Lesson Notes By Weeks and Term v5 - Grade 9

Lesson Video

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Performance objectives

• Apply angle sum properties for triangles and quadrilaterals.
• Identify and apply properties of different triangles.
• Identify and apply properties of different quadrilaterals.
• Solve problems using geometric reasoning.
• Prove simple geometric theorems.

Lesson summary

Geometry is all around us, from the shape of our classrooms to the design of buildings. Understanding the properties of triangles and quadrilaterals is fundamental to solving practical problems in construction, design, and even sports. In South Africa, construction projects are booming, and understanding geometry is crucial for professionals involved in these fields. Even seemingly simple tasks like calculating the amount of paint needed for a wall requires geometrical knowledge. This week, we'll focus on theorems related to triangles and quadrilaterals, equipping you with the skills to analyze shapes and solve related problems.

Lesson notes

2. 1.

Triangles Definition: A triangle is a closed two-dimensional shape with three straight sides and three angles.

Types of Triangles (based on sides): Equilateral Triangle: All three sides are equal in length, and all three angles are equal (60° each).

Property:* All angles are 60°.

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

Property:* Base angles are equal.

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

Types of Triangles (based on 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 (the longest side).

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

Angle Sum Property of a Triangle: The sum of the interior angles of any triangle is always 180°. If ∠A, ∠B, and ∠C are the angles of a triangle, then ∠A + ∠B + ∠C = 180°.* Exterior Angle Theorem: An exterior angle of a triangle is equal to the sum of the two opposite interior angles. If 'd' is an exterior angle to a triangle, opposite interior angles 'a' and 'b', then d = a + b Example 1: In triangle ABC, ∠A = 50° and ∠B = 70°. Find ∠

C. Solution: Using the angle sum property of a triangle: ∠A + ∠B + ∠C = 180° 50° + 70° + ∠C = 180° 120° + ∠C = 180° ∠C = 180° - 120° ∠C = 60° Example 2: In a right-angled triangle, one of the acute angles is 35°. Find the other acute angle.

Solution: Let the other acute angle be x. Since it's a right-angled triangle, one angle is 90°.

Using the angle sum property: 90° + 35° + x = 180° 125° + x = 180° x = 180° - 125° x = 55° Example 3: An exterior angle of a triangle measures 110°. One of the interior opposite angles measures 40°. Find the other interior opposite angle.

Solution: Let the other interior opposite angle be y. 110° = 40° + y (Exterior Angle Theorem) y = 110° - 40° y = 70° 2.

2. Quadrilaterals Definition: A quadrilateral is a closed two-dimensional shape with four straight sides and four angles.

Types of Quadrilaterals: Parallelogram: A quadrilateral with both pairs of opposite sides parallel.

Properties:* Opposite sides are equal in length. Opposite angles are equal. Diagonals bisect each other (cut each other in half).

Rectangle: A parallelogram with all four angles equal to 90°.

Properties:* All properties of a parallelogram apply. All angles are right angles. Diagonals are equal in length.

Square: A rectangle with all four sides equal in length.

Properties:* All properties of a rectangle apply. All sides are equal. Diagonals are perpendicular bisectors of each other.

Rhombus: A parallelogram with all four sides equal in length.

Properties:* All properties of a parallelogram apply. All sides are equal. Diagonals bisect the angles. Diagonals are perpendicular bisectors of each other.

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

Property:* Only one pair of sides is parallel.

Kite: A quadrilateral with two pairs of adjacent sides equal in length.

Properties:* One pair of opposite angles is equal. Diagonals are perpendicular. The longer diagonal bisects the shorter diagonal and the angles at its endpoints.

Angle Sum Property of a Quadrilateral: The sum of the interior angles of any quadrilateral is always 360°. If ∠A, ∠B, ∠C, and ∠D are the angles of a quadrilateral, then ∠A + ∠B + ∠C + ∠D = 360°.* Example 4: In a parallelogram ABCD, ∠A = 110°. Find ∠C and ∠

B. Solution: In a parallelogram, opposite angles are equal.

Therefore, ∠C = ∠A = 110°. Also, adjacent angles are supplementary (add up to 180°).

Therefore, ∠B = 180° - ∠A = 180° - 110° = 70°.

Therefore, ∠C = 110° and ∠B = 70°.

Example 5: The angles of a quadrilateral are x, 2x, 3x, and 4x. Find the value of x.

Solution: Using the angle sum property of a quadrilateral: x + 2x + 3x + 4x = 360° 10x = 360° x = 360° / 10 x = 36° Example 6: In a kite, one angle between unequal sides measures 70°. The other angle between unequal sides measures 110°. Determine the other two angles of the kite, knowing that one pair of opposite angles in a kite are equal.

Solution: Let the other two angles both be a. The sum of the angles in a quadrilateral is 360°. 70° + 110° + a + a = 360° 180° + 2a = 360° 2a = 180° a = 90° Guided Practice (With Solutions)

Question 1: In triangle PQR, PQ = PR and ∠P = 40°. Find ∠Q and ∠

R. Solution: Since PQ = PR, triangle PQR is an isosceles triangle.

Therefore, ∠Q = ∠

R. Using the angle sum property of a triangle: ∠P + ∠Q + ∠R = 180° 40° + ∠Q + ∠Q = 180° (Since ∠Q = ∠R) 2∠Q = 180° - 40° 2∠Q = 140° ∠Q = 70° Therefore, ∠Q = ∠R = 70°.

Commentary: This question combines the properties of an isosceles triangle with the angle sum property. We first identify the type of triangle and then use its properties to simplify the problem.

Question 2: ABCD is a rectangle. If ∠BAC = 32°, find ∠BCA and ∠CD

A. Solution: In a rectangle, all angles are 90°.

Therefore, ∠CDA = 90°.