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:
- \( \angle a \) and \( \angle a' \) are corresponding angles
- \( \angle b \) and \( \angle b' \) are corresponding angles
- \( \angle c \) and \( \angle c' \) are corresponding angles
- \( \angle d \) and \( \angle d' \) are corresponding angles
Thus:
\[ m\angle a = m\angle a', \quad m\angle b = m\angle b', \quad m\angle c = m\angle c', \quad m\angle d = m\angle d' \]
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:
- \( \angle d \) and \( \angle b' \) are alternate interior angles
- \( \angle c \) and \( \angle a' \) are alternate interior angles
Thus:
\[ m\angle d = m\angle b', \quad m\angle c = m\angle a' \]
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:
- \( \angle a \) and \( \angle c' \) are alternate exterior angles
- \( \angle b \) and \( \angle d' \) are alternate exterior angles
Thus:
\[ m\angle a = m\angle c', \quad m\angle b = m\angle d' \]
Fig.3 - Alternate exterior angles are congruent
Practice Problems & Solutions
Problem 1 & Solution
Problem: Given that \( L_1 \parallel L_2 \), find the measures of angles \( b, c, d, e, f, g, \) and \( h \).
Fig.4 - Find all unknown angles
Solution:
Given: \( m\angle a = 74^\circ \), \( L_1 \parallel L_2 \)
Step 1: \( \angle a \) and \( \angle b \) are supplementary angles (linear pair):
\[ m\angle a + m\angle b = 180^\circ \implies 74^\circ + m\angle b = 180^\circ \implies m\angle b = 106^\circ \]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: \( \angle b = 106^\circ, \angle c = 74^\circ, \angle d = 106^\circ, \angle e = 106^\circ, \angle f = 74^\circ, \angle g = 74^\circ, \angle h = 106^\circ \)
Problem 2 & Solution
Problem: In the figure below, \( L_1 \parallel L_2 \) and \( L_3 \parallel L_4 \), forming a parallelogram. Show that:
- Opposite angles are congruent: \( m\angle d = m\angle e' \) and \( m\angle g = m\angle c' \)
- Consecutive angles are supplementary: \( \angle d \) and \( \angle g \) are supplementary, \( \angle c' \) and \( \angle e' \) are supplementary.
Fig.5 - Parallelogram formed by two pairs of parallel lines
Solution:
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:
\[ m\angle c' + m\angle e' = 180^\circ \quad \text{(supplementary)} \]