Worked Examples

Each example shows the reasoning line by line. Read the model solution, then close it and complete the matching “Try it yourself” task.

Example 1

Find the value of x

AB ∩ CD = {M}

In the opposite figure, AB ∩ CD = {M}. Find the value of x.

Solution

m(∠DMB) = m(∠AMC) — vertically opposite angles

2x − 7 = 129

2x = 129 + 7 = 136

x = 136 ÷ 2 = 68

XY ∩ LZ = {M}

Try it yourself 1

In the opposite figure XY ∩ LZ = {M}. Find the value of x.

Example 2

Find several angle measures in one figure

BE ∩ AD = {M}, with ray MC between them

Find m(∠AMB), m(∠DME) and m(∠AME).

Solution

m(∠AMB) = 180° − (115° + 28°) = 37° (straight line)

m(∠DME) = m(∠AMB) = 37° (vertically opposite angles)

m(∠AME) = m(∠BMD) = 28° + 115° = 143° (V.O.A.)

Check: 37° + 143° = 180°, as the two angles lie on a straight line.

Try it yourself 2

Two straight lines AB and CD intersect at M, and the ray ME lies inside ∠BMD so that m(∠CMA) = 70° and m(∠BME) = 120° − 70° = 50°. Find m(∠CMB) and m(∠EMD).

Example 3

Angles accumulated at a point

Figure 1 — find m(∠BMC)

m(∠BMC) = 360° − (90° + 150°) = 120°

Figure 2 — find m(∠AMB) where MB bisects ∠AMC

m(∠AMC) = 360° − (140° + 50° + 100°) = 70°

m(∠AMB) = m(∠BMC) = 70° ÷ 2 = 35°

Figure 3 — find the value of x

2x − 1 + 90 + 5x + 110 = 360

7x + 199 = 360

7x = 161 → x = 23

Try it yourself 3 — part 1

Find m(∠AMD)

Try it yourself 3 — part 2

Find the value of x