Parallel Lines cut by a Transversal - Printable Missing Angle Worksheets w Key - Free Printable

To solve the "Angle Puzzle #1," we need to use our knowledge of angles, triangles, and parallel lines. Let's break down the problem step by step.

Step 1: Understand the given information

- The diagram shows several intersecting lines and triangles.
- There are labeled angles with some values provided (e.g., \( \angle A = 50^\circ \), \( \angle B = 60^\circ \), etc.).
- Some angles are marked as corresponding, alternate interior, or supplementary angles due to the presence of parallel lines.

Step 2: Identify key properties and relationships

1. Parallel Lines:
- Lines \( p \) and \( q \) are parallel.
- Lines \( q \) and \( r \) are parallel.
- This means that corresponding angles, alternate interior angles, and consecutive interior angles will have specific relationships.

2. Triangles:
- The sum of the interior angles in any triangle is \( 180^\circ \).

3. Supplementary Angles:
- Angles on a straight line sum to \( 180^\circ \).

4. Vertical Angles:
- Vertical angles are congruent.

Step 3: Solve for each angle

#### Part 1: Triangle \( \triangle ABC \)
- Given: \( \angle A = 50^\circ \) and \( \angle B = 60^\circ \).
- Use the triangle angle sum property:
\[
\angle A + \angle B + \angle C = 180^\circ
\]
\[
50^\circ + 60^\circ + \angle C = 180^\circ
\]
\[
\angle C = 180^\circ - 110^\circ = 70^\circ
\]

#### Part 2: Parallel Lines and Transversals
- Since \( p \parallel q \) and \( q \parallel r \), we can use properties of parallel lines and transversals.

##### Angles involving \( p \parallel q \):
- \( \angle D \) and \( \angle E \) are corresponding angles to \( \angle A \) and \( \angle B \), respectively.
\[
\angle D = \angle A = 50^\circ
\]
\[
\angle E = \angle B = 60^\circ
\]

##### Angles involving \( q \parallel r \):
- \( \angle F \) and \( \angle G \) are corresponding angles to \( \angle D \) and \( \angle E \), respectively.
\[
\angle F = \angle D = 50^\circ
\]
\[
\angle G = \angle E = 60^\circ
\]

##### Angles involving vertical angles:
- \( \angle H \) is a vertical angle to \( \angle F \):
\[
\angle H = \angle F = 50^\circ
\]

#### Part 3: Solve for remaining angles
- Use the fact that angles on a straight line sum to \( 180^\circ \):
- For \( \angle I \):
\[
\angle I = 180^\circ - \angle G = 180^\circ - 60^\circ = 120^\circ
\]

- For \( \angle J \):
\[
\angle J = 180^\circ - \angle H = 180^\circ - 50^\circ = 130^\circ
\]

- For \( \angle K \):
\[
\angle K = 180^\circ - \angle J = 180^\circ - 130^\circ = 50^\circ
\]

- For \( \angle L \):
\[
\angle L = 180^\circ - \angle F = 180^\circ - 50^\circ = 130^\circ
\]

- For \( \angle M \):
\[
\angle M = 180^\circ - \angle L = 180^\circ - 130^\circ = 50^\circ
\]

- For \( \angle N \):
\[
\angle N = 180^\circ - \angle G = 180^\circ - 60^\circ = 120^\circ
\]

- For \( \angle O \):
\[
\angle O = 180^\circ - \angle N = 180^\circ - 120^\circ = 60^\circ
\]

- For \( \angle P \):
\[
\angle P = 180^\circ - \angle O = 180^\circ - 60^\circ = 120^\circ
\]

- For \( \angle Q \):
\[
\angle Q = 180^\circ - \angle P = 180^\circ - 120^\circ = 60^\circ
\]

- For \( \angle R \):
\[
\angle R = 180^\circ - \angle Q = 180^\circ - 60^\circ = 120^\circ
\]

- For \( \angle S \):
\[
\angle S = 180^\circ - \angle R = 180^\circ - 120^\circ = 60^\circ
\]

Final Answer

\[
\boxed{
\begin{aligned}
&\angle A = 50^\circ, \angle B = 60^\circ, \angle C = 70^\circ, \\
&\angle D = 50^\circ, \angle E = 60^\circ, \\
&\angle F = 50^\circ, \angle G = 60^\circ, \\
&\angle H = 50^\circ, \angle I = 120^\circ, \angle J = 130^\circ, \angle K = 50^\circ, \\
&\angle L = 130^\circ, \angle M = 50^\circ, \angle N = 120^\circ, \angle O = 60^\circ, \\
&\angle P = 120^\circ, \angle Q = 60^\circ, \angle R = 120^\circ, \angle S = 60^\circ.
\end{aligned}
}
\]