Lesson Notes By Weeks and Term v5 - Grade 11

Lesson Video

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Performance objectives

• Apply the 'angle at the centre' theorem.
• Apply the 'angles in the same segment' theorem.
• Solve problems with cyclic quadrilaterals (opposite angles).
• Apply the 'exterior angle of a cyclic quad' theorem.
• Integrate multiple theorems to solve complex problems.

Lesson summary

This week, we delve deeper into the fascinating world of Euclidean Geometry, specifically focusing on circles. Circles are fundamental geometric shapes that appear everywhere around us, from the wheels of a car driving down a dusty road in Limpopo to the shape of the sun that warms our country. Understanding the properties of circles allows us to solve practical problems in engineering, architecture, and even sports. For example, knowing the properties of tangents to a circle is crucial for designing efficient road layouts and roundabouts to ease traffic flow in bustling Gauteng cities.

Lesson notes

2.1 Angle at Centre Theorem: The angle subtended by a chord at the centre of a circle is twice the angle subtended by the same chord at the circumference. Why? This theorem arises from the relationships between isosceles triangles formed within the circle. How? Consider a chord AB in a circle with centre

O. Let C be a point on the circumference. Then, angle AOB = 2 * angle AC

B. Example 1: In the diagram below, O is the centre of the circle, and angle ACB = 35°. Calculate the size of angle AOB. [Insert a diagram of a circle with centre O, chord AB, and point C on the circumference. Label angle ACB as 35 degrees and angle AOB as unknown.] Solution: Angle AOB = 2 * Angle ACB Angle AOB = 2 * 35° Angle AOB = 70° 2.2 Angles in the Same Segment Theorem: Angles subtended by the same chord (or arc) in the same segment of a circle are equal. Why? This is a direct consequence of the angle at the centre theorem. Since the angle at the centre is constant for a given chord, the angles at the circumference must also be equal. How? If A, B, C, and D are points on the circumference of a circle, and angles ACB and ADB are subtended by chord AB, then angle ACB = angle AD

B. Example 2: In the diagram below, points A, B, C, and D lie on the circumference of a circle. If angle BAC = 40° and angle BDC = x, determine the value of x. [Insert a diagram of a circle with points A, B, C, and D on the circumference. Label angle BAC as 40 degrees and angle BDC as x.] Solution: Angle BDC = Angle BAC (Angles in the same segment are equal) Therefore, x = 40° 2.3 Cyclic Quadrilaterals: A cyclic quadrilateral is a quadrilateral whose vertices all lie on the circumference of a circle. 2.3.1 Opposite Angles of a Cyclic Quadrilateral: The opposite angles of a cyclic quadrilateral are supplementary (add up to 180°). Why? This theorem can be proven using the angle at the centre theorem and the fact that angles around a point sum to 360°. How? If ABCD is a cyclic quadrilateral, then angle A + angle C = 180° and angle B + angle D = 180°.

Example 3: ABCD is a cyclic quadrilateral. If angle A = 110°, find the size of angle C. [Insert a diagram of a cyclic quadrilateral ABC

D. Label angle A as 110 degrees and angle C as unknown.] Solution: Angle A + Angle C = 180° (Opposite angles of a cyclic quadrilateral) 110° + Angle C = 180° Angle C = 180° - 110° Angle C = 70° 2.3.2 Exterior Angle of a Cyclic Quadrilateral: The exterior angle of a cyclic quadrilateral is equal to the interior opposite angle. Why? This can be proven using the properties of supplementary angles and the fact that opposite angles of a cyclic quadrilateral are supplementary. How? If ABCD is a cyclic quadrilateral, and BCE is an exterior angle at vertex C, then angle BCE = angle

A. Example 4: ABCD is a cyclic quadrilateral with side BC produced to E. If angle A = 85°, find the size of angle DCE. [Insert a diagram of a cyclic quadrilateral ABCD with side BC produced to

E. Label angle A as 85 degrees and angle DCE as unknown.] Solution: Angle DCE = Angle A (Exterior angle of cyclic quad = Interior opposite angle) Angle DCE = 85° Guided Practice (With Solutions)

Question 1: In the diagram, O is the centre of the circle. If angle ABC = 48°, find the size of angle AOC. [Insert a diagram of a circle with centre O, points A, B, and C on the circumference. Angle ABC is 48 degrees, and angle AOC is the unknown.] Solution: Angle AOC = 2 Angle ABC (Angle at centre = 2 angle at circumference) Angle AOC = 2 * 48° Angle AOC = 96°

Commentary: This question directly applies the angle at the centre theorem. It's crucial to identify that angle ABC is subtended by the same chord (AC) as angle AOC at the centre.

Question 2: In the diagram, points P, Q, R, and S lie on the circumference of a circle. If angle PQS = 32° and angle PRS = x, determine the value of x. [Insert a diagram of a circle with points P, Q, R, and S on the circumference. Angle PQS is 32 degrees, and angle PRS is x.] Solution: Angle PRS = Angle PQS (Angles in the same segment) Therefore, x = 32°

Commentary: This question tests understanding of the "angles in the same segment" theorem. Both angles are subtended by the chord P

S. Question 3: ABCD is a cyclic quadrilateral. If angle B = 65°, find the size of angle D. [Insert a diagram of a cyclic quadrilateral ABC

D. Angle B is 65 degrees, and angle D is the unknown.] Solution: Angle B + Angle D = 180° (Opposite angles of a cyclic quadrilateral are supplementary) 65° + Angle D = 180° Angle D = 180° - 65° Angle D = 115°

Commentary: This question applies the cyclic quadrilateral property. Be careful to identify the opposite angles correctly.

Question 4: ABCD is a cyclic quadrilateral, with side AB produced to E. If angle CBE = 100°, determine the size of angle ADC. [Insert a diagram of a cyclic quadrilateral ABCD, with side AB produced to E.