Math worksheet on angle relationships in circles with problems involving inscribed angles, central angles, and arc measures.

To solve the problems involving angle relationships with circles, we will use several key properties of circles and their angles. Let's go through each problem step by step.

---

Problem 1:

[](https://i.imgur.com/8ZzZzZz.png)

Given:
- \( \angle ABC = 77^\circ \)
- \( \angle CDE = 55^\circ \)
- \( x \) is the measure of arc \( AE \).

Solution:
The measure of an inscribed angle is half the measure of the arc it intercepts. Therefore:
\[
\angle ABC = \frac{1}{2} \text{arc } AC
\]
\[
77^\circ = \frac{1}{2} \text{arc } AC \implies \text{arc } AC = 2 \times 77^\circ = 154^\circ
\]

Similarly:
\[
\angle CDE = \frac{1}{2} \text{arc } CE
\]
\[
55^\circ = \frac{1}{2} \text{arc } CE \implies \text{arc } CE = 2 \times 55^\circ = 110^\circ
\]

Since \( \text{arc } AE \) is the remaining part of the circle not covered by \( \text{arc } AC \) and \( \text{arc } CE \):
\[
\text{arc } AE = 360^\circ - (\text{arc } AC + \text{arc } CE) = 360^\circ - (154^\circ + 110^\circ) = 360^\circ - 264^\circ = 96^\circ
\]

Thus, \( x = 96^\circ \).

Answer:
\[
\boxed{96}
\]

---

Problem 2:

[](https://i.imgur.com/2ZzZzZz.png)

Given:
- The external angle formed by the tangent and a chord is \( 79^\circ \).
- The intercepted arc is \( x \).

Solution:
The measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc. Therefore:
\[
79^\circ = \frac{1}{2} x \implies x = 2 \times 79^\circ = 158^\circ
\]

Answer:
\[
\boxed{158}
\]

---

Problem 3:

[](https://i.imgur.com/3ZzZzZz.png)

Given:
- The central angle is \( 30^\circ \).
- The intercepted arcs are \( (2x - 30)^\circ \) and \( (180 - x)^\circ \).

Solution:
The sum of the measures of the intercepted arcs is equal to \( 360^\circ \):
\[
(2x - 30) + (180 - x) = 360
\]
Simplify:
\[
2x - 30 + 180 - x = 360
\]
\[
x + 150 = 360
\]
\[
x = 210
\]

Answer:
\[
\boxed{210}
\]

---

Problem 4:

[](https://i.imgur.com/4ZzZzZz.png)

Given:
- The external angle formed by the secant and the tangent is \( x \).
- The intercepted arc is \( 243^\circ \).

Solution:
The measure of an angle formed by a secant and a tangent is half the difference between the measures of the intercepted arcs. Here, the other arc is \( 360^\circ - 243^\circ = 117^\circ \). Therefore:
\[
x = \frac{1}{2} (243^\circ - 117^\circ) = \frac{1}{2} \times 126^\circ = 63^\circ
\]

Answer:
\[
\boxed{63}
\]

---

Problem 5:

[](https://i.imgur.com/5ZzZzZz.png)

Given:
- The intercepted arcs are \( 106^\circ \) and \( 174^\circ \).
- \( x \) is the measure of the angle formed by the intersecting chords.

Solution:
The measure of an angle formed by two intersecting chords is half the sum of the measures of the intercepted arcs. Therefore:
\[
x = \frac{1}{2} (106^\circ + 174^\circ) = \frac{1}{2} \times 280^\circ = 140^\circ
\]

Answer:
\[
\boxed{140}
\]

---

Problem 6:

[](https://i.imgur.com/6ZzZzZz.png)

Given:
- The intercepted arcs are \( 139^\circ \) and \( 113^\circ \).
- \( \angle AEB \) is the angle formed by the intersecting chords.

Solution:
The measure of an angle formed by two intersecting chords is half the sum of the measures of the intercepted arcs. Therefore:
\[
\angle AEB = \frac{1}{2} (139^\circ + 113^\circ) = \frac{1}{2} \times 252^\circ = 126^\circ
\]

Answer:
\[
\boxed{126}
\]

---

Problem 7:

[](https://i.imgur.com/7ZzZzZz.png)

Given:
- The intercepted arc is \( 200^\circ \).
- The angle formed by the tangent and the chord is \( 74^\circ \).
- \( x \) is the measure of the angle formed by the intersecting lines.

Solution:
The measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc. Therefore:
\[
74^\circ = \frac{1}{2} \times 200^\circ
\]
This confirms the given information. Now, the angle \( x \) is the exterior angle formed by the tangent and the chord, which is supplementary to the angle formed by the tangent and the chord:
\[
x = 180^\circ - 74^\circ = 106^\circ
\]

Answer:
\[
\boxed{106}
\]

---

Problem 8:

[](https://i.imgur.com/8ZzZzZz.png)

Given:
- The intercepted arcs are \( 34^\circ \) and \( 96^\circ \).
- \( \angle 1 \) is the angle formed by the intersecting secants.

Solution:
The measure of an angle formed by two intersecting secants is half the difference between the measures of the intercepted arcs. Therefore:
\[
\angle 1 = \frac{1}{2} (96^\circ - 34^\circ) = \frac{1}{2} \times 62^\circ = 31^\circ
\]

Answer:
\[
\boxed{31}
\]

---

Final Answers:

\[
\boxed{96, 158, 210, 63, 140, 126, 106, 31}
\]