Complementary & Supplementary Angles worksheet with geometric diagrams and algebraic expressions.

To solve the problems involving complementary and supplementary angles, we need to use the following definitions:

1. Complementary Angles: Two angles are complementary if their measures add up to 90°.
\[
\text{If } \angle A + \angle B = 90^\circ, \text{ then } \angle A \text{ and } \angle B \text{ are complementary.}
\]

2. Supplementary Angles: Two angles are supplementary if their measures add up to 180°.
\[
\text{If } \angle A + \angle B = 180^\circ, \text{ then } \angle A \text{ and } \angle B \text{ are supplementary.}
\]

Let's solve each problem step by step.

---

Problem 1:

[Image shows two angles forming a right angle. One angle is labeled \( x \), and the other is labeled \( 35^\circ \).]

#### Solution:
The two angles form a right angle, so they are complementary.
\[
x + 35^\circ = 90^\circ
\]
Solve for \( x \):
\[
x = 90^\circ - 35^\circ = 55^\circ
\]

Answer:
\[
\boxed{55^\circ}
\]

---

Problem 2:

[Image shows two angles forming a straight line. One angle is labeled \( x \), and the other is labeled \( 40^\circ \).]

#### Solution:
The two angles form a straight line, so they are supplementary.
\[
x + 40^\circ = 180^\circ
\]
Solve for \( x \):
\[
x = 180^\circ - 40^\circ = 140^\circ
\]

Answer:
\[
\boxed{140^\circ}
\]

---

Problem 3:

[Image shows two angles forming a right angle. One angle is labeled \( x \), and the other is labeled \( 60^\circ \).]

#### Solution:
The two angles form a right angle, so they are complementary.
\[
x + 60^\circ = 90^\circ
\]
Solve for \( x \):
\[
x = 90^\circ - 60^\circ = 30^\circ
\]

Answer:
\[
\boxed{30^\circ}
\]

---

Problem 4:

[Image shows two angles forming a straight line. One angle is labeled \( x \), and the other is labeled \( 75^\circ \).]

#### Solution:
The two angles form a straight line, so they are supplementary.
\[
x + 75^\circ = 180^\circ
\]
Solve for \( x \):
\[
x = 180^\circ - 75^\circ = 105^\circ
\]

Answer:
\[
\boxed{105^\circ}
\]

---

Problem 5:

[Image shows two angles forming a right angle. One angle is labeled \( (x - 10)^\circ \), and the other is labeled \( 35^\circ \).]

#### Solution:
The two angles form a right angle, so they are complementary.
\[
(x - 10)^\circ + 35^\circ = 90^\circ
\]
Solve for \( x \):
\[
x - 10 + 35 = 90
\]
\[
x + 25 = 90
\]
\[
x = 90 - 25 = 65
\]

Answer:
\[
\boxed{65^\circ}
\]

---

Problem 6:

[Image shows two angles forming a straight line. One angle is labeled \( (2x + 15)^\circ \), and the other is labeled \( 125^\circ \).]

#### Solution:
The two angles form a straight line, so they are supplementary.
\[
(2x + 15)^\circ + 125^\circ = 180^\circ
\]
Solve for \( x \):
\[
2x + 15 + 125 = 180
\]
\[
2x + 140 = 180
\]
\[
2x = 180 - 140
\]
\[
2x = 40
\]
\[
x = \frac{40}{2} = 20
\]

Answer:
\[
\boxed{20^\circ}
\]

---

Problem 7:

[Image shows two angles forming a right angle. One angle is labeled \( x \), and the other is labeled \( 2x \).]

#### Solution:
The two angles form a right angle, so they are complementary.
\[
x + 2x = 90^\circ
\]
Solve for \( x \):
\[
3x = 90^\circ
\]
\[
x = \frac{90^\circ}{3} = 30^\circ
\]

Answer:
\[
\boxed{30^\circ}
\]

---

Problem 8:

[Image shows two angles forming a straight line. One angle is labeled \( x \), and the other is labeled \( 3x \).]

#### Solution:
The two angles form a straight line, so they are supplementary.
\[
x + 3x = 180^\circ
\]
Solve for \( x \):
\[
4x = 180^\circ
\]
\[
x = \frac{180^\circ}{4} = 45^\circ
\]

Answer:
\[
\boxed{45^\circ}
\]

---

Final Answers:

1. \( \boxed{55^\circ} \)
2. \( \boxed{140^\circ} \)
3. \( \boxed{30^\circ} \)
4. \( \boxed{105^\circ} \)
5. \( \boxed{65^\circ} \)
6. \( \boxed{20^\circ} \)
7. \( \boxed{30^\circ} \)
8. \( \boxed{45^\circ} \)