Angles in Parallel Lines and Transversals

Tutorial on angles formed when a transversal L3 intersects two parallel lines L1 and L2 (see figure 1 below). Questions with solutions are also included. You may also want to solve problems related to angles in parallel lines and transversals.
In what follows the size of measure of an angle, \( a \) for example, is written as: \( m \angle a \)

Corresponding Angles


The following are pairs of
corresponding angles : \( (a , a') \) , \( (b , b')\) , \( (c , c') \) and \( (d , d')\).(see figure 1 below).
All pairs of corresponding angles are congruent and therefore have equal measures; hence
\( m\angle a = m\angle a' \)
\( m\angle b = m\angle b' \)
\( m\angle c = m\angle c' \)
\( m\angle d = m\angle d' \)
Angles in parallel and transverse lines
Fig.1 - Angles in Parallel Lines and Transversal



Alternate Interior Angles

The pairs of alternate interior angles in the figure below are \( (d,b') \) and \( (c,a') \). The alternate interior angles are congruent and equal in measure. (see figure 2 below)
\( m\angle d = m\angle b' \)
\( m\angle c = m\angle a' \)
Alternate Interior Angles
Fig.2 - Alternate Interior Angles



Alternate Exterior Angles

The pairs of alternate exterior angles in the figure below are \( (a,c') \) and \( (b,d') \). The alternate exterior angles are congruent and equal in measure. See figure 3 below)
\( m\angle a = m\angle c' \)
\( m\angle b = m\angle d' \)
Alternate Exterior Angles
Fig.3 - Alternate Exterior Angles



Question with Solution

Question 1
Given that \( L_1 \) and \( L_2 \) are parallel lines, find the measures of angles \( b ,c ,d , e, f, g \) and \( h \).

Angles in parallel and transverse lines, question 1
Fig.4 - Find Angles


Solution to Question 1
\( a \) and \( b \) are
supplementary angles and therefore: \( \quad m \angle a + m \angle b = 180^{\circ}\)
Substitute \( m \angle a \) by \( 74^{\circ} \) in the above equation: \( \quad 74^{\circ} + m \angle b = 180^{\circ}\)
The above equation gives: \( \quad m \angle b = 180^{\circ} - 74^{\circ} = 106^{\circ} \)
\( b \) and \( d \) are
vertical angles and are therefore congruent, hence: \( \quad m \angle d = m \angle b = 106^{\circ} \)
\( a \) and \( c \) are
vertical angles and are therefore congruent, hence: \( \quad m \angle c = m \angle a = 74^{\circ} \)
The remaining angles \( f, g, h \) and \( e \) are
corresponding angles to angles \( a, b, c \) and \( d \) respectively; hence
\( m \angle f = m \angle a = 74^{\circ} \)
\( m \angle e = m \angle b = 106^{\circ} \)
\( m \angle g = m \angle c = 74^{\circ} \)
\( m \angle h = m \angle d = 106^{\circ} \)
As an exercise , check that the alternate interior and exterior angles are congruent.



Question 2
In figure 5 below, line \( L_1 \) is parallel to line \( L_2 \) and line \( L_3 \) is parallel to line \( L_4 \) which by definition makes a parallelogram. Show that:
1) the pairs of opposite angles \( (d , e') \) and \( (g , c') \) are congruent or \( m \angle d = m \angle e' \) and \( m \angle g = m \angle c'\)
2) the pairs of angles \( (d,g) \) and \( (c',e') \) are supplementary.

two pairs of parallel lines
Fig.5 - Angles in Parallelogram


Solution to Question 2
1)
Since \( L_1 \) and \( L_2 \) are parallel and \(L_3 \) is a transverse, angles \( a, b, c , d\) and angles angles \( g, e, f , h\) are corresponding angles and therefore congruent in pairs and in particular \( m \angle d = m \angle h \).
Since \( L_1 \) and \( L_2 \) are parallel and \(L_4 \) is a transverse, angles \( a', b', c' , d'\) and angles angles \( g', e', f' , h'\) are corresponding angles and therefore congruent in pairs and in particular \( m \angle d' = m \angle h' \).
Since \( L_3 \) and \( L_4 \) are parallel and \(L_1 \) is a transverse, angles \( a, b, c , d\) and angles angles \( a', b', c' , d'\) are corresponding angles and therefore congruent in pairs and in particular \( m \angle d = m \angle d' \).
Since \( L_3 \) and \( L_4 \) are parallel and \(L_2 \) is a transverse, angles \( g, e, f , h\) and angles angles \( g', e', f' , h'\) are corresponding angles and congruent therefore in pairs and in particular \( m \angle h = m \angle h' \).
Hence we may conclude that: \( m \angle d = m \angle h' \)
\( \angle h' \) and \( \angle e' \) are vertical angles and therefore congruent, hence \( m \angle d = m \angle h' = m \angle e' \)
In a very similar way, it can be shown that \( m \angle g = m \angle a' = m \angle c' \)

2)
\( \angle a \) and \( \angle d \) are supplementary angles, and \( \angle a \) and \( \angle g\) are congruent; hence angles \( \angle d \) and \( \angle g\) are supplementary.
\( \angle b' \) and \( \angle c' \) are supplementary angles, and \( \angle b' \) and \( \angle e'\) are congruent; hence angles \( \angle c' \) and \( \angle e'\) are supplementary.

More angles in parallel lines and transversals are included.

More References and Links to Geometry

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