|Lines and Angles: Intersecting, Parallel and Transversal Lines,
Exterior, Interior, and Corresponding Angles
If two lines or segments lie in a plane, they can have only two types of
relationships. Either they are parallel to or they intersect each other. If they
intersect without completely overlapping, they form four angles around the point of intersection (Figure
Look at Figure 1.11. Here, we named the angles formed by the intersecting
segments IQ and CD as
4. These angles can be grouped in two sets
Pairs of angles in each group only share a common vertex. They do not have
any common sides. Such angles are called vertical angles. Vertical angles are always are congruent. That is,
For example, if m1 = 125˚, then m3
If two sides of an angle lie on a straight line, it is called a straight angle (Figure 1.12). The measure of any straight angle is always
180˚. So by knowing m1 in Figure
1.11, we can determine m3 and m4.
We see that 1 and
together form a straight angle we can calculate
m3 = 180˚ – 125˚ = 65˚. Similarly, m4
= 180˚ – 125˚ = 65˚.
Two angles are called adjacent if:
- they have a common vertex and side
- their non-common sides lie opposite areas of the common side.
MKN are adjacent angles (Figure
If the measures of two angles sum up 180˚, they are called supplementary
angles. Each of a pair of supplementary angles is called a supplement
of the other.
If the measures of two angles sum up 90˚, they are called
complementary angles. Each of the two complementary angles is called a
complement of the other.
Parallel Lines and Transversals
If two or more lines or segments lie on a plane and never intersect each
other, they are called parallel lines or parallel segments.
Two parallel lines are always in the same distance from each other. We
denote parallel lines using ||. In Figure 1.14,
are parallel. We denote these
segments as .
If a line or a segment intersects two or more parallel lines, it is
called a transversal. For example,
is a transversal segment with
respect to parallel segments
Exterior and Interior Angles
When a transversal intersect two parallel lines, eight angles are formed
around the points of intersection (Figure 1.15). The angles that lie outside
the area bounded by the parallel lines are called exterior angles.
For example, 1,
in Figure 1.15 are exterior angles. The angles that lie inside the area
bounded by the parallel lines are called interior angles. In Figure
are interior angles.
Alternate Interior Angles
If two interior angles located on the opposite sides of the transversal,
they are called alternate interior angles. In Figure 1.15, pairs of (4,
5) and (3,
8) are alternate interior angles. Each pair
of alternate interior angles are always congruent. That is,
Alternate Exterior Angles
If two exterior angles lie on the opposite sides of the transversal,
they are called alternate exterior angles. In Figure 1.15, (1,
6) and (2,
7) are pairs of alternate exterior angles.
Each pair of alternate exterior angles are always congruent. That is,
If an alternate exterior angle and an alternate interior angle lie
on the same side of the transversal, they are called corresponding angles.
In Figure 1.15, pairs (1,
6), and (4,
7) are corresponding angles. Each pair of
corresponding angles are supplementary.
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for Students, Parents and Teachers
Now let's do Practice
Exercise 1-3 (top).