back to geometry

Angle pairs, perpendicular and parallel lines

Straight angle
A angle with a measure of 180o is a straight angle.
A angle with a measure of 180 degrees is a straight angle.
In the figure above, the angle AOB = 180o. The angle AOB is a straight angle.
Right angle
A right angle has a measure of 90o.
In the figure below, the angle AOB = 90o. The angle AOB is a straight angle.
A right angle has a measure of 90 degrees.
The symbol in the interior of an angle means that the angle is a right angle.
Acute angle
An acute angle is an angle whose measure is less than 90o.
An acute angle is an angle whose measure is less than 90 degrees.
In the figure above, the angle AOB < 90o. The angle AOB is an acute angle.
Obtuse angle
An obtuse angle is an angle whose measure is more than 90o and less than 180o.
An obtuse angle is a angle whose measure is more than 90 degrees but less than 180 degrees.
In the figure above, 90o < angle AOB < 180o. The angle AOB is an obtuse angle.
Adjacent angles
The adjacent angles are any two angles that share a common side and share a common vertex.
An adjacent angles are any two angles that share a common side and share a common vertex.
In the figure above, angle AOB and angle BOC have the common side OB.
Angle AOB and angle BOC have the same common vertex O.
Angle AOB and angle BOC do not share any common interior points.
So angle AOB and angle BOC are adjacent angles.
Vertical angles
Vertical angles are formed when two lines intersect. Any two of them which are not adjacent angles are called vertical angles.
Vertical angles are formed when two lines intersect. Any two of them that are not adjacent angles are called vertex angles.
In the figure above, line m and line n intersect at point O.
Angle 1 and angle 2 are vertical angles.
Angle 3 and angle 4 are vertical angles.
Property of vertical angles
Vertical angles are equal in measure. For example, since angle 1 and angle 2 are vertical angles, so angle 1 = angle 2. Since angle 3 and angle 4 are vertical angles, so angle 3 = angle 4.
Vertical angles are equal in measure.
Supplementary angles
Supplementary angles are two angles whose sum is 180o.
Supplementary angles are two angles whose sum is 180 degrees.
In the figure above, since angle 1 + angle 2 = 180o, so angle a and angle 2 are supplementary angles.
Supplementary angles are two angles whose sum is 180 degrees.
In the figure above, angle 3 = 70o, angle 4 = 110o. Since angle 3 + angle 4 = 180o, so angle 3 and angle 4 are supplementary angles.
Complementary angles
Complementary angles are two angles whose sum is 90o.
Complementary angles are two angles whose sum is 90 degrees.
In the figure above, since angle 1 + angle 2 = 90o, so angle 1 and angle 2 are complementary angles.
Complementary angles are two angles whose sum is 90 degrees.
In the figure above, angle 3 = 30o, angle 4 = 60o. Since angle 3 + agle 4 = 90o, so angle 3 and angle 4 are complementary angles.
Angle bisector
Angle bisector is a ray that divides an angle into two equal angles.
Angle bisector is a ray that divides an angle into two equal angles.
In the figure above, ray OC is an angle bisector so that angle AOC = angle COB.
Angle addition
If ray OS is in the interior of the angle AOB, then angle AOS + angle SOB = angle AOB.
If ray OS is in the interior of angle AOB, then angle AOR + angle ROB = angle AOB.
Perpendicular lines
Two lines that intersect and form right angles are called perpendicular lines.
Two lines that intersect and form right angles are called perpendicular lines.
In the figure above, line m and line n intersect at point O, if angle AOB = 90o, then line m and line n are perpendicular each other at point O.
Distance from a point to a line
The shortest distance from a point to a line is measured by the length of the segment drawn from the point perpendicular to the line.
The shortest distance from a point not on a line to the line is measured by the length of the segment drawn from the 
                        point perpendicular to the line.
In the figure above, m is a line. P is a point not on the line m.
d is the distance from the point P to the line m.
PA is the shortest distance from point P to line m.
Parall lines
Parallel lines are those lines that are in the same plane but never intersect.
Parallel lines are those lines that are in the same plane but never intersect.
In the figure above, line m and line n never intersect.
m // n read as line m is parallel to line n.
Example 1:
Rays OA, OB and OC lie in the same plane. Find the degree measure of the angle AOB.
If ray OS is in the interior of angle AOB, then angle AOR + angle ROB = angle AOB.
Solution:
angle AOB = angle AOC + angle COB = 32o + 30o = 62o
Example 2:
Ray OS is the bisector of the angle AOB. If the degree measure of the angle SOB is 55o, what is the degree measure of the angle AOB?
Angle bisector is a ray that divides an angle into two equal angles.
Solution:
Since ray OS is the bisector of the angle AOB, so angle AOS = angle SOB.
angle AOB = angle AOS + angle SOB = 2 angle SOB = 2 × 55o = 110o
Example 3:
Rays l, m and n intersect at the same point. Line m is perpendicular to line n. If the angle 1 = 38o, what is the degree measure of the angle 5?
Complementary angles are two angles whose sum is 90 degrees.
Solution:
Since angle 1 and angle 4 are vertical angles, so angle 1 = angle 4 = 38o
Since n is a straight line, so angle 3 + angle 4 + angle 5 = 180o
Since angle 3 = 90o, so angle 4 + angle 5 = 90o angle 4 and angle 5 are complementary angles.
angle 5 = 90o - angle 4 = 90o - 38o = 52o