Lesson Notes By Weeks and Term v4 - JHS 2

Lesson Video

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Performance objectives

• Construct angles of 120°, 105°, 135°, and 150°.
• Bisect these angles accurately.
• Explain the steps for construction and bisection.
• Apply these concepts to real-life situations.

Lesson summary

This lesson builds upon our previous knowledge of constructing basic angles like 60° and 90°. Today, we will learn how to construct and bisect more complex angles: 120°, 105°, 135°, and 150°. This skill is not just for examinations; it is a practical skill used by many professionals in Ghana. Carpenters use it to build strong roof trusses, tailors and fashion designers use it to cut fabric for beautiful clothes, artists use it to create patterns in Kente and Adinkra designs, and architects use it to design our homes and schools. Mastering this skill helps us understand the world around us better and improves our logical thinking.

Lesson notes

A. Foundational Knowledge (Recap) Before we begin, let's remember what we already know: Construction: In geometry, this means drawing shapes and angles accurately using only two tools: a straightedge (ruler without markings) and a pair of compasses. We do not use a protractor to create the angle. Bisection: This means to divide something into two perfectly equal parts. To bisect an angle means to draw a line through its vertex that splits it into two smaller, equal angles. Key Angles we can already construct: 60°: The angle formed by constructing an equilateral triangle. 90°: The angle formed by bisecting a straight line (180°). Key Idea: All the angles we are learning today (120°, 105°, 135°, 150°) are made from combinations or bisections of 60° and 90°.

B. Constructing the Angles (Step-by-Step)

i. How to Construct a 120° Angle The idea here is that 120° = 60° + 60°. We will simply construct a 60° angle, and then construct another 60° angle right next to it. Step 1: Draw a straight line segment and label it AB. We will construct the angle at point A. Step 2: Place the compass point at A and draw a large arc that cuts the line AB at a point, let's call it P. Step 3: Without changing the compass width, move the compass point to P and draw another arc that cuts the first arc at a point, let's call it Q. (If you draw a line from A to Q, you have just made a 60° angle ∠QAP). Step 4: Now, move the compass point to Q (again, do not change the width) and draw another arc that cuts the very first big arc at a new point, R. Step 5: Draw a straight line from the vertex A through the point R. Result: The angle ∠RAB is exactly 120°.

ii. How to Construct a 135° Angle The idea here is that 135° = 90° + 45°. Since 45° is half of 90°, we first construct a 90° angle, and then bisect the adjacent 90° angle on the straight line. Step 1: Draw a straight line XY and mark a point A on it. Step 2: Construct a 90° angle at point A. (Place compass at A, draw a semicircle. From the two points where the semicircle hits the line, draw two intersecting arcs above A. Draw a line from A through the intersection). Let's call the top of this 90° line point M. So, ∠MAY = 90°. Step 3: The angle on the other side, ∠MAX, is also 90°. We need to add 45° to our first 90°. We get this 45° by bisecting ∠MAX. Step 4: To bisect ∠MAX, place your compass point at A and draw an arc that cuts AM and AX. Step 5: From where the arc cuts AM and AX, draw two new intersecting arcs in the middle of the angle. Call the intersection point N. Step 6: Draw a line from A through N. This line AN has bisected the 90° angle, so ∠NAX = 45°. Result: The complete angle we want is ∠NAY. This is ∠NAX + ∠XAY = 45° + 90° = 135°.