Lines & Angles
Lines and angles are the fundamentals of geometry; almost every geometric drawing contains an angle and a line.
Parallel Lines
Two or more lines that, when extended indefinitely, never intersect or touch each other. These lines run side by side and maintain a consistent distance apart from each other at all points.
Equidistant: The distance between two parallel lines remains constant at any point along their length.
Never Intersect: No matter how far they are extended, parallel lines will never meet.
Intersecting Lines
Two or more lines that cross each other at a specific point. This point is known as the point of intersection. Unlike parallel lines, which never meet, intersecting lines do meet, and they do so exactly once (unless the lines are coincident, meaning they lie on top of each other).
We use intersecting lines to represent angles. So, what is an angle, and how is it represented?
Angles
An angle is a measurement of a turn and we represent this turning with lines (intersecting rays).
A Full Turn - 360°
The direction of the rotation, whether clockwise or counterclockwise, is not important; for now, we are only concerned with the amount of rotation.
Straight Angle - Half Turn - 180°
Since a full rotation is 360 degrees, a half turn is equal to 180 degrees. This means that two rays or lines are in exactly opposite directions from each other.
Quarter Turn 90°
Since a full rotation is 360 degrees, a quarter turn is equal to 90 degrees. This means that two rays or lines are perpendicular to each other.
Perpendicular Lines
When the angle between two lines is 90 degrees, they are perpendicular to each other. In mathematical notation, this relationship can be shown as:
If lines l and m are perpendicular to each other, this relationship can be denoted as "l ⊥ m."
The symbol "⊥" represents the perpendicular relationship between the lines.
The small square at the intersection point is used to show that the lines are perpendicular to each other.
Acute Angle
An angle that is less than 90 degrees .
Acute Angle
Obtuse Angle
An obtuse angle measures between 90 and 180 degrees.
Reflex Angle
A reflex angle measures between 180 and 360 degrees.
Angle Types
In geometry, apart from these angles, we often use some special relationships between angles, such as the sum of two angles equalling 90 degree or 180 degree exct.
Complementary & Supplementary Angles
Complementary - measures add up to 90 degrees
his means that the sum of their angle measurements is equal to a right angle. If you know the measurement of one of the complementary angles, you can easily find the other by subtracting the known angle from 90 degrees. For example, if one angle measures 30 degrees, its complement is 60 degrees, because 90 - 30 = 60.
Supplementary - measures add up to 180 degrees
This means that the sum of their angle measurements is equal to a straight angle. If you know the measurement of one of the supplementary angles, you can find the other by subtracting the known angle from 180 degrees. For instance, if one angle measures 120 degrees, its supplement is 60 degrees, because 180 - 120 = 60.
Vertical Angles ( Opposite Angles)
Vertical angles, also known as opposite angles, are angles formed by the intersection of two straight lines. These angles are always congruent, meaning they have the same measure. When two lines intersect, they create two pairs of vertical angles. For any intersecting lines, the vertical angles are always equal to each other.
A and B are opposite angles.
C and D are opposite angles.
When you move one arm of an angle, the other also moves. For instance, in a figure, if we move the arm of angle A counterclockwise, angle A decreases or narrows. Similarly, since the arm of angle B will also move, angle B narrows as well. As much as angle A narrows, angle B narrows by the same amount. Therefore, the measures of vertical angles are always equal and remain equal.
Adjacent Angles ( Neighbouring Angles )
Angles that are next to each other or side by side, formed by two intersecting lines.
They might be complementary or supplementary angles, but they don't have to be.
Naming Angles
Using Three Points:
Angles can be named using three points: one on each ray and the vertex in the middle. For instance, if you have an angle formed by points A, B, and C, with B being the vertex, the angle can be named as ∠ABC or ∠CBA.
Using Vertex:
Just draw an arc on the arms of the angle you want to show and give it a name in capital letters
With the same rays in the same position, I can represent two angles.The arc indicates which angle we are referring to. The arc clarifies this.
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