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

To solve the missing angles in the given worksheet, we need to 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 problem step by step.

---

Section A: Calculate the missing angles

#### Problem 1:
- Given: \(51^\circ\)
- To find: \(a\)

Solution:
- The angle \(a\) is a vertical angle to the given \(51^\circ\) angle.
- Vertical angles are equal.
- Therefore, \(a = 51^\circ\).

#### Problem 2:
- Given: \(117^\circ\)
- To find: \(b\) and \(c\)

Solution:
- The angle \(b\) is an alternate interior angle to the given \(117^\circ\) angle.
- Alternate interior angles are equal when the lines are parallel.
- Therefore, \(b = 117^\circ\).
- The angle \(c\) is a corresponding angle to the given \(117^\circ\) angle.
- Corresponding angles are equal when the lines are parallel.
- Therefore, \(c = 117^\circ\).

#### Problem 3:
- Given: \(67^\circ\) and \(79^\circ\)
- To find: \(d\) and \(e\)

Solution:
- The angle \(d\) is a vertical angle to the given \(67^\circ\) angle.
- Vertical angles are equal.
- Therefore, \(d = 67^\circ\).
- The angle \(e\) is an exterior angle formed by the transversal and the parallel lines.
- The exterior angle is equal to the sum of the two non-adjacent interior angles.
- Therefore, \(e = 67^\circ + 79^\circ = 146^\circ\).

#### Problem 4:
- Given: \(104^\circ\)
- To find: \(f\) and \(g\)

Solution:
- The angle \(f\) is a corresponding angle to the given \(104^\circ\) angle.
- Corresponding angles are equal when the lines are parallel.
- Therefore, \(f = 104^\circ\).
- The angle \(g\) is an alternate interior angle to the given \(104^\circ\) angle.
- Alternate interior angles are equal when the lines are parallel.
- Therefore, \(g = 104^\circ\).

#### Problem 5:
- Given: \(124^\circ\)
- To find: \(h\)

Solution:
- The angle \(h\) is a supplementary angle to the given \(124^\circ\) angle.
- Supplementary angles add up to \(180^\circ\).
- Therefore, \(h = 180^\circ - 124^\circ = 56^\circ\).

#### Problem 6:
- Given: \(43^\circ\) and \(119^\circ\)
- To find: \(i\), \(j\), \(k\), and \(l\)

Solution:
- The angle \(i\) is an alternate interior angle to the given \(43^\circ\) angle.
- Alternate interior angles are equal when the lines are parallel.
- Therefore, \(i = 43^\circ\).
- The angle \(j\) is a corresponding angle to the given \(43^\circ\) angle.
- Corresponding angles are equal when the lines are parallel.
- Therefore, \(j = 43^\circ\).
- The angle \(k\) is a vertical angle to the given \(119^\circ\) angle.
- Vertical angles are equal.
- Therefore, \(k = 119^\circ\).
- The angle \(l\) is an exterior angle formed by the transversal and the parallel lines.
- The exterior angle is equal to the sum of the two non-adjacent interior angles.
- Therefore, \(l = 43^\circ + 119^\circ = 162^\circ\).

---

Final Answers:

\[
\boxed{
\begin{aligned}
&\text{a} = 51^\circ, \quad \text{b} = 117^\circ, \quad \text{c} = 117^\circ, \\
&\text{d} = 67^\circ, \quad \text{e} = 146^\circ, \quad \text{f} = 104^\circ, \\
&\text{g} = 104^\circ, \quad \text{h} = 56^\circ, \quad \text{i} = 43^\circ, \\
&\text{j} = 43^\circ, \quad \text{k} = 119^\circ, \quad \text{l} = 162^\circ
\end{aligned}
}
\]