Angles in Parallel Lines and Transversals

When a transversal line intersects two parallel lines, it creates specific angle relationships that are always consistent. This tutorial explores these relationships with diagrams, definitions, and solved examples.

You may also want to solve additional problems on angles in parallel lines and transversals.

Corresponding Angles

When two parallel lines are cut by a transversal, the angles in matching corners are called corresponding angles and are congruent:

Thus:

\[ m\angle a = m\angle a', \quad m\angle b = m\angle b', \quad m\angle c = m\angle c', \quad m\angle d = m\angle d' \]

Diagram showing corresponding angles a-a', b-b', c-c', d-d' formed by two parallel lines and a transversal
Fig.1 - Corresponding angles formed by parallel lines and a transversal

Alternate Interior Angles

Alternate interior angles lie on opposite sides of the transversal and between the two parallel lines. They are always congruent:

Thus:

\[ m\angle d = m\angle b', \quad m\angle c = m\angle a' \]

Diagram highlighting alternate interior angles d and b', c and a' between parallel lines
Fig.2 - Alternate interior angles are congruent

Alternate Exterior Angles

Alternate exterior angles lie on opposite sides of the transversal and outside the two parallel lines. They are always congruent:

Thus:

\[ m\angle a = m\angle c', \quad m\angle b = m\angle d' \]

Diagram highlighting alternate exterior angles a and c', b and d' outside parallel lines
Fig.3 - Alternate exterior angles are congruent

Practice Problems

Problem 1

Given that \( L_1 \parallel L_2 \), find the measures of angles \( b, c, d, e, f, g, \) and \( h \).

Parallel lines L1 and L2 cut by transversal with angle a=74°, find other angles
Fig.4 - Find all unknown angles

Problem 2

In the figure below, \( L_1 \parallel L_2 \) and \( L_3 \parallel L_4 \), forming a parallelogram. Show that:

  1. Opposite angles are congruent: \( m\angle d = m\angle e' \) and \( m\angle g = m\angle c' \)
  2. Consecutive angles are supplementary: \( \angle d \) and \( \angle g \) are supplementary, \( \angle c' \) and \( \angle e' \) are supplementary.
Two pairs of parallel lines forming a parallelogram with various angles labeled
Fig.5 - Parallelogram formed by two pairs of parallel lines

Solutions

Solution to Problem 1

Given: \( m\angle a = 74^\circ \), \( L_1 \parallel L_2 \)

Step 1: \( \angle a \) and \( \angle b \) are supplementary angles (linear pair):

\[ m\angle a + m\angle b = 180^\circ \Rightarrow 74^\circ + m\angle b = 180^\circ \Rightarrow m\angle b = 106^\circ \]

Step 2: \( \angle b \) and \( \angle d \) are vertical angles, so:

\[ m\angle d = m\angle b = 106^\circ \]

Step 3: \( \angle a \) and \( \angle c \) are vertical angles, so:

\[ m\angle c = m\angle a = 74^\circ \]

Step 4: Using corresponding angles (since \( L_1 \parallel L_2 \)):

\[ m\angle f = m\angle a = 74^\circ,\quad m\angle e = m\angle b = 106^\circ \]

\[ m\angle g = m\angle c = 74^\circ,\quad m\angle h = m\angle d = 106^\circ \]

Answer: \( \angle b = 106^\circ, \angle c = 74^\circ, \angle d = 106^\circ, \angle e = 106^\circ, \angle f = 74^\circ, \angle g = 74^\circ, \angle h = 106^\circ \)

Solution to Problem 2

Part 1: Show opposite angles are congruent

Since \( L_1 \parallel L_2 \) and \( L_3 \) is a transversal, corresponding angles give:

\[ m\angle d = m\angle h \]

Since \( L_1 \parallel L_2 \) and \( L_4 \) is a transversal, corresponding angles give:

\[ m\angle d' = m\angle h' \]

Since \( L_3 \parallel L_4 \) and \( L_1 \) is a transversal, corresponding angles give:

\[ m\angle d = m\angle d' \]

Since \( L_3 \parallel L_4 \) and \( L_2 \) is a transversal, corresponding angles give:

\[ m\angle h = m\angle h' \]

Thus \( m\angle d = m\angle h' \). But \( \angle h' \) and \( \angle e' \) are vertical angles, so \( m\angle h' = m\angle e' \). Therefore:

\[ m\angle d = m\angle e' \]

Similarly, \( m\angle g = m\angle a' \). Since \( \angle a' \) and \( \angle c' \) are vertical angles, \( m\angle a' = m\angle c' \), so:

\[ m\angle g = m\angle c' \]

Part 2: Show consecutive angles are supplementary

Angles \( \angle a \) and \( \angle d \) are supplementary (linear pair). But \( \angle a \) and \( \angle g \) are corresponding angles (since \( L_1 \parallel L_2 \)), so \( m\angle a = m\angle g \). Thus:

\[ m\angle d + m\angle g = 180^\circ \quad \text{(supplementary)} \]

Similarly, \( \angle b' \) and \( \angle c' \) are supplementary, and \( \angle b' \) and \( \angle e' \) are corresponding angles, so \( m\angle b' = m\angle e' \). Thus:

\[ m\angle c' + m\angle e' = 180^\circ \quad \text{(supplementary)} \]

More practice problems on parallel lines and transversals are available.

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