Worksheet on angles formed by parallel lines and transversals, with problems to find the sizes of lettered angles.

To solve the problem of finding the sizes of the lettered 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 part step by step.

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Question 1: Write down the sizes of the lettered angles.

#### (a)
- The diagram shows two parallel lines with a transversal.
- The given angle is \(112^\circ\).
- The angle \(x\) is the alternate interior angle to the given \(112^\circ\) angle.
- Alternate interior angles are equal when the lines are parallel.
- Therefore, \(x = 112^\circ\).

Answer for (a): \(x = 112^\circ\)

#### (b)
- The diagram shows two parallel lines with a transversal.
- The given angle is \(75^\circ\).
- The angle \(x\) is the corresponding angle to the given \(75^\circ\) angle.
- Corresponding angles are equal when the lines are parallel.
- Therefore, \(x = 75^\circ\).

Answer for (b): \(x = 75^\circ\)

#### (c)
- The diagram shows two parallel lines with a transversal.
- The given angle is \(150^\circ\).
- The angle \(x\) is the co-interior (or consecutive interior) angle to the given \(150^\circ\) angle.
- Co-interior angles are supplementary when the lines are parallel, meaning their sum is \(180^\circ\).
- Therefore, \(x + 150^\circ = 180^\circ\).
- Solving for \(x\):
\[
x = 180^\circ - 150^\circ = 30^\circ
\]

Answer for (c): \(x = 30^\circ\)

#### (d)
- The diagram shows two perpendicular lines intersected by a transversal.
- The given angle is \(99^\circ\).
- The angle \(x\) is the adjacent angle to the given \(99^\circ\) angle on the straight line.
- Adjacent angles on a straight line are supplementary, meaning their sum is \(180^\circ\).
- Therefore, \(x + 99^\circ = 180^\circ\).
- Solving for \(x\):
\[
x = 180^\circ - 99^\circ = 81^\circ
\]
- The angle \(y\) is the vertical angle to the given \(99^\circ\) angle.
- Vertical angles are equal.
- Therefore, \(y = 99^\circ\).
- The angle \(z\) is the adjacent angle to \(y\) on the straight line.
- Therefore, \(z + 99^\circ = 180^\circ\).
- Solving for \(z\):
\[
z = 180^\circ - 99^\circ = 81^\circ
\]

Answer for (d): \(x = 81^\circ\), \(y = 99^\circ\), \(z = 81^\circ\)

#### (e)
- The diagram shows two parallel lines with a transversal.
- The given angle is \(74^\circ\).
- The angle \(x\) is the alternate interior angle to the given \(74^\circ\) angle.
- Alternate interior angles are equal when the lines are parallel.
- Therefore, \(x = 74^\circ\).
- The angle \(y\) is the corresponding angle to the given \(74^\circ\) angle.
- Corresponding angles are equal when the lines are parallel.
- Therefore, \(y = 74^\circ\).

Answer for (e): \(x = 74^\circ\), \(y = 74^\circ\)

#### (f)
- The diagram shows two parallel lines with a transversal.
- The given angles are \(123^\circ\) and \(110^\circ\).
- The angle \(x\) is the co-interior (or consecutive interior) angle to the given \(123^\circ\) angle.
- Co-interior angles are supplementary when the lines are parallel, meaning their sum is \(180^\circ\).
- Therefore, \(x + 123^\circ = 180^\circ\).
- Solving for \(x\):
\[
x = 180^\circ - 123^\circ = 57^\circ
\]
- The angle \(y\) is the alternate interior angle to the given \(110^\circ\) angle.
- Alternate interior angles are equal when the lines are parallel.
- Therefore, \(y = 110^\circ\).

Answer for (f): \(x = 57^\circ\), \(y = 110^\circ\)

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Final Answers:

\[
\boxed{
\begin{array}{ll}
(a) & x = 112^\circ \\
(b) & x = 75^\circ \\
(c) & x = 30^\circ \\
(d) & x = 81^\circ, y = 99^\circ, z = 81^\circ \\
(e) & x = 74^\circ, y = 74^\circ \\
(f) & x = 57^\circ, y = 110^\circ \\
\end{array}
}
\]