Geometry I

Grade 10 Math - Geometry

Lesson Objectives

Identify and classify plane figures.
Measure and calculate angles within polygons.
Understand the properties of polygons including triangles and quadrilaterals.
Interpret and create scale drawings accurately.

Lesson Introduction

Geometry deals with shapes, sizes, and the properties of space. This lesson explores different plane figures, how to calculate their angles, understand their properties, and create scale drawings used in technical drawings and maps.

Core Lesson Content

1. Plane Figures

Plane figures are two-dimensional shapes like triangles, rectangles, and circles. They lie on a flat surface.

2. Angles in Polygons

The sum of interior angles of a polygon with $n$ sides is given by:

$\text{Sum of interior angles} = (n - 2) \times 180^\circ$

3. Properties of Common Polygons

Triangle: 3 sides, sum of angles = $180^\circ$
Quadrilateral: 4 sides, sum of angles = $360^\circ$
Regular Polygon: All sides and angles are equal.

4. Scale Drawing

Scale drawings are accurate drawings of real objects, reduced or enlarged by a scale factor.

Example: If a map uses a scale of 1:100, it means 1 cm on the map represents 100 cm in real life.

Worked Example

Example 1: Find the sum of interior angles of a pentagon.

(5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ

Example 2: Calculate one interior angle of a regular hexagon.

\frac{(6 - 2) \times 180^\circ}{6} = \frac{720^\circ}{6} = 120^\circ

Example 3: What is the measure of the unknown angle in a triangle with angles

50^\circ

and

60^\circ

180^\circ - (50^\circ + 60^\circ) = 70^\circ

Example 4: A rectangle has a scale of 1:50. What is the real length if the drawing shows 6 cm?

6 \times 50 = 300 \text{ cm} = 3 \text{ m}

Example 5: A quadrilateral has three angles:

90^\circ, 85^\circ, 100^\circ

. Find the fourth angle.

360^\circ - (90^\circ + 85^\circ + 100^\circ) = 85^\circ

Example 6: Draw a triangle with sides 3 cm, 4 cm, and 5 cm using a scale of 1:1.
Use a ruler and compass to construct the triangle accurately with those lengths.

Example 7: How many sides does a polygon have if the sum of interior angles is

1260^\circ

(n - 2) \times 180^\circ = 1260^\circ \Rightarrow n - 2 = 7 \Rightarrow n = 9

Example 8: [WAEC] A regular polygon has an exterior angle of

30^\circ

. How many sides does it have?

\frac{360^\circ}{30^\circ} = 12

Example 9: What is the scale factor if a 2 cm line represents 10 m?
Convert 10 m = 1000 cm. So, scale =

\frac{2}{1000} = 1:500

Example 10: What is the measure of each exterior angle of a regular octagon?

\frac{360^\circ}{8} = 45^\circ

Exercises

Find the sum of interior angles of a decagon.
[WAEC] Calculate one interior angle of a regular nonagon. (Past Question)
Find the number of sides in a polygon with interior angles summing to $1440^\circ$ .
[NECO] A map scale is 1:200. What is the real distance if 5 cm is measured? (Past Question)
[WAEC] A quadrilateral has angles: $95^\circ, 85^\circ, 90^\circ$ . Find the fourth angle. (Past Question)
What is the size of each exterior angle of a regular dodecagon?
[JAMB] Draw a scale drawing of a room that is 6 m by 4 m using a 1:100 scale. (Past Question)
How many triangles are formed when diagonals are drawn from one vertex of a heptagon?
Find the measure of an angle in an equilateral triangle.
Determine the total number of diagonals in an octagon.

Conclusion/Recap

This lesson covered identifying and classifying plane figures, computing angles, properties of polygons, and drawing objects using scale. These concepts are widely used in design, architecture, and engineering. Next, we will explore constructions involving compass and ruler.