Lines and Angles

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Lines and Angles

Ancient mathematicians introduced the concept of lines to represent one-dimensional objects without width or depth. Lines serve as fundamental building blocks in geometry, providing a straightforward understanding of straight objects. On the other hand, an angle is a geometric shape that arises from the intersection of two line segments, lines, or rays. When two rays intersect within the same plane, they form an angle. Angles provide a means to measure the amount of rotation or inclination between two intersecting lines or line segments.

Line

A line is a straight path that extends infinitely in both directions. Lines are often represented by a straight line with arrows on both ends to indicate that it goes on forever.

Ray

A line with one endpoint and another endpoint that extends infinitely in one direction is known as a ray.

Line Segment

A line segment is a part of a line that consists of two endpoints. It can be defined as a straight line with a starting and ending point.

Other Types of Lines

Curved Line

A line that bends and has a distinct shape, such as a circle or an ellipse.

Vertical Line

A line that runs straight up and down and is perpendicular to the horizon.

Horizontal Line

A line that runs straight across and is parallel to the horizon.

Perpendicular Lines

Perpendicular lines are two lines that intersect each other at a 90-degree angle. They form an "L" shape at their point of intersection.

Parallel Lines

Parallel lines are two or more lines that are always the same distance apart and never intersect, no matter how far they are extended in either direction.

Intersecting Lines

Intersecting lines are two or more lines that meet at a common point. The point of intersection is the point where the lines meet.

Properties of Lines

1. A line has an infinite length and no width or thickness.
2. A line is made up of an infinite number of points.
3. Two distinct lines cannot intersect at more than one point.
4. If two lines intersect, then the opposite angles formed are equal.
5. If two lines are parallel, then the corresponding angles formed are equal.
6. If two lines are parallel, then the alternate angles formed are equal.
7. If a line intersects two parallel lines, then the interior angles on the same side of the transversal are supplementary (add up to 180 degrees).
8. The sum of the three angles in a triangle is always 180 degrees.
9. The exterior angle of a triangle is equal to the sum of the two opposite interior angles.

Angles

An angle is the measure of the amount of rotation between two lines that intersect at a point. Angles are often represented by a small arc, with a point at the vertex (the point where the end point of rays meet) and two rays (lines) coming out of the vertex. The angle is measured in degrees (°) or radians (rad).

Classification of Angles

Zero Angle: An angle that measures exactly 0°.

Acute Angle: An angle that measures less than 90°, typically between 0° and 89°.

Right Angle: An angle that measures exactly 90°.

Obtuse Angle: An angle that measures greater than 90° and less than 180°, typically between 91° and 179°.

Straight Angle: An angle that measures exactly 180°.

Reflex Angle: An angle that measures greater than 180° and less than 360°, typically between 181° and 359°.

Complete Angle: An angle that measures exactly 360°.

Types of Angles based on Relation of Angles

1. Complementary Angles

Two angles whose measures combine to form 90°. If two angles a and b are complementary, then their sum is equal to 90°, i.e., ∠ a + ∠ b = 90°.

In this case, ∠ a is the complement of ∠ b, and ∠ b is the complement of ∠ a.

2. Supplementary Angles

Two angles whose measures combine to form 180°. If two angles a and b are supplementary, then their sum is equal to 180°, i.e., ∠ a + ∠ b = 180°.

In this case, ∠ a is the supplement of ∠ b, and ∠ b is the supplement of ∠ a.

3. Vertically Opposite Angles

Two angles are formed by intersecting lines that are opposite each other and have the same measure.

4. Linear Pair of Angles

Two angles are adjacent and supplementary.

Angles formed by Parallel Lines and Transversal Lines

1. Alternate Interior Angles: Two angles that are on opposite sides of the transversal and inside the alternate interior angles.

∠ 4 = ∠ 5
∠ 3 = ∠ 6

2. Alternate Exterior Angles: Two angles that are on opposite sides of the transversal and outside the alternate interior angles.

∠ 2 = ∠ 7
∠ 1 = ∠ 8

3. Corresponding Angles: Two angles that are on the same side of the transversal and correspond to each other.

∠ 1 = ∠ 5
∠ 2 = ∠ 6
∠ 3 = ∠ 7
∠4 = ∠ 8

4. Co-interior Angles: Two interior angles that are on the same side of the transversal make 180° together.

∠ 3 + ∠ 5 = 180°
∠ 4 + ∠ 6 = 180°

5. Exterior Angle Property: The exterior angle property states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two interior angles that are not adjacent to it.