Lesson Notes By Weeks and Term v5 - Grade 8

Lesson Video

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Performance objectives

• Classify triangles based on their sides and angles.
• Apply the angle sum property of triangles to find unknown angles.
• Identify the properties of different quadrilaterals.
• Calculate the area and perimeter of squares, rectangles, and triangles.
• Solve problems involving angle properties of quadrilaterals.

Lesson summary

This week, we delve deeper into the fascinating world of geometry, focusing on the properties of triangles and quadrilaterals. Geometry is more than just shapes; it's the foundation for understanding spatial relationships, architectural designs, and even the layout of our communities. From the tiles on your kitchen floor to the roof of your school, geometry is everywhere! Understanding these properties will help you analyze the world around you and build a strong foundation for future mathematics courses. In South Africa, a good grasp of geometry is essential for careers in fields like engineering, architecture, surveying, and even construction.

Lesson notes

2.1 Triangles A triangle is a polygon with three sides and three angles.

Classification by Sides: Equilateral Triangle: All three sides are equal in length, and all three angles are equal (60° each).

Isosceles Triangle: Two sides are equal in length, and the angles opposite these sides are equal.

Scalene Triangle: All three sides are of different lengths, and all three angles are different.

Classification by Angles: Acute-angled Triangle: All three angles are less than 90°.

Right-angled Triangle: One angle is exactly 90°. The side opposite the right angle is called the hypotenuse.

Obtuse-angled Triangle: One angle is greater than 90°.

Angle Sum Property: The sum of the interior angles of any triangle is always 180°. This is a fundamental property that allows us to calculate unknown angles.

Example 1: In triangle ABC, angle A = 70° and angle B = 50°. Find angle

C. Solution: Angle A + Angle B + Angle C = 180° 70° + 50° + Angle C = 180° 120° + Angle C = 180° Angle C = 180° - 120° Angle C = 60° Example 2: An isosceles triangle has one angle measuring 40°. The other two angles are equal. Find the measure of the equal angles.

Solution: Let the two equal angles be x. 40° + x + x = 180° 40° + 2x = 180° 2x = 180° - 40° 2x = 140° x = 140° / 2 x = 70° Therefore, each of the equal angles measures 70°. 2.2 Quadrilaterals A quadrilateral is a polygon with four sides and four angles. The sum of the interior angles of any quadrilateral is 360°.

Parallelogram: A quadrilateral with two pairs of parallel sides. Opposite sides are equal in length. Opposite angles are equal. Diagonals bisect each other (cut each other in half).

Rectangle: A parallelogram with four right angles (90°). Opposite sides are equal in length. All angles are 90°. Diagonals are equal in length and bisect each other.

Square: A rectangle with all sides equal in length. All sides are equal in length. All angles are 90°. Diagonals are equal in length, bisect each other at right angles (90°), and bisect the angles of the square (45°).

Rhombus: A parallelogram with all sides equal in length. All sides are equal in length. Opposite angles are equal. Diagonals bisect each other at right angles (90°) and bisect the angles of the rhombus.

Trapezium (Trapezoid): A quadrilateral with only one pair of parallel sides.

Kite: A quadrilateral with two pairs of adjacent sides that are equal in length. Diagonals intersect at right angles (90°). One diagonal bisects the other diagonal. One diagonal bisects a pair of opposite angles.

Example 3: In a parallelogram ABCD, angle A = 110°. Find angle C and angle

B. Solution: In a parallelogram, opposite angles are equal, so angle C = angle A = 110°. Adjacent angles in a parallelogram are supplementary (add up to 180°).

Therefore, angle B = 180° - angle A = 180° - 110° = 70°. Angle D = angle B = 70°.

Example 4: The perimeter of a square is 24 cm. Find the length of each side.

Solution: A square has four equal sides. Perimeter = 4 side length 24 cm = 4 side length Side length = 24 cm / 4 Side length = 6 cm Guided Practice (With Solutions)

Question 1: Classify the triangle with sides 5cm, 7cm, and 9cm.

Solution: Since all three sides have different lengths, this is a scalene triangle.

Question 2: In a right-angled triangle, one of the acute angles is 35°. Find the measure of the other acute angle.

Solution: Let the other acute angle be x. Since it's a right-angled triangle, one angle is 90°. 90° + 35° + x = 180° 125° + x = 180° x = 180° - 125° x = 55° The other acute angle is 55°. The logic here is using the angle sum property and the definition of a right-angled triangle.

Question 3: A quadrilateral has angles of 80°, 100°, and 70°. What is the measure of the fourth angle?

Solution: Let the fourth angle be y. 80° + 100° + 70° + y = 360° 250° + y = 360° y = 360° - 250° y = 110° The fourth angle is 110°. Remember that the sum of interior angles in any quadrilateral is always 360°.

Question 4: A rectangle has a length of 8cm and a width of 5cm. Calculate its perimeter.

Solution: Perimeter of a rectangle = 2 (length + width) Perimeter = 2 (8cm + 5cm) Perimeter = 2 13cm Perimeter = 26cm The perimeter of the rectangle is 26cm. Independent Practice (Questions Only) Classify the triangle with angles 60°, 60°, and 60°. In triangle PQR, angle P = 30° and angle Q = 80°. Find angle R. What is the name of a quadrilateral with all sides equal and all angles equal to 90 degrees? The perimeter of an equilateral triangle is 36cm. Find the length of each side. A kite has angles of 70°, 80° and 70°. Calculate the size of the fourth angle. One angle of a rhombus is 120°. Find the size of the opposite angle. Calculate the area of a rectangle with length 12cm and width 7cm. One of the angles in a parallelogram is 65°. What are the sizes of the other three angles? The area of a square is 81 cm². Find the length of its side. A trapezium has parallel sides of length 10 cm and 14 cm, and a height of 5 cm. Calculate its area.