Chapter Notes: Angles
Angles are one of the most important ideas in geometry. They appear everywhere around us-in the corners of rooms, the hands of a clock, the paths of roads that meet, and even in the movements of a basketball player pivoting on the court. An angle is formed when two rays share a common endpoint. Understanding angles helps us measure turns, describe shapes, and solve real-world problems involving direction and rotation. In this chapter, you will learn what angles are, how to measure and classify them, and how they relate to each other in different situations.
What Is an Angle?
An angle is formed by two rays that have a common endpoint. The common endpoint is called the vertex of the angle. The two rays are called the sides or arms of the angle.
Think of an angle like the opening of a door. The hinge is the vertex, and the door and the door frame are the two sides of the angle. As you open or close the door, the size of the angle changes.
When naming an angle, we use three points: one point on each side and the vertex in the middle. For example, if the vertex is point B and the sides pass through points A and C, we write the angle as ∠ABC or ∠CBA. The vertex letter is always in the middle. Sometimes, if there is no confusion, we can name the angle using just the vertex letter, like ∠B.
Measuring Angles
The size of an angle is measured by the amount of rotation between its two sides. We measure angles in units called degrees, which we write with the symbol °. A full rotation around a point is 360°.
To measure an angle, we use a tool called a protractor. A protractor is a semicircular or circular tool marked with degree measurements from 0° to 180° (or 0° to 360° for a full circle protractor).
How to Use a Protractor
- Place the center point of the protractor (usually a small hole or mark at the base) exactly on the vertex of the angle.
- Align the baseline of the protractor (the 0° line) with one side of the angle.
- Read the degree measurement where the other side of the angle crosses the protractor scale.
- Make sure to use the correct scale-most protractors have two sets of numbers, one going left to right and one going right to left.
Example: You measure an angle with a protractor.
One side is aligned with the 0° mark.
The other side crosses the protractor at 65°.What is the measure of the angle?
Solution:
The angle measure is read directly from the protractor.
The measure is 65°.
The angle measures 65 degrees.
Types of Angles
Angles are classified based on their measures. There are several important types of angles you should know.
Acute Angle
An acute angle is an angle that measures greater than 0° but less than 90°. Acute angles are sharp and narrow.
Examples of acute angles: 30°, 45°, 60°, 89°.
Right Angle
A right angle is an angle that measures exactly 90°. Right angles are often marked with a small square symbol at the vertex. Right angles appear in the corners of squares and rectangles.
When you stand up straight, your body forms a right angle with the floor.
Obtuse Angle
An obtuse angle is an angle that measures greater than 90° but less than 180°. Obtuse angles are wider than right angles.
Examples of obtuse angles: 100°, 120°, 150°, 179°.
Straight Angle
A straight angle is an angle that measures exactly 180°. A straight angle looks like a straight line, with the two sides pointing in opposite directions.
Think of a straight angle as a line that has been "folded flat."
Reflex Angle
A reflex angle is an angle that measures greater than 180° but less than 360°. Reflex angles measure the "outside" or larger rotation between two rays.
Examples of reflex angles: 200°, 270°, 315°.
Full Rotation
A full rotation or complete angle measures exactly 360°. This is when one ray rotates all the way around and comes back to its starting position.
Example: Classify each angle based on its measure:
a) 42°
b) 90°
c) 135°
d) 180°What type is each angle?
Solution:
a) 42° is less than 90°, so it is an acute angle.
b) 90° is exactly a right angle, so it is a right angle.
c) 135° is greater than 90° but less than 180°, so it is an obtuse angle.
d) 180° is exactly a straight angle, so it is a straight angle.
The angles are acute, right, obtuse, and straight, respectively.
Angle Relationships
Angles can relate to each other in special ways. Understanding these relationships helps us solve problems and find unknown angle measures.
Adjacent Angles
Adjacent angles are two angles that share a common vertex and a common side, but do not overlap. The angles are next to each other.
Think of two slices of pizza sharing a crust edge-they are adjacent.
Complementary Angles
Complementary angles are two angles whose measures add up to 90°. The angles do not need to be adjacent to be complementary.
If angle A and angle B are complementary, then:
\[ m\angle A + m\angle B = 90° \]where \( m\angle A \) means "the measure of angle A."
Example: One angle measures 35°.
Find the measure of its complement.What is the complement of 35°?
Solution:
Complementary angles add up to 90°.
Let the unknown angle be \( x \).
\( 35° + x = 90° \)
\( x = 90° - 35° = 55° \)
The complement of 35° is 55°.
Supplementary Angles
Supplementary angles are two angles whose measures add up to 180°. Like complementary angles, supplementary angles do not need to be adjacent.
If angle C and angle D are supplementary, then:
\[ m\angle C + m\angle D = 180° \]Example: One angle measures 110°.
Find the measure of its supplement.What is the supplement of 110°?
Solution:
Supplementary angles add up to 180°.
Let the unknown angle be \( y \).
\( 110° + y = 180° \)
\( y = 180° - 110° = 70° \)
The supplement of 110° is 70°.
Vertical Angles
Vertical angles (also called vertically opposite angles) are formed when two lines intersect. They are the angles that are across from each other at the intersection point. Vertical angles are always equal in measure.
When two lines intersect, they form four angles. The angles that are opposite each other are vertical angles, and they have the same measure.
Imagine two pencils crossing to form an X. The angles at the top and bottom are vertical angles, and the angles on the left and right are also vertical angles.
Example: Two lines intersect to form four angles.
One angle measures 75°.
The angle directly across from it is a vertical angle.What is the measure of the vertical angle?
Solution:
Vertical angles are equal in measure.
If one angle is 75°, then the vertical angle is also 75°.
The vertical angle measures 75°.
Linear Pair
A linear pair is a pair of adjacent angles formed when two lines intersect. The two angles share a common side and their non-common sides form a straight line. The angles in a linear pair are always supplementary (they add up to 180°).
Example: Two angles form a linear pair.
One angle measures 130°.What is the measure of the other angle?
Solution:
Angles in a linear pair are supplementary.
They add up to 180°.
\( 130° + x = 180° \)
\( x = 180° - 130° = 50° \)
The other angle measures 50°.
Angles and Parallel Lines
When a line intersects two parallel lines, special angle relationships are created. The line that crosses the parallel lines is called a transversal.
Corresponding Angles
Corresponding angles are angles that are in the same relative position at each intersection where the transversal crosses the parallel lines. When two parallel lines are cut by a transversal, corresponding angles are equal.
Think of corresponding angles as being in matching positions-like the top-right corner at both intersections.
Alternate Interior Angles
Alternate interior angles are angles that are on opposite sides of the transversal and between (inside) the two parallel lines. When two parallel lines are cut by a transversal, alternate interior angles are equal.
Alternate Exterior Angles
Alternate exterior angles are angles that are on opposite sides of the transversal and outside the two parallel lines. When two parallel lines are cut by a transversal, alternate exterior angles are equal.
Same-Side Interior Angles
Same-side interior angles (also called consecutive interior angles or co-interior angles) are angles that are on the same side of the transversal and between the two parallel lines. When two parallel lines are cut by a transversal, same-side interior angles are supplementary (they add up to 180°).
Example: Two parallel lines are cut by a transversal.
One alternate interior angle measures 65°.What is the measure of the other alternate interior angle?
Solution:
Alternate interior angles formed by parallel lines and a transversal are equal.
If one alternate interior angle is 65°, the other is also 65°.
The other alternate interior angle measures 65°.
Drawing and Constructing Angles
You can draw angles of specific measures using a protractor and a straightedge (like a ruler).
Steps to Draw an Angle
- Draw a ray starting from a point. This point will be the vertex of your angle.
- Place the center of the protractor on the vertex.
- Align the 0° mark on the protractor with the ray you just drew.
- Find the desired degree measure on the protractor scale and make a small mark.
- Remove the protractor and use a straightedge to draw a ray from the vertex through your mark.
- Label the angle with the appropriate degree measure or angle name.
Finding Unknown Angle Measures
Many geometry problems require you to find unknown angle measures using the relationships you have learned. You will often set up and solve equations.
Example: Two angles are complementary.
One angle is 15° more than twice the other angle.Find the measure of each angle.
Solution:
Let the smaller angle be \( x \).
Then the larger angle is \( 2x + 15 \).
Complementary angles add up to 90°, so:
\( x + (2x + 15) = 90 \)
\( 3x + 15 = 90 \)
\( 3x = 75 \)
\( x = 25 \)
The smaller angle is 25°. The larger angle is \( 2(25) + 15 = 50 + 15 = 65° \).
Check: \( 25° + 65° = 90° \) ✓
The two angles measure 25° and 65°.
Example: Two angles form a linear pair.
One angle is four times the measure of the other.Find the measure of each angle.
Solution:
Let the smaller angle be \( a \).
Then the larger angle is \( 4a \).
Angles in a linear pair are supplementary, so they add to 180°:
\( a + 4a = 180 \)
\( 5a = 180 \)
\( a = 36 \)
The smaller angle is 36°. The larger angle is \( 4(36) = 144° \).
Check: \( 36° + 144° = 180° \) ✓
The two angles measure 36° and 144°.
Angles in Polygons
Polygons are closed figures made of straight sides. The angles inside a polygon are called interior angles. Understanding the angles in polygons helps us analyze shapes and solve geometric problems.
Sum of Interior Angles
The sum of the interior angles of a polygon depends on the number of sides. For a polygon with \( n \) sides, the sum of the interior angles is:
\[ \text{Sum of interior angles} = (n - 2) \times 180° \]where \( n \) is the number of sides.
- Triangle (3 sides): \( (3 - 2) \times 180° = 1 \times 180° = 180° \)
- Quadrilateral (4 sides): \( (4 - 2) \times 180° = 2 \times 180° = 360° \)
- Pentagon (5 sides): \( (5 - 2) \times 180° = 3 \times 180° = 540° \)
- Hexagon (6 sides): \( (6 - 2) \times 180° = 4 \times 180° = 720° \)
Example: A quadrilateral has four angles.
Three of the angles measure 85°, 95°, and 110°.What is the measure of the fourth angle?
Solution:
The sum of interior angles in a quadrilateral is 360°.
Let the fourth angle be \( x \).
\( 85° + 95° + 110° + x = 360° \)
\( 290° + x = 360° \)
\( x = 360° - 290° = 70° \)
The fourth angle measures 70°.
Exterior Angles
An exterior angle of a polygon is formed by extending one side of the polygon. For any polygon, the sum of the exterior angles (one at each vertex) is always 360°.
For a triangle, an important property is that an exterior angle equals the sum of the two non-adjacent (remote) interior angles.
Angle Bisectors
An angle bisector is a ray that divides an angle into two equal parts. Each part has half the measure of the original angle.
If ray BD bisects ∠ABC, then:
\[ m\angle ABD = m\angle DBC = \frac{1}{2} m\angle ABC \]Example: An angle measures 80°.
A ray bisects the angle into two equal parts.What is the measure of each part?
Solution:
An angle bisector divides an angle into two equal angles.
Each part is half of the original angle.
\( \text{Each part} = \frac{80°}{2} = 40° \)
Each part measures 40°.
