Calculating Angles on Parallel Lines with Transversals (C ... - Free Printable

To solve the problem of finding the missing angles in the given diagrams, we will use properties of parallel lines and transversals, as well as basic angle relationships (such as supplementary, complementary, and corresponding angles). Let's go through each diagram step by step.

---

Diagram 1:

- Given: \(51^\circ\)
- To find: \(a\)

#### Solution:
- The angle \(a\) is a corresponding angle to the given \(51^\circ\) angle because they are on the same side of the transversal and above the parallel lines.
- Therefore, \(a = 51^\circ\).

Answer: \(a = 51^\circ\)

---

Diagram 2:

- Given: \(104^\circ\) and \(117^\circ\)
- To find: \(b\) and \(c\)

#### Solution:
1. Finding \(b\):
- The angle \(b\) is a vertical angle to the given \(104^\circ\) angle.
- Vertical angles are equal, so \(b = 104^\circ\).

2. Finding \(c\):
- The angle \(c\) is a corresponding angle to the given \(117^\circ\) angle because they are on the same side of the transversal and below the parallel lines.
- Therefore, \(c = 117^\circ\).

Answers: \(b = 104^\circ\), \(c = 117^\circ\)

---

Diagram 3:

- Given: \(67^\circ\) and \(75^\circ\)
- To find: \(d\) and \(e\)

#### Solution:
1. Finding \(d\):
- The angle \(d\) is an alternate interior angle to the given \(67^\circ\) angle because they are on opposite sides of the transversal and between the parallel lines.
- Therefore, \(d = 67^\circ\).

2. Finding \(e\):
- The angle \(e\) is a corresponding angle to the given \(75^\circ\) angle because they are on the same side of the transversal and above the parallel lines.
- Therefore, \(e = 75^\circ\).

Answers: \(d = 67^\circ\), \(e = 75^\circ\)

---

Diagram 4:

- Given: A triangle with one exterior angle labeled \(f\) and one interior angle labeled \(g\).
- To find: \(f\) and \(g\)

#### Solution:
1. Finding \(g\):
- The angle \(g\) is an alternate interior angle to the given \(67^\circ\) angle because they are on opposite sides of the transversal and between the parallel lines.
- Therefore, \(g = 67^\circ\).

2. Finding \(f\):
- The angle \(f\) is the exterior angle of the triangle formed by the transversal and the parallel lines.
- The exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles.
- Here, the two non-adjacent interior angles are \(67^\circ\) and \(75^\circ\).
- Therefore, \(f = 67^\circ + 75^\circ = 142^\circ\).

Answers: \(f = 142^\circ\), \(g = 67^\circ\)

---

Diagram 5:

- Given: \(124^\circ\)
- To find: \(h\) and \(i\)

#### Solution:
1. Finding \(h\):
- The angle \(h\) is a corresponding angle to the given \(124^\circ\) angle because they are on the same side of the transversal and above the parallel lines.
- Therefore, \(h = 124^\circ\).

2. Finding \(i\):
- The angle \(i\) is the interior angle adjacent to the given \(124^\circ\) angle.
- Since they form a linear pair, their sum is \(180^\circ\).
- Therefore, \(i = 180^\circ - 124^\circ = 56^\circ\).

Answers: \(h = 124^\circ\), \(i = 56^\circ\)

---

Diagram 6:

- Given: \(41^\circ\) and \(119^\circ\)
- To find: \(j\), \(k\), and \(l\)

#### Solution:
1. Finding \(j\):
- The angle \(j\) is an alternate interior angle to the given \(41^\circ\) angle because they are on opposite sides of the transversal and between the parallel lines.
- Therefore, \(j = 41^\circ\).

2. Finding \(k\):
- The angle \(k\) is the interior angle adjacent to the given \(119^\circ\) angle.
- Since they form a linear pair, their sum is \(180^\circ\).
- Therefore, \(k = 180^\circ - 119^\circ = 61^\circ\).

3. Finding \(l\):
- The angle \(l\) is a corresponding angle to the given \(41^\circ\) angle because they are on the same side of the transversal and above the parallel lines.
- Therefore, \(l = 41^\circ\).

Answers: \(j = 41^\circ\), \(k = 61^\circ\), \(l = 41^\circ\)

---

Final Answers:

\[
\boxed{
\begin{aligned}
& a = 51^\circ, \quad b = 104^\circ, \quad c = 117^\circ, \\
& d = 67^\circ, \quad e = 75^\circ, \\
& f = 142^\circ, \quad g = 67^\circ, \\
& h = 124^\circ, \quad i = 56^\circ, \\
& j = 41^\circ, \quad k = 61^\circ, \quad l = 41^\circ.
\end{aligned}
}
\]