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Angles
An angle is a space between two intersecting lines or surfaces, measured in degrees or radians. It represents the amount of rotation between the two lines or rays, and it is widely used in geometry and trigonometry.
1.0Exploring Angles
An angle is formed when two rays or lines meet at a common endpoint called the vertex. The arms or sides of the angle are the rays.
Components of an Angle
- Vertex: The point common through which two rays meet.
- Arms: The two rays that make up the angle.
- Angle Measure: The degree of rotation between the two rays. The unit is usually measured in degrees or radian units.
2.0Types of Angles
Angles are different from each other based on their measurements. Here are the types of angles
3.0Angle Relationship
Pair of Angles
Any given two angles can be related to each other in various ways, such as:
- Complementary Angles: When the sum of any two angles is equal to 90°, then the given angles are said to be complementary angles. Mathematically, it is written as ∠A+∠B = 90°.
- Supplementary Angles: When the sum of any two angles is equal to 180° then the given angles are said to be complementary angles. Mathematically, it is written as: ∠A+∠B = 180°. This result can also be said to be in a linear pair.
- Adjacent Angles: Angles that are adjacent have a common vertex and a common arm. They don't overlap at all.
- Vertical Angles: The opposite angles, or non-adjacent angles, that are formed by two intersecting lines are called vertical angles. Vertical angles are always equal.
4.0Angle Pairs in Parallel Lines
When two parallel lines are cut by a transversal, the following angle pairs result:
- Alternate Interior Angles: These are angles that appear on opposite sides of the transversal line ‘t’ and lie between the two parallel lines. This is a congruent set of angles; that is, each of these angles has an equal measure. Suppose two parallel lines get intersected by a transversal. Then, the alternate interior angles formed are the angles inside the parallel lines but on opposite sides of the transversal.
- Alternate Exterior Angles: These are the angles that are on opposite sides of below transversal line ‘t’ outside the parallel lines. They are congruent.
- Corresponding Angles: Angles formed by any pair of points on the transversal in the same location where they cross each of the two parallel lines. The angle measures are congruent to corresponding angles. Example: Given below is a transversal of two parallel lines. Draw it and mark a couple of the corresponding angles.
- Consecutive Interior Angles: The angles that lie on the same side of the transversal inside the parallel lines. Such angles are supplementary to each other (add up to 180°).
5.0Solved Problems
Problem 1: Two angles are complementary. If one angle is 35°, find the measure of the other angle.
Solution: Let the other angle = x
The Sum of complementary angles = 90
x + 35 = 90
x = 90 – 35 = 55
Hence, the other angle is 55.
Problem 2: Two parallel lines are cut by a transversal. One of the alternate interior angles measures 3x+20, and the corresponding angle on the other side of the transversal measures 4x−10. Find the value of x.
Solution: by the property of Alternate interior angle, we know:
3x+20 = 4x–10
3x–4x = –10 – 20
–x = –30
x = 30°
Problem 3: Two lines intersect, forming two pairs of opposite (vertical) angles. One of the angles is 5x+15, and the vertical angle is 4x+25. Find the value of x.
Solution: By the property of vertical angles, opposite angles are equal hence,
5x+15 = 4x+25
5x–4x = 25 – 15
x = 10°