angles

A line is a perfectly straight path extending indefinitely in both directions. A line has infinite length. i.e. it has no endpoints. A line segment is a part of a line. It has a definite length and has two endpoints.

• Concurrent lines - Three or more lines lying in a plane are called concurrent lines if all of them pass through the same point.
• Intersecting lines - Any two lines lying in the same plane which meet each other at a point are called intersecting lines.
• Parallel lines - Two non-intersecting lines lying in the same plane are called parallel lines.
• Perpendicular lines - Any two lines lying on the same plane which intersect at right angles are called perpendicular lines.

Angle

In geometry, an angle can be defined as the figure formed by two rays meeting at a common endpoint called a vertex. An angle is represented by the symbol ∠. The angle below is ∠AOB. Point O is the vertex of ∠AOB. \(OA\) and \(OB\) are the arms of ∠AOB.

Angles are measured in degrees, using a protractor.  The angle can range from 0° to 360°.

Classification of Angles

Angle	Figure
Acute angle - An angle whose measure is greater than 0° but less than 90° is called an acute angle.
Right angle - An angle that measures 90° is called a right angle.
Obtuse angle - An angle whose measure is greater than 90° but less than 180° is called an obtuse angle.
Straight angle - An angle whose measure is 180° is called a straight angle.
Reflex angle - An angle whose measure is greater than 180° but less than 360° is called a reflex angle.
Complete angle - An angle whose measure is 360° is called a complete angle.

Related Angles

Complementary Angles: Two angles are said to be complementary if the sum of their measures is 90°. In the below figure \(\angle 1+ \angle 2 = 90°\).

We say \(\angle 1 \) is a complement of \(\angle 2 \) and vice versa.

Supplementary Angles: Two angles are said to be supplementary if the sum of their measures is 180°. In the below figure \(\angle 3+ \angle 4 = 180°\). \(\angle 3\) and \(\angle4\) are supplementary angles.

 \(\angle 3\) is the supplement of \(\angle4\) and vice versa.

Adjacent Angles: A pair of angles that meets below three conditions is called a pair of adjacent angles.
- Both the angles have the same vertex.
- Both the angles have a common arm.
- Both the angles are on the opposite sides of the common arm.

A is the common vertex. \(AD\) is the common arm. \(\angle 7\) and \(\angle8\) are pairs of adjacent angles.

Vertically Opposite Angles: Two angles formed by two intersecting lines and with no common arm are called vertically opposite angles.

\(\angle 1 \) and \(\angle 2 \) are vertically opposite angles, also  \(\angle 3\) and\(\angle4\) are vertically opposite angles.

Vertically opposite angles are equal, i.e. \(\angle 1 \) = \(\angle 2 \), \(\angle 3\) = \(\angle4\)

Alternate, Corresponding, Interior, and Exterior Angles

When a transversal (a line that passes through two lines in the same plane at two distinct points) intersects two lines, eight angles are formed. These eight angles can be classified into four groups as below:

1. Angles 3, and 4; angles 5 and 6 are called interior angles. Angles 4 and 6 and angles 3 and 5 forms a pair of co-interior angles.
2. Angles 1 and 5; angles 2 and 6; angles 4 and 8 and angles 3 and 7 forms a pair of corresponding angles.
3. Angles 1, 2, 7, and 8 are exterior angles.
4. Angles 4 and 5; angles 3 and 6 form a pair of alternate angles.

When a transversal intersects two parallel lines then the following holds true:

1. The Sum of the measure of all four interior angles is 360°, i.e.   \(\angle 3 + \angle 4 + \angle 5 + \angle 6 = 360°\)
2. The Sum of the measure of a co-interior angle is 180°, i.e. \(\angle 3 + \angle5 = 180°, \angle 4 + \angle 6 = 180°\)
3. The Sum of the measure of all four exterior angles is 360°, i.e.   \(\angle1 + \angle2 + \angle7 + \angle8 = 360°\)
4. Alternate angles are equal, i.e. \(\angle 4 = \angle 5, \angle 3 = \angle 6\)
5. Corresponding angles are equal, i.e. \(\angle 2 = \angle 6, \angle 1 = \angle 5, \angle 4 = \angle 8, \angle 3 = \angle 7\)

Conversely, the following statements also hold true:

If two lines are cut by a transversal in such a way that any two corresponding angles are of equal measure then the two lines are parallel. If two lines are cut by a transversal in such a way that any two alternate angles are of equal measure then the two lines are parallel. If two lines are cut by a transversal in such a way that the sum of co-interior angle is 180º then the two lines are parallel.