Angles ( Definition, types and measures of angles)

Definition:

An angle is a geometric figure formed by two rays or two line segments with a common endpoint called the vertex. The rays or segments are called the sides of the angle.

Types of Angles:

1. Acute Angle: An angle that measures less than 90 degrees.
2. Right Angle: An angle that measures exactly 90 degrees.
3. Obtuse Angle: An angle that measures more than 90 degrees but less than 180 degrees.
4. Straight Angle: An angle that measures exactly 180 degrees.
5. Reflex Angle: An angle that measures more than 180 degrees but less than 360 degrees.
6. Full Angle: An angle that measures exactly 360 degrees.
7. Complementary Angles: Two angles whose measures add up to 90 degrees.
8. Supplementary Angles: Two angles whose measures add up to 180 degrees.
9. Adjacent Angles: Angles that share a common vertex and a common side, but do not overlap.
10. Vertical Angles: Pair of opposite angles formed by the intersection of two lines. They are congruent.

Measurement of Angles:

Angles can be measured in degrees, minutes, and seconds.

• 1 degree (°) = 60 minutes (‘)
• 1 minute (‘) = 60 seconds (”)

Notation:

Angles are often denoted by Greek letters (such as α, β, γ) or by three letters, where the vertex is the middle letter (such as ∠ABC).

Protractor:

A protractor is a tool used to measure and draw angles. It usually consists of a semicircular or circular disc marked with degree graduations.

Understanding angles and their properties is fundamental in various fields, including geometry, trigonometry, and physics. If you have any specific questions or need further clarification, feel free to ask!

Worked Samples

Example 1: Definition of Angles An angle formed by two intersecting lines AB and CD measures 45 degrees. Identify the angle.

Solution: The angle formed by lines AB and CD is a right angle, as it measures 90 degrees.

Example 2: Classifying Angles Classify the following angles as acute, obtuse, or right angles: a) ∠PQR = 30 degrees b) ∠XYZ = 110 degrees c) ∠LMN = 90 degrees

Solution: a) ∠PQR is an acute angle. b) ∠XYZ is an obtuse angle. c) ∠LMN is a right angle.

Example 3: Measuring Angles Measure the angle formed by the hands of a clock when the time is 6:30.

Solution: The minute hand points at the 6, while the hour hand is halfway between 6 and 7. Each hour mark represents 30 degrees. So, the angle formed is 30 degrees × 6 = 180 degrees.

Example 4: Complementary and Supplementary Angles If two angles are complementary and one of them measures 45 degrees, find the measure of the other angle.

Solution: Complementary angles add up to 90 degrees. So, the other angle measures 90 degrees – 45 degrees = 45 degrees.

Example 5: Adjacent Angles Identify the adjacent angles in the figure below:

Solution: ∠ABC and ∠CBA are adjacent angles.

Example 6: Vertical Angles In the figure below, identify the pairs of vertical angles:

Solution: ∠ABC and ∠BAC are vertical angles. ∠CBA and ∠BAC are also vertical angles.

Example 7: Finding the Measure of Unknown Angles If ∠X and ∠Y are complementary angles, and ∠X measures 40 degrees, find the measure of ∠Y.

Solution: Since complementary angles add up to 90 degrees, ∠Y = 90 degrees – 40 degrees = 50 degrees.

Example 8: Using a Protractor Measure the angle formed by the lines AB and CD using a protractor, where the angle is ∠ABC.

Solution: Place the center of the protractor on point B, align the base line of the protractor with line AB, and read the measurement where line CD intersects the protractor.

Example 9: Applying Trigonometric Concepts In a right triangle, if one acute angle measures 30 degrees, find the measure of the other acute angle.

Solution: Since the sum of the angles in a triangle is 180 degrees, the other acute angle measures 180 degrees – 90 degrees – 30 degrees = 60 degrees.

Example 10: Real-Life Application A ladder leans against a wall, forming an angle of 60 degrees with the ground. If the bottom of the ladder is 12 feet away from the wall, find the length of the ladder.

Solution: Using trigonometric ratios, the length of the ladder (hypotenuse) can be found using the cosine of the angle: cos(60∘)=adjacenthypotenusecos(60∘)=hypotenuseadjacent​. Thus, the length of the ladder is 12cos(60∘)cos(60∘)12​. Calculate the value to find the length of the ladder.

These examples cover various aspects of angles, from basic definitions to real-life applications. Let me know if you need further explanation on any of them!

1. The sum of the interior angles of a triangle is ______ degrees. a) 90 b) 180 c) 270 d) 360
2. Two angles whose measures add up to 90 degrees are called ______ angles. a) complementary b) supplementary c) vertical d) adjacent
3. An angle that measures exactly 90 degrees is called a ______ angle. a) acute b) obtuse c) right d) straight
4. The angle formed by the hands of a clock at 3:00 is ______ degrees. a) 90 b) 120 c) 180 d) 360
5. The type of angles formed by two intersecting lines that are opposite each other is called ______ angles. a) complementary b) supplementary c) vertical d) adjacent
6. The measure of an angle can be determined using a ______. a) ruler b) protractor c) compass d) calculator
7. Two angles that share a common vertex and a common side but have no common interior points are called ______ angles. a) complementary b) supplementary c) vertical d) adjacent
8. The angle of elevation of a ladder leaning against a wall is 60 degrees. If the bottom of the ladder is 10 meters away from the wall, the length of the ladder is ______ meters. a) 10 b) 20 c) 10√3 d) 20√3
9. If two angles are supplementary and one of them measures 60 degrees, the measure of the other angle is ______ degrees. a) 60 b) 120 c) 180 d) 240
10. The sum of the measures of exterior angles of any polygon is always ______ degrees. a) 90 b) 180 c) 360 d) 720
11. A triangle with all angles measuring less than 90 degrees is called an ______ triangle. a) acute b) obtuse c) right d) equilateral
12. The angle between the hour hand and the minute hand of a clock at 9:00 is ______ degrees. a) 30 b) 45 c) 60 d) 90
13. An angle whose measure is greater than 90 degrees but less than 180 degrees is called an ______ angle. a) acute b) obtuse c) right d) straight
14. The measure of a straight angle is ______ degrees. a) 90 b) 180 c) 270 d) 360
15. Two angles that add up to 180 degrees are called ______ angles. a) complementary b) supplementary c) vertical d) adjacent