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

Problem: Solving Angles on Parallel Lines

The task involves calculating missing angles in various diagrams involving parallel lines and transversals. We will use the properties of corresponding angles, alternate angles, interior angles, vertically opposite angles, and angles on a straight line to solve the problems.

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Section A

#### 1. Angle \( a \)

- Diagram: Two parallel lines with a transversal.
- Given: One angle is \( 48^\circ \).
- Reason: The angle \( a \) is a corresponding angle to the given \( 48^\circ \) angle.
- Solution: Corresponding angles are equal.
\[
a = 48^\circ
\]

#### 2. Angle \( b \)

- Diagram: Two parallel lines with a transversal.
- Given: One angle is \( 52^\circ \).
- Reason: The angle \( b \) is an interior angle with the given \( 52^\circ \) angle.
- Solution: Interior angles on the same side of the transversal add up to \( 180^\circ \).
\[
b + 52^\circ = 180^\circ \implies b = 180^\circ - 52^\circ = 128^\circ
\]

#### 3. Angle \( c \)

- Diagram: Two parallel lines with a transversal.
- Given: One angle is \( 65^\circ \).
- Reason: The angle \( c \) is a corresponding angle to the given \( 65^\circ \) angle.
- Solution: Corresponding angles are equal.
\[
c = 65^\circ
\]

#### 4. Angle \( d \)

- Diagram: Two parallel lines with a transversal.
- Given: One angle is \( 71^\circ \).
- Reason: The angle \( d \) is an alternate angle to the given \( 71^\circ \) angle.
- Solution: Alternate angles are equal.
\[
d = 71^\circ
\]

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Section B

#### 1. Angle \( a \)

- Diagram: Two parallel lines with a transversal.
- Given: One angle is \( 108^\circ \).
- Reason: The angle \( a \) is an interior angle with the given \( 108^\circ \) angle.
- Solution: Interior angles on the same side of the transversal add up to \( 180^\circ \).
\[
a + 108^\circ = 180^\circ \implies a = 180^\circ - 108^\circ = 72^\circ
\]

#### 2. Angle \( b \)

- Diagram: Two parallel lines with a transversal.
- Given: One angle is \( 43^\circ \).
- Reason: The angle \( b \) is a corresponding angle to the given \( 43^\circ \) angle.
- Solution: Corresponding angles are equal.
\[
b = 43^\circ
\]

#### 3. Angle \( c \)

- Diagram: Two intersecting lines.
- Given: One angle is \( 114^\circ \).
- Reason: The angle \( c \) is a vertically opposite angle to the given \( 114^\circ \) angle.
- Solution: Vertically opposite angles are equal.
\[
c = 114^\circ
\]

#### 4. Angle \( d \)

- Diagram: Two parallel lines with a transversal.
- Given: One angle is \( 79^\circ \).
- Reason: The angle \( d \) is an alternate angle to the given \( 79^\circ \) angle.
- Solution: Alternate angles are equal.
\[
d = 79^\circ
\]

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Final Answers

- Section A:
- Angle \( a \): \( 48^\circ \) (Corresponding angle)
- Angle \( b \): \( 128^\circ \) (Interior angles add to \( 180^\circ \))
- Angle \( c \): \( 65^\circ \) (Corresponding angle)
- Angle \( d \): \( 71^\circ \) (Alternate angle)

- Section B:
- Angle \( a \): \( 72^\circ \) (Interior angles add to \( 180^\circ \))
- Angle \( b \): \( 43^\circ \) (Corresponding angle)
- Angle \( c \): \( 114^\circ \) (Vertically opposite angle)
- Angle \( d \): \( 79^\circ \) (Alternate angle)

\[
\boxed{
\begin{array}{ll}
\text{Section A:} & \text{Angle } a = 48^\circ, \text{Angle } b = 128^\circ, \text{Angle } c = 65^\circ, \text{Angle } d = 71^\circ \\
\text{Section B:} & \text{Angle } a = 72^\circ, \text{Angle } b = 43^\circ, \text{Angle } c = 114^\circ, \text{Angle } d = 79^\circ
\end{array}
}
\]