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Angle pairs, perpendicular and parallel lines

Straight angle

A angle with a measure of 180^o is a straight angle.

In the figure above, the angle AOB = 180^o. The angle AOB is a straight angle.

Right angle

A right angle has a measure of 90^o. In the figure below, the angle AOB = 90^o. The angle AOB is a straight angle.

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 90^o.

In the figure above, the angle AOB < 90^o. The angle AOB is an acute angle.

Obtuse angle

An obtuse angle is an angle whose measure is more than 90^o and less than 180^o.

In the figure above, 90^o < angle AOB < 180^o. 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.

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.

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.

Supplementary angles

Supplementary angles are two angles whose sum is 180^o.

In the figure above, since angle 1 + angle 2 = 180^o, so angle a and angle 2 are supplementary angles.

In the figure above, angle 3 = 70^o, angle 4 = 110^o. Since angle 3 + angle 4 = 180^o, so angle 3 and angle 4 are supplementary angles.

Complementary angles

Complementary angles are two angles whose sum is 90^o.

In the figure above, since angle 1 + angle 2 = 90^o, so angle 1 and angle 2 are complementary angles.

In the figure above, angle 3 = 30^o, angle 4 = 60^o. Since angle 3 + agle 4 = 90^o, so angle 3 and angle 4 are complementary angles.

Angle bisector

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.

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 = 90^o, 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.

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.

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.

Solution

angle AOB = angle AOC + angle COB = 32^o + 30^o = 62^o

Example 2

Ray OS is the bisector of the angle AOB. If the degree measure of the angle SOB is 55^o, what is the degree measure of the angle AOB?

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 × 55^o = 110^o

Example 3

Rays l, m and n intersect at the same point. Line m is perpendicular to line n. If the angle 1 = 38^o, what is the degree measure of the angle 5?

Solution

Since angle 1 and angle 4 are vertical angles, so angle 1 = angle 4 = 38^o

Since n is a straight line, so angle 3 + angle 4 + angle 5 = 180^o

Since angle 3 = 90^o, so angle 4 + angle 5 = 90^o angle 4 and angle 5 are complementary angles.

angle 5 = 90^o - angle 4 = 90^o - 38^o = 52^o