Introduction to Polygons
A polygon is a closed two-dimensional shape with straight sides. Polygons are named according to the number of sides they have:
Common Polygons:
- Triangle: 3 sides
- Quadrilateral: 4 sides
- Pentagon: 5 sides
- Hexagon: 6 sides
- Heptagon: 7 sides
- Octagon: 8 sides
Regular vs Irregular:
A regular polygon has all sides and all angles equal. An irregular polygon has sides and/or angles that are not equal.
The sum of interior angles in any polygon can be calculated using the formula:
\[ \text{Sum of interior angles} = (n - 2) \times 180° \]where \( n \) is the number of sides.
Angles in Triangles
Sum of Angles in a Triangle
For a triangle (\( n = 3 \)):
\[ \begin{align*} \text{Sum of angles} &= (3 - 2) \times 180° \\ &= 1 \times 180° \\ &= 180° \end{align*} \]Example 1: Find the missing angle \( x \) in the triangle below:
Given angles: 50° and 60°
\[ \begin{align*} 50° + 60° + x &= 180° \\ 110° + x &= 180° \\ x &= 180° - 110° \\ x &= 70° \end{align*} \]Key Concept
The sum of interior angles in any triangle is always 180°. This property can be used to find missing angles when two angles are known.
Angles in Quadrilaterals
Sum of Angles in a Quadrilateral
For a quadrilateral (\( n = 4 \)):
\[ \begin{align*} \text{Sum of angles} &= (4 - 2) \times 180° \\ &= 2 \times 180° \\ &= 360° \end{align*} \]Example 2: Find the missing angle \( x \) in the quadrilateral below:
Given angles: 80°, 95°, and 110°
\[ \begin{align*} 80° + 95° + 110° + x &= 360° \\ 285° + x &= 360° \\ x &= 360° - 285° \\ x &= 75° \end{align*} \]Key Concept
The sum of interior angles in any quadrilateral is always 360°. This property can be used to find missing angles when three angles are known.
Angles in Polygons
General Formula for Polygons
The sum of interior angles for any polygon with \( n \) sides is given by:
\[ \text{Sum of interior angles} = (n - 2) \times 180° \]Example 3: Find the sum of interior angles in a pentagon (\( n = 5 \)):
\[ \begin{align*} \text{Sum of angles} &= (5 - 2) \times 180° \\ &= 3 \times 180° \\ &= 540° \end{align*} \]Example 4: Find the sum of interior angles in a hexagon (\( n = 6 \)):
\[ \begin{align*} \text{Sum of angles} &= (6 - 2) \times 180° \\ &= 4 \times 180° \\ &= 720° \end{align*} \]Key Concept
The formula \((n - 2) \times 180°\) works for any polygon, whether regular or irregular. For regular polygons, each interior angle can be found by dividing the sum by \( n \).
Finding Missing Angles
Solving for Missing Angles
To find a missing angle in a polygon:
- Calculate the sum of interior angles using \((n - 2) \times 180°\)
- Add up all the known angles
- Subtract the sum of known angles from the total sum
Example 5: Find the missing angle \( x \) in a pentagon with angles 100°, 110°, 120°, and 130°:
\[ \begin{align*} \text{Total sum} &= (5 - 2) \times 180° = 540° \\ \text{Sum of known angles} &= 100° + 110° + 120° + 130° = 460° \\ x &= 540° - 460° = 80° \end{align*} \]Example 6: Find two missing angles \( x \) and \( y \) in a hexagon with angles 120°, 115°, 130°, and 125° (given \( x = y \)):
\[ \begin{align*} \text{Total sum} &= (6 - 2) \times 180° = 720° \\ \text{Sum of known angles} &= 120° + 115° + 130° + 125° = 490° \\ x + y &= 720° - 490° = 230° \\ \text{Since } x = y, \quad 2x &= 230° \\ x &= 115°, \quad y = 115° \end{align*} \]Practice Exercise
Question 1
Find the sum of interior angles in a heptagon (7-sided polygon).
Question 2
A triangle has angles measuring 45° and 60°. Find the measure of the third angle.
Question 3
A quadrilateral has angles measuring 80°, 95°, and 110°. Find the measure of the fourth angle.
Question 4
A regular pentagon has five equal angles. Calculate the measure of each interior angle.
Question 5
An irregular hexagon has angles measuring 120°, 130°, 140°, 110°, and 115°. Find the measure of the sixth angle.
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