Lesson Notes By Weeks and Term v5 - Grade 8

Lesson Video

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Performance objectives

• Classify triangles by their sides and angles.
• Calculate unknown angles in triangles using the 180° rule.
• Describe the properties of different quadrilaterals.
• Calculate unknown angles in quadrilaterals using the 360° rule.
• Apply these properties to solve problems.

Lesson summary

This week, we delve into the fascinating world of geometry, specifically focusing on the properties of triangles and quadrilaterals. Understanding these properties is crucial not only for success in mathematics but also for developing critical thinking and problem-solving skills applicable in many real-life situations. From designing buildings and bridges to understanding the layout of our homes and communities, geometric principles are all around us. In South Africa, with its rich architectural heritage and diverse landscape, a strong grasp of geometry allows us to appreciate and understand the structures and spaces that shape our lives.

Lesson notes

Triangles: Definition: A triangle is a closed two-dimensional shape with three straight 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 those 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. Why?* Because if you tear off the three angles of a triangle and place them adjacent to each other, they will form a straight line (180°). Let the angles of a triangle be a, b, and c.

Then: a + b + c = 180°* Example 1: In a triangle, two angles are 50° and 70°. Find the third angle. Let the third angle be x. 50° + 70° + x = 180° 120° + x = 180° x = 180° - 120° x = 60° Example 2: An isosceles triangle has one angle of 40° that is NOT one of the equal angles. Find the other two angles. Let each of the equal angles be y. 40° + y + y = 180° 40° + 2y = 180° 2y = 180° - 40° 2y = 140° y = 70° Quadrilaterals: Definition: A quadrilateral is a closed two-dimensional shape with four straight sides and four angles. Types of Quadrilaterals and their Properties: Square: All four sides are equal, all four angles are right angles (90°). Diagonals are equal and bisect each other at right angles.

Rectangle: Opposite sides are equal, all four angles are right angles (90°). Diagonals are equal and bisect each other.

Parallelogram: Opposite sides are equal and parallel, opposite angles are equal. Diagonals bisect each other.

Rhombus: All four sides are equal, opposite angles are equal. Diagonals bisect each other at right angles.

Trapezium (or Trapezoid): At least one pair of opposite sides is parallel.

Kite: Two pairs of adjacent sides are equal. Diagonals intersect at right angles, and one diagonal bisects the other.

Angle Sum Property: The sum of the interior angles of any quadrilateral is always 360°. Why?* You can divide any quadrilateral into two triangles, and each triangle has an angle sum of 180°. Let the angles of a quadrilateral be p, q, r, and s.

Then: p + q + r + s = 360°* Example 3: In a quadrilateral, three angles are 80°, 90°, and 100°. Find the fourth angle. Let the fourth angle be z. 80° + 90° + 100° + z = 360° 270° + z = 360° z = 360° - 270° z = 90° Example 4: A parallelogram has one angle of 60°. Find the measure of the other angles. In a parallelogram, opposite angles are equal. So, one opposite angle is also 60°. Let the other two equal angles each be w. 60° + 60° + w + w = 360° 120° + 2w = 360° 2w = 360° - 120° 2w = 240° w = 120° Therefore, the angles of the parallelogram are 60°, 120°, 60°, and 120°. Guided Practice (With Solutions)

Question 1: A triangle has angles of 30° and 80°. What is the size of the third angle? Is this triangle acute, obtuse or right angled?

Solution: Let the third angle be x. 30° + 80° + x = 180° 110° + x = 180° x = 180° - 110° x = 70° All angles are less than 90 degrees, therefore it is an acute triangle.

Commentary: This question reinforces the angle sum property of triangles. The additional question forces students to classify it based on its angles.

Question 2: ABCD is a rectangle. Angle BAC is 35°. Find angle BC

A. Solution: In rectangle ABCD, angle ABC is 90°. Triangle ABC is a right-angled triangle. The angles in triangle ABC add up to 180°. Angle BAC + Angle ABC + Angle BCA = 180° 35° + 90° + Angle BCA = 180° 125° + Angle BCA = 180° Angle BCA = 180° - 125° Angle BCA = 55°

Commentary: This problem combines knowledge of rectangles and triangles. Students need to recall that all angles in a rectangle are 90 degrees.

Question 3: In a quadrilateral PQRS, angle P = 70°, angle Q = 110°, and angle R = 80°. Find the measure of angle

S. Solution: The sum of the angles in a quadrilateral is 360°. Angle P + Angle Q + Angle R + Angle S = 360° 70° + 110° + 80° + Angle S = 360° 260° + Angle S = 360° Angle S = 360° - 260° Angle S = 100°

Commentary: A straightforward application of the angle sum property of quadrilaterals.

Question 4: One angle of a rhombus is 120°. What are the measures of the other three angles?

Solution: In a rhombus, opposite angles are equal.

Therefore, one other angle is also 120°. Let the other two equal angles be a. 120° + 120° + a + a = 360° 240° + 2a = 360° 2a = 360° - 240° 2a = 120° a = 60° Therefore, the angles are 120°, 60°, 120°, and 60°.

Commentary: This question uses the angle properties specific to a rhombus. Independent Practice (Questions Only) What type of triangle has sides of length 5cm, 12cm, and 13cm? (Hint: Use the Pythagorean theorem).