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1. Complementary, supplementary and vertical angles

Warm-up

Three quick questions from earlier lessons.

Discover

Two angles sit side by side on a straight line. One is $130^\circ$ . What is the other?

$50^\circ$
$-40^\circ$

Key idea

Angle facts give equations.

Complementary angles add to

90^\circ

; supplementary angles add to

180^\circ

(angles on a straight line). Vertical angles (opposite at a crossing) are equal. Write an equation and solve for the unknown.

Worked example

Angles $2x^\circ$ and $40^\circ$ are complementary.

2x + 40 = 90

Add to 90.

x = 25

Solve.

Your turn, guided

Your turn: the supplement of $72^\circ$ .

108

180 − 72.

Practice ★

An angle measures $33^\circ$ . What is its complement (degrees)?
An angle measures $61^\circ$ . What is its complement (degrees)?

An angle measures $81^\circ$ . What is its supplement (degrees)?

Watch out!

Practice ★★

Two angles on a straight line measure $68^\circ$ and $(x + 22)^\circ$ . Find $x$ .
Two angles on a straight line measure $34^\circ$ and $(x + 9)^\circ$ . Find $x$ .

Exit ticket

Three questions to check you have got it.

An angle measures $33^\circ$ . What is its complement (degrees)?
An angle measures $61^\circ$ . What is its complement (degrees)?
An angle measures $39^\circ$ . What is its complement (degrees)?

Summary card

Angle relationships

Complementary: 90°. Supplementary: 180°. Vertical: equal.

Watch out: complement 90, supplement 180.

2. Building triangles: the triangle inequality

Warm-up

Three quick questions from earlier lessons.

Discover

Can you make a triangle from sticks of 2 cm, 3 cm and 7 cm?

Key idea

The triangle inequality.

Three lengths make a triangle only if the two shorter ones add to more than the longest. So a third side lies strictly between the difference and the sum of the other two:

a - b < c < a + b

Worked example

Sides 8 and 5.

Third side

> 3

Difference.

Third side

< 13

Sum.

From 4 to 12

Whole numbers.

Your turn, guided

Your turn: sides 10 and 4. The largest whole-number third side.

13

Less than 14.

Practice ★

Can three sticks of lengths 8, 9 and 18 cm form a triangle?
1. No: the two shorter sticks are too short to meet
2. Yes: any three lengths make a triangle
3. Only as a right triangle
Can three sticks of lengths 9, 2 and 12 cm form a triangle?
1. No: the two shorter sticks are too short to meet
2. Yes: any three lengths make a triangle
3. Only as a right triangle

Two sides of a triangle are 15 cm and 3 cm. What is the largest possible whole-number length of the third side?

Watch out!

Practice ★★

Two sides of a triangle are 14 cm and 4 cm. What is the smallest possible whole-number length of the third side?
Two sides of a triangle are 6 cm and 4 cm. What is the smallest possible whole-number length of the third side?

Exit ticket

Three questions to check you have got it.

Two sides of a triangle are 15 cm and 3 cm. What is the largest possible whole-number length of the third side?
Two sides of a triangle are 6 cm and 3 cm. What is the largest possible whole-number length of the third side?
Two sides of a triangle are 15 cm and 4 cm. What is the largest possible whole-number length of the third side?

Summary card

Building triangles

Two shorter sides must add to more than the longest.

Watch out: equal is not enough.

3. Cross sections and solids of rotation

Warm-up

Three quick questions from earlier lessons.

Discover

A loaf of bread shaped like a cylinder is sliced straight across. What shape is each slice?

A circle
A rectangle

Key idea

Solids from slices and from rotations.

Not to scale

A cross section is the shape where a plane cuts a solid. Slices parallel to the base of a prism or cylinder copy the base. Rotating a rectangle about a side gives a cylinder; a right triangle about a leg gives a cone; a semicircle about its diameter gives a sphere.

Worked example

A square pyramid.

Horizontal slice: a smaller square

Parallel to the base.

Vertical slice through the apex: a triangle

Through the apex.

Your turn, guided

Your turn: how many faces does a triangular prism have?

5

2 triangles + 3 rectangles.

Practice ★

A cylinder of radius 4 is sliced by a plane parallel to its base. What shape is the cross section?
1. A circle of radius 4
2. A rectangle
3. An ellipse

A 9 by 6 rectangle is rotated about one of its sides. Which solid does it sweep out?
1. A cylinder
2. A cone
3. A sphere

Watch out!

Practice ★★

A cone of height 3 is sliced by a vertical plane through its apex. What shape is the cross section?
1. An isosceles triangle
2. A circle
3. A rectangle

Exit ticket

Three questions to check you have got it.

A cylinder of radius 4 is sliced by a plane parallel to its base. What shape is the cross section?
1. A circle of radius 4
2. A rectangle
3. An ellipse
A cylinder of radius 2 is sliced by a plane parallel to its base. What shape is the cross section?
1. A circle of radius 2
2. A rectangle
3. An ellipse
A cylinder of radius 5 is sliced by a plane parallel to its base. What shape is the cross section?
1. A circle of radius 5
2. A rectangle
3. An ellipse

Summary card

Cross sections and rotations

Parallel slices copy the base; rectangle → cylinder, right triangle → cone, semicircle → sphere.

Watch out: the angle of the slice decides the shape.

4. Volume of prisms and cylinders

Warm-up

Three quick questions from earlier lessons.

Discover

A stack of identical coins forms a cylinder. If each coin has area $A$ and the stack is $h$ high, what is its volume?

$A \times h$
$A + h$

Key idea

Base area times height.

For any prism or cylinder:

V = B \cdot h

, with

B

the area of the base. Cylinder:

V = \pi r^2 h

Worked example

A cylinder with $r = 3$ , $h = 10$ .

Not to scale

B = 9\pi

Base area.

V = 90\pi \approx 282.7

Times height.

Your turn, guided

Your turn: a rectangular prism 4 by 5 by 6.

120

4 × 5 × 6.

Practice ★

Find the volume of a cylinder with radius 3 cm and height 4 cm (exact, in terms of π).Not to scale
Find the volume of a cylinder with radius 8 cm and height 11 cm (exact, in terms of π).Not to scale

A prism has a triangular base with base 2 cm and height 7 cm, and the prism is 7 cm long. Find its volume (cm³).Not to scale

Watch out!

Practice ★★

A cylinder of radius 5 cm has volume $175\pi$ cm³. Find its height (cm).Not to scale
A cylinder of radius 6 cm has volume $360\pi$ cm³. Find its height (cm).Not to scale

Exit ticket

Three questions to check you have got it.

Find the volume of a cylinder with radius 3 cm and height 4 cm (exact, in terms of π).Not to scale
Find the volume of a cylinder with radius 8 cm and height 11 cm (exact, in terms of π).Not to scale
Find the volume of a cylinder with radius 5 cm and height 5 cm (exact, in terms of π).Not to scale

Summary card

Prisms and cylinders

$V = Bh$ ; cylinder $V = \pi r^2 h$ .

Watch out: base area, not circumference.

5. Surface area of prisms

Warm-up

Three quick questions from earlier lessons.

Discover

To wrap a box you need to cover its outside. Which measure is that?

Surface area
Volume

Not to scale

Turn and tilt the box. Count its faces, the hidden ones too.

6 faces, in 3 matching pairs
4 faces: the ones on the sides

Key idea

Surface area adds the faces.

The surface area of a prism is the sum of the areas of all its faces: two bases plus the side rectangles (perimeter of the base × length). For a box:

2(lw + lh + wh)

. Volume is base area × height, in cubic units.

Worked example

A box 5 by 3 by 2.

Not to scale

2(15 + 10 + 6)

Pairs of faces.

62

square units

Add.

Your turn, guided

Your turn: the surface area of a cube with edge 3.

54

6 · 9.

Practice ★

Find the surface area of a box (rectangular prism) 6 cm by 9 cm by 5 cm (cm²).Not to scale
Find the surface area of a box (rectangular prism) 7 cm by 3 cm by 7 cm (cm²).Not to scale

Find the surface area of a cube with edge 10 cm (cm²).

Watch out!

Practice ★★

A prism has right-triangle bases with sides 3, 4 and 5 cm, and length 7 cm. Find its surface area (cm²).Not to scale
A prism has right-triangle bases with sides 9, 12 and 15 cm, and length 2 cm. Find its surface area (cm²).Not to scale

Boss problem. A gift box is 6 in long, 4 in wide and 5 in tall.

Exit ticket

Three questions to check you have got it.

Find the surface area of a box (rectangular prism) 6 cm by 9 cm by 5 cm (cm²).Not to scale
Find the surface area of a box (rectangular prism) 7 cm by 3 cm by 7 cm (cm²).Not to scale
Find the surface area of a box (rectangular prism) 8 cm by 2 cm by 3 cm (cm²).Not to scale

Summary card

Surface area

Add the areas of all faces (square units).

Watch out: covering is area; filling is volume.

Answers: Angles, triangles and prisms

1. Complementary, supplementary and vertical angles

$57$
$29$

$99$

$90$
$137$

$57$
$29$
$51$

2. Building triangles: the triangle inequality

No: the two shorter sticks are too short to meet
No: the two shorter sticks are too short to meet

$17$

$11$
$3$

$17$
$8$
$18$

3. Cross sections and solids of rotation

A circle of radius 4

A cylinder

An isosceles triangle

A circle of radius 4
A circle of radius 2
A circle of radius 5

4. Volume of prisms and cylinders

$36 \pi$
$704 \pi$

$49$

$7$
$10$

$36 \pi$
$704 \pi$
$125 \pi$

5. Surface area of prisms

$258$
$182$

$600$

$96$
$180$

$124$

$258$
$182$
$92$