Comprehensive Guide to Angles
Table of Contents
1. Introduction to Angles
An angle is formed when two rays share a common endpoint called the vertex. Angles are measured in degrees (°) or radians (rad).
Key Terms:
- Vertex: The common endpoint of the two rays
- Arms/Sides: The two rays that form the angle
- Measure: The amount of rotation from one arm to another
Units of Measurement:
- Degrees (°): A full circle is 360°
- Radians (rad): A full circle is 2π radians
- Conversion: 180° = π radians
2. Types of Angles
Acute Angle
Measures less than 90°
Right Angle
Measures exactly 90°
Obtuse Angle
Measures between 90° and 180°
Straight Angle
Measures exactly 180°
Reflex Angle
Measures between 180° and 360°
Complete Angle
Measures exactly 360°
Example Problem:
If an angle measures 135°, what type of angle is it?
Solution: Since 135° is greater than 90° but less than 180°, it is an obtuse angle.
3. Special Angle Pairs
Complementary Angles
Two angles whose sum is 90°
α + β = 90°
If α = 30°, then β = 60°
Supplementary Angles
Two angles whose sum is 180°
α + β = 180°
If α = 110°, then β = 70°
Vertical Angles
Opposite angles formed by two intersecting lines
Vertical angles are equal
α = α
Adjacent Angles
Angles that share a common vertex and side
α + β = 180° (if they form a straight line)
If α = 45°, then β = 135°
Parallel Lines & Transversal
Corresponding Angles: 1&5, 2&6, 3&7, 4&8 (equal)
Alternate Interior: 3&6, 4&5 (equal)
Alternate Exterior: 1&8, 2&7 (equal)
Same-Side Interior: 3&5, 4&6 (supplementary)
Example Problem:
If two parallel lines are cut by a transversal and one of the alternate interior angles measures 65°, what is the measure of the corresponding angle?
Solution: Since alternate interior angles are equal when lines are parallel, the other alternate interior angle is also 65°. The corresponding angle to this alternate interior angle is also equal to 65°.
4. Angles in Triangles
Triangle Angle Properties:
- The sum of interior angles in any triangle is 180°
- The exterior angle of a triangle equals the sum of the two non-adjacent interior angles
- In an equilateral triangle, all angles are 60°
- In an isosceles triangle, the angles opposite the equal sides are equal
- In a right-angled triangle, one angle is 90° and the other two angles are complementary
Acute Triangle
All angles are less than 90°
Right Triangle
One angle is 90°
Obtuse Triangle
One angle is greater than 90°
Example Problem:
In a triangle, if two angles measure 45° and 60°, what is the measure of the third angle?
Solution: Using the triangle angle sum property:
α + β + γ = 180°
45° + 60° + γ = 180°
105° + γ = 180°
γ = 75°
5. Angles in Polygons
Polygon Angle Properties:
- The sum of interior angles in a polygon with n sides = (n-2) × 180°
- Each interior angle of a regular polygon with n sides = (n-2) × 180° ÷ n
- The sum of exterior angles of any polygon = 360°
- Each exterior angle of a regular polygon = 360° ÷ n
| Polygon | Sides | Interior Angle Sum | Regular Interior Angle |
|---|---|---|---|
| Triangle | 3 | 180° | 60° |
| Quadrilateral | 4 | 360° | 90° |
| Pentagon | 5 | 540° | 108° |
| Hexagon | 6 | 720° | 120° |
Regular Pentagon
Exterior Angles
Example Problem:
Find the sum of interior angles in a decagon (10-sided polygon).
Solution: Using the formula for the sum of interior angles:
Sum = (n-2) × 180°
Sum = (10-2) × 180°
Sum = 8 × 180°
Sum = 1440°
6. Angles in Circles
Circle Angle Properties:
- The angle at the center of a circle is twice the angle at the circumference when both angles are subtended by the same arc
- Angles in the same segment of a circle are equal
- The angle in a semicircle is always 90° (right angle)
- Opposite angles in a cyclic quadrilateral sum to 180° (supplementary)
- The angle between a tangent and a radius is 90° (right angle)
Inscribed Angle
Angle formed by two chords meeting on the circle
Measured by half the arc it subtends
Central Angle
Angle formed at the center of a circle
Measured by the arc it subtends
Angle in a Semicircle
Any angle inscribed in a semicircle
Always equals 90° (right angle)
Tangent-Radius Angle
Angle between a tangent and a radius
Always equals 90° (right angle)
Example Problem:
If the central angle in a circle measures 120°, what is the measure of the inscribed angle that subtends the same arc?
Solution: Using the relationship between central and inscribed angles:
Inscribed angle = Central angle ÷ 2
Inscribed angle = 120° ÷ 2
Inscribed angle = 60°
7. Methods to Solve Angle Problems
Common Approaches:
- Identify angle relationships (complementary, supplementary, vertical, etc.)
- Use angle sum properties (triangle = 180°, quadrilateral = 360°)
- Apply parallel line properties (corresponding, alternate, etc.)
- Use circular angle theorems for problems involving circles
- Draw auxiliary lines to create useful triangles or angles
- Work step-by-step, using known angles to find unknown ones
Problem-Solving Strategies:
- Label all angles (use variables for unknown angles)
- Mark equal angles with the same symbol
- Set up equations based on angle relationships
- Work systematically from what you know to what you need to find
- Check your answer by verifying all angle properties are satisfied
Comprehensive Example Problem:
In the figure, two parallel lines are cut by a transversal. If one of the angles is 115° as shown, find the values of x, y, and z.
Solution:
Step 1: Since the given angle is 115°, its supplement is 180° - 115° = 65°.
Step 2: Angle x is supplementary to 115° (they form a straight line)
Therefore, x = 180° - 115° = 65°
Step 3: Angle y corresponds to the 115° angle (when parallel lines are cut by a transversal, corresponding angles are equal)
Therefore, y = 115°
Step 4: Angle z corresponds to the 65° angle
Therefore, z = 65°
8. Test Your Knowledge: Angles Quiz
Test your understanding of angle concepts with this interactive quiz. Select your answer and click "Submit" to check if you're correct!
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