Lines and Angles

What is a Line?

Do you know what a line is, or perhaps a line segment or a ray?

• Line is a one-dimensional figure that extends infinitely on either sides (i.e. in both directions). That is, a line only has length (no breadth, no height) and it has no end points.

• If a line has one end point, then it’s called a Ray. So, a Ray is a one-dimensional figure that extends infinitely in one direction.

• A Line segment is basically a portion of a line with two end points. So, a Line segment is a one-dimensional figure that has an end point on either side. In the above figure, AB or BA is a line segment.

We hope, now you can tell the difference between line, line segment and ray.

What is an Angle?

Angle is the angular distance between two lines. It is measured in degrees or radians.

An angle is formed when two lines/line-segments/rays converge at a point. An angle is denoted by the symbol ∠

In the above figure, the angle may be denoted as ∠BAC or ∠CAB.

• Arms of an angle - two lines/line-segments/rays forming the angle.
• Vertex of an angle - the common point where the two lines/line-segments/rays meet (which is A in the above figure).

Now, let us first of all see the various types of lines and angles. Thereafter, we will have a look at their properties too.

Types of Lines

• Parallel lines - Two lines on a plane are parallel if they never meet, even if they are extended infinitely on either sides. We denote them using the symbol ∥
  In the above figure, the lines AB and CD are parallel. So, we can denote them as: AB ∥ CD

• Perpendicular lines - Two lines are perpendicular to each other, if they form an angle of 90° with each other. We denote them using the symbol ⊥ In the above figure, the lines AB and CD are perpendicular. So, we can denote them as: AB ⊥ CD

• Transversal line - A line which cuts two or more given lines at different points. In the above figure, XY is a transversal line.

Types of Angles

According to measurement of angle, we have the following types of angles.

• Acute Angle - An angle measuring less than 90°.

• Right Angle - An angle measuring exactly 90°. As you can see, the arms of a right angle are perpendicular to each others.

• Obtuse Angle - An angle measuring more than 90°, but less than 180°.

• Straight Angle - An angle measuring exactly 180°.

• Reflex Angle - An angle measuring more than 180°, but less than 360°.

• Complete Angle - An angle measuring exactly 360°.

Angle-Pairs

There are some angle-pairs that you should be aware of.

• Complementary Angles - If the sum of two angles is 90°, then they are called complementary angles.

• Supplementary Angles - If the sum of two angles is 180°, then they are called supplementary angles.

• Adjacent Angles - If two angles have a common vertex and a common arm (between two other arms), then they are called adjacent angles. In the above figure, ∠AVB and ∠BVC are adjacent angles.

• Linear Pair Angles - A pair of adjacent angles will form a linear pair, if their outer arms lie on one straight line. So, the sum of linear pair angles will be 180°. (∠AVB + ∠BVC = 180°)

• Vertically Opposite Angles - Consider the common vertex formed by the intersection of two lines. Here, the pair of angles having no common arm, are called vertically opposite angles. They appear opposite to each other.
  Vertically opposite angles are always equal to each other. For example, in the above figure there are two pairs of vertically opposite angles:
  I. ∠AVC and ∠BVD form the first pair of vertically opposite angles. So, ∠AVC = ∠BVD.
  II. ∠AVD and ∠BVC form the second pair of vertically opposite angles. So, ∠AVD = ∠BVC.

Properties of Lines

Properties related to Perpendicular lines

Property 1: Perpendicular Bisector

If a line (say CD) passes through the mid-point of a line segment (say AB) and is perpendicular to it, then the line is called the perpendicular bisector of the line segment.

In the above figure, CD is perpendicular bisector of AB, that is:

• CD is perpendicular to AB (CD ⊥ AB) and
• CD bisects AB in two equal halves (AD = BD).

Every point on a perpendicular bisector is equidistant from both ends of the line. In the above figure, AQ = QB, and AP = PB

Properties related to Angles

Property 1: Angle Bisector

Angle bisector is a line that bisects an angle. In the above figure, line AD bisects the angle ∠BAC. So, ∠BAD = ∠CAD

Every point on an angle bisector is equidistant from both arms of the angle that it bisects. In the above figure, QW = QY, and PX = PZ

Properties related to Parallel lines

Property 1: Parallel lines are always equidistant

The perpendicular distance between two parallel lines always remains the same, no matter where we measure it. For example, in the following figure, AB ∥ CD.

Property 2

If a line makes the same angle with a couple of lines (or planes), then those lines (or planes) must be parallel to each other. For example, in the following figure, AB ∥ CD.

Property 3: Corresponding angles

Corresponding angles are equal.

In the following figure, AB ∥ CD & XY is a transversal line.

In the above figure, the corresponding angles are:

• ∠Angle 1 = ∠Angle 5 = x°
• ∠Angle 2 = ∠Angle 6 = y°
• ∠Angle 3 = ∠Angle 7 = x°
• ∠Angle 4 = ∠Angle 8 = y°

Property 4: Alternate angles

Pairs of alternate (interior or exterior) angles are equal.

In the above figure, the alternate angles are:

• ∠Angle 3 = ∠Angle 5 = x° (interior alternate angles)
• ∠Angle 4 = ∠Angle 6 = y° (interior alternate angles)
• ∠Angle 2 = ∠Angle 8 = y° (exterior alternate angles)
• ∠Angle 1 = ∠Angle 7 = x° (exterior alternate angles)

Property 5

Sum of interior angles or exterior angles on the same side of the transversal line is equal to 180°. In the above figure:
∠Angle 3 + ∠Angle 6 = ∠Angle 4 + ∠Angle 5 = ∠Angle 2 + ∠Angle 7 = ∠Angle 1 + ∠Angle 8 = 180°

Property 6

Bisectors of interior angles intersect at 90°.

Property 6a

Bisectors of interior angles form a rectangle. In the above figure, PQRS is a rectangle.

Property 6b

If transversal line is perpendicular to the two parallel lines, then bisectors of its interior angle form a square. In the above figure, PQRS is a square.