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
\( m\angle a = m\angle a' \)
\( m\angle b = m\angle b' \)
\( m\angle c = m\angle c' \)
\( m\angle d = m\angle d' \)
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)
\( m\angle d = m\angle b' \)
\( m\angle c = m\angle a' \)
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)
\( m\angle a = m\angle c' \)
\( m\angle b = m\angle d' \)
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 \( h \).
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.
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.
