7.1 Angles and Polygons
Understanding Angles
An angle measures the amount of turn between two lines. Angles are measured in degrees (°).
An angle measures the amount of turn between two lines. Angles are measured in degrees (°).
⚡ Key Angle Facts:
Angles on a Straight Line
Add up to 180°Angles at a Point
Add up to 360°Vertically Opposite Angles
Are equal
Example 1: Angles on a Straight Line
Find the value of $x$.
Rule: Angles on a straight line add up to 180°
$x + 120° = 180°$
$x = 180° - 120°$
$x = 60°$
Check: $60° + 120° = 180°$ ✓
Rule: Angles on a straight line add up to 180°
$x + 120° = 180°$
$x = 180° - 120°$
$x = 60°$
Check: $60° + 120° = 180°$ ✓
Example 2: Angles at a Point
Find the value of $x$.
Rule: Angles at a point add up to 360°
$80° + 95° + 110° + x = 360°$
$285° + x = 360°$
$x = 360° - 285°$
$x = 75°$
Rule: Angles at a point add up to 360°
$80° + 95° + 110° + x = 360°$
$285° + x = 360°$
$x = 360° - 285°$
$x = 75°$
Example 3: Vertically Opposite Angles
Rule: Vertically opposite angles are equal
When two straight lines cross, they create two pairs of equal angles.
The angles opposite each other are always the same!
When two straight lines cross, they create two pairs of equal angles.
The angles opposite each other are always the same!
💡 Angle Types:
Acute
< 90°
Smaller than a right angle
Right-angles
= 90°
A quarter turn
Obtuse
90° - 180°
Bigger than right angle
Straight
= 180°
A straight line
Reflex
180° - 360°
More than half a turn
🎯 Find the Missing Angle
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Parallel Lines and Transversals
When a line crosses two parallel lines, it creates special angle relationships.
When a line crosses two parallel lines, it creates special angle relationships.
⚡ The Three Rules:
Alternate Angles
ARE EQUAL
(Z-pattern)
Corresponding Angles
ARE EQUAL
(F-pattern)
Co-interior Angles
ADD TO 180°
(C-pattern or U-pattern)
Example 1: Alternate Angles (Z-pattern)
Two parallel lines are cut by a transversal. One angle is 65°. Find the alternate angle.
Rule: Alternate angles are equal (Z-pattern)
$x = 65°$
Remember: Look for the Z shape!
$x = 65°$
Remember: Look for the Z shape!
Example 2: Corresponding Angles (F-pattern)
Find angle $y$ when corresponding angle is 110°.
Rule: Corresponding angles are equal (F-pattern)
$y = 110°$
Remember: Look for the F shape!
$y = 110°$
Remember: Look for the F shape!
Example 3: Co-interior Angles (C-pattern)
Find angle $z$ when the other co-interior angle is 120°.
Rule: Co-interior angles add up to 180° (C-pattern)
$120° + z = 180°$
$z = 180° - 120°$
$z = 60°$
Remember: Look for the C or U shape!
$120° + z = 180°$
$z = 180° - 120°$
$z = 60°$
Remember: Look for the C or U shape!
🎯 Parallel Lines Practice
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Understanding Polygons
A polygon is a closed shape made of straight lines.
A polygon is a closed shape made of straight lines.
Triangle
3 sides
Quadrilateral
4 sides
Pentagon
5 sides
Hexagon
6 sides
Octagon
8 sides
⚡ Interior Angles Formula:
For a polygon with $n$ sides:
Sum of interior angles:
$$\text{Sum} = (n - 2) \times 180°$$
Each interior angle (regular polygon):
$$\text{Each angle} = \frac{(n - 2) \times 180°}{n}$$
Sum of interior angles:
$$\text{Sum} = (n - 2) \times 180°$$
Each interior angle (regular polygon):
$$\text{Each angle} = \frac{(n - 2) \times 180°}{n}$$
Example 1: Sum of Interior Angles
Find the sum of interior angles in a hexagon (6 sides).
Formula: $(n - 2) \times 180°$
$n = 6$ (hexagon has 6 sides)
Sum $= (6 - 2) \times 180°$
Sum $= 4 \times 180°$
Sum $= 720°$
Answer: The sum of interior angles in a hexagon is 720°
Formula: $(n - 2) \times 180°$
$n = 6$ (hexagon has 6 sides)
Sum $= (6 - 2) \times 180°$
Sum $= 4 \times 180°$
Sum $= 720°$
Answer: The sum of interior angles in a hexagon is 720°
Example 2: Each Interior Angle (Regular Polygon)
Find each interior angle of a regular pentagon.
Step 1: Find the sum
Sum $= (5 - 2) \times 180° = 3 \times 180° = 540°$
Step 2: Divide by number of angles
Each angle $= 540° \div 5 = 108°$
Answer: Each interior angle is 108°
Step 1: Find the sum
Sum $= (5 - 2) \times 180° = 3 \times 180° = 540°$
Step 2: Divide by number of angles
Each angle $= 540° \div 5 = 108°$
Answer: Each interior angle is 108°
Example 3: Finding a Missing Angle
A quadrilateral has angles 85°, 110°, 95°, and $x$. Find $x$.
Step 1: Sum of angles in quadrilateral
$(4 - 2) \times 180° = 360°$
Step 2: Write equation
$85° + 110° + 95° + x = 360°$
$290° + x = 360°$
$x = 70°$
Step 1: Sum of angles in quadrilateral
$(4 - 2) \times 180° = 360°$
Step 2: Write equation
$85° + 110° + 95° + x = 360°$
$290° + x = 360°$
$x = 70°$
⚡ Exterior Angles:
Key facts about exterior angles:
• Exterior angles always sum to 360° (for any polygon)
• Each exterior angle of a regular polygon: $\frac{360°}{n}$
• Interior + Exterior = 180° (on a straight line)
• Exterior angles always sum to 360° (for any polygon)
• Each exterior angle of a regular polygon: $\frac{360°}{n}$
• Interior + Exterior = 180° (on a straight line)
Example 4: Exterior Angles
Find each exterior angle of a regular octagon.
Formula: $\frac{360°}{n}$
Each exterior angle $= \frac{360°}{8} = 45°$
Check: Interior angle $= 180° - 45° = 135°$
Using interior formula: $\frac{(8-2) \times 180°}{8} = \frac{1080°}{8} = 135°$ ✓
Formula: $\frac{360°}{n}$
Each exterior angle $= \frac{360°}{8} = 45°$
Check: Interior angle $= 180° - 45° = 135°$
Using interior formula: $\frac{(8-2) \times 180°}{8} = \frac{1080°}{8} = 135°$ ✓
Example 5: Finding Number of Sides
A regular polygon has each exterior angle equal to 40°. How many sides does it have?
Formula: Each exterior angle $= \frac{360°}{n}$
$40° = \frac{360°}{n}$
$40n = 360°$
$n = 360° \div 40°$
$n = 9$
Answer: The polygon has 9 sides (it's a nonagon)
Formula: Each exterior angle $= \frac{360°}{n}$
$40° = \frac{360°}{n}$
$40n = 360°$
$n = 360° \div 40°$
$n = 9$
Answer: The polygon has 9 sides (it's a nonagon)
🧮 Polygon Calculator:
💡 Common Polygons:
| Name | Sides | Sum of Interior Angles | Each Interior (Regular) |
|---|---|---|---|
| Triangle | 3 | 180° | 60° |
| Quadrilateral | 4 | 360° | 90° |
| Pentagon | 5 | 540° | 108° |
| Hexagon | 6 | 720° | 120° |
| Octagon | 8 | 1080° | 135° |
🎯 Polygon Angles
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Real Life Uses:
• Architecture: Designing buildings with angles (roofs, staircases)
• Engineering: Bridge construction and support angles
• Art & Design: Creating patterns and tessellations
• Navigation: Using angles for directions and bearings
• Carpentry: Cutting wood at correct angles for joints
• Sports: Understanding angles in ball trajectories
• Architecture: Designing buildings with angles (roofs, staircases)
• Engineering: Bridge construction and support angles
• Art & Design: Creating patterns and tessellations
• Navigation: Using angles for directions and bearings
• Carpentry: Cutting wood at correct angles for joints
• Sports: Understanding angles in ball trajectories