide more details. Otherwise, the problem as stated does not have a unique solution beyond the equality of the angles.

#### Final Answer for Problem 4 (assuming no additional context):
\[
\boxed{\text{Insufficient information}}
\]

---

Final Summary of All Answers:

1. \(\boxed{\text{a. Complementary, b. Equal, c. Equal, d. Neither}}\)
2. \(\boxed{\text{a. } 42^\circ, \text{ b. } 37.5^\circ, \text{ c. } 180^\circ}\)
3. \(\boxed{110^\circ \text{ and } 70^\circ}\)
4. \(\boxed{\text{Insufficient information}}\)