Geometry worksheet focusing on calculating missing angles formed by parallel lines and transversals.

To solve 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.

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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 between the parallel lines.
- Therefore, \(a = 51^\circ\).

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

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Diagram 2:

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

Solution:
- The angle \(b\) is a corresponding angle to the given \(104^\circ\) angle.
\[
b = 104^\circ
\]
- The angle \(c\) is a corresponding angle to the given \(117^\circ\) angle.
\[
c = 117^\circ
\]

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

---

Diagram 3:

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

Solution:
- The angle \(d\) is an alternate interior angle to the given \(75^\circ\) angle.
\[
d = 75^\circ
\]
- The angle \(e\) is a supplementary angle to the given \(67^\circ\) angle (since they form a linear pair).
\[
e = 180^\circ - 67^\circ = 113^\circ
\]

Answers: \(d = 75^\circ\), \(e = 113^\circ\)

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Diagram 4:

- Given: A triangle with one angle \(g\) and another angle formed by a transversal.
- To find: \(f\) and \(g\)

Solution:
- The angle \(f\) is an alternate interior angle to the given \(g\) angle.
\[
f = g
\]
- In the triangle, the sum of the interior angles is \(180^\circ\). The triangle has one right angle (\(90^\circ\)) and another angle that is supplementary to \(f\):
\[
g + 90^\circ + (180^\circ - f) = 180^\circ
\]
Since \(f = g\), we have:
\[
g + 90^\circ + (180^\circ - g) = 180^\circ
\]
Simplifying:
\[
g = 45^\circ
\]
Therefore:
\[
f = 45^\circ
\]

Answers: \(f = 45^\circ\), \(g = 45^\circ\)

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Diagram 5:

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

Solution:
- The angle \(h\) is a corresponding angle to the given \(124^\circ\) angle.
\[
h = 124^\circ
\]
- The angle \(i\) is an alternate interior angle to the given \(124^\circ\) angle.
\[
i = 124^\circ
\]

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

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Diagram 6:

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

Solution:
- The third angle of the triangle is:
\[
180^\circ - 41^\circ - 119^\circ = 20^\circ
\]
- The angle \(j\) is a corresponding angle to the \(20^\circ\) angle.
\[
j = 20^\circ
\]
- The angle \(k\) is a supplementary angle to the \(41^\circ\) angle.
\[
k = 180^\circ - 41^\circ = 139^\circ
\]
- The angle \(l\) is a corresponding angle to the \(119^\circ\) angle.
\[
l = 119^\circ
\]

Answers: \(j = 20^\circ\), \(k = 139^\circ\), \(l = 119^\circ\)

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Final Answers:

\[
\boxed{
\begin{aligned}
& a = 51^\circ, \quad b = 104^\circ, \quad c = 117^\circ, \\
& d = 75^\circ, \quad e = 113^\circ, \quad f = 45^\circ, \quad g = 45^\circ, \\
& h = 124^\circ, \quad i = 124^\circ, \quad j = 20^\circ, \quad k = 139^\circ, \quad l = 119^\circ
\end{aligned}
}
\]