Geometry and Measurement · Shape and Space · Lesson 3

Adjacent, Supplementary and Vertically Opposite Angles

When two straight lines cross, like two roads at a roundabout or the blades of a pair of scissors, four angles are made. They are not random: they follow neat rules. In this lesson you will learn the names of the angle pairs, the three key facts about angles on a line, at a point and between crossing lines, and use them to find missing angles.

NaCCA B7.3.1.1.3 ⏱ about 45 min

By the end you will be able to

✔Identify adjacent angles and vertically opposite angles in a diagram.
✔State and use the facts: angles on a straight line add up to 180°, angles at a point add up to 360°, and vertically opposite angles are equal.
✔Find unknown angles, including those given as algebraic expressions.
✔Give reasons for each step in an angle calculation.

Key words

Adjacent angles: Two angles that share a common vertex and a common arm, with no overlap.
Vertically opposite angles: The pairs of equal angles that are opposite each other when two straight lines cross.
Angles on a straight line: Angles that together make a straight line; their sum is 180°.
Angles at a point: Angles that together fill the whole space round a point; their sum is 360°.

Curriculum reference

Content standard: B7.3.1.1 Demonstrate understanding of angles including adjacent, vertically opposite, complementary, supplementary and use them to solve problems.

Indicator: Use adjacent, supplementary and vertically opposite angles to solve problems.

Starter

Scissors at the tailor's shop

Auntie Dzifa is a seamstress in Ho. When she opens her big cutting scissors wide, the blades make an angle. If she opens them more, the angle on the other side of the pivot opens by exactly the same amount, even though it is on the opposite side. The handles and the blades always match in this way. That surprising partnership between opposite angles is one of the most useful facts in geometry, and today you will see why it is always true.

1. Adjacent angles and angles on a straight line

Two angles are adjacent when they sit side by side: they share one vertex (corner point) and one arm (line), and they do not overlap. Think of two slices of a cut pizza or two neighbouring doors of a wardrobe opened from the same hinge.

When adjacent angles together form a straight line, they are also supplementary. Angles on a straight line always add up to 180°. So if one angle on a line is 130°, the adjacent angle is 180° − 130° = 50°.

▶ The two adjacent angles lie on a straight line. If one is 60°, the other is 180° − 60° = 120°.

Adjacent angles do not always lie on a straight line. If three or more angles are packed round one point, each neighbouring pair is adjacent, but their total is different. That brings us to the next fact.

2. Angles at a point

Stand still and turn all the way round, ending facing the same way. You have turned through one full turn, 360°. In the same way, all the angles that surround one point and fill the space completely add up to 360°.

For example, four angles at a point are 100°, 120°, 90° and $x$ . Then $100 + 120 + 90 + x = 360$ , so $310 + x = 360$ and $x = 50°$ .

Three sums to know
Situation	The angles add up to	Reason
On a straight line	180°	half a full turn
At a point	360°	a full turn
In a right-angled corner	90°	a quarter turn

3. Vertically opposite angles

When two straight lines cross, four angles are formed. The angles that are directly opposite each other, touching only at the crossing point, are called vertically opposite angles. The word 'vertically' does not mean up and down. It refers to the vertex, the point where the lines cross.

The key fact is: vertically opposite angles are equal. Here is why. Call the four angles $a$ , $b$ , $c$ and $d$ going round the point. Angles $a$ and $b$ are on a straight line, so $a + b = 180°$ . Angles $b$ and $c$ are also on a straight line, so $b + c = 180°$ . Therefore $a + b = b + c$ , and subtracting $b$ from both sides gives $a = c$ . The same argument shows $b = d$ .

▶ Directly opposite angles are equal. Neighbouring angles add up to 180°.

4. Using the facts together

Most exam questions use more than one fact. A good approach is to write a reason beside each line of working, such as 'angles on a straight line', 'angles at a point' or 'vertically opposite angles'. Follow this plan:

1Look at the diagram and decide which fact connects the known angle to the unknown angle.
2Write the equation or the equality, with the reason.
3Solve for the unknown, using algebra if the angles contain $x$ .
4Check that every angle on a line adds to 180° and every angle round a point adds to 360°.

Suppose two lines cross, and the angles opposite each other are $(2 x + 10) °$ and $(3 x - 20) °$ . Because they are vertically opposite, they are equal: $2 x + 10 = 3 x - 20$ . So $30 = x$ . Each angle is $2 (30) + 10 = 70°$ . The angles next to them are $180° - 70° = 110°$ .

To identify pairs from a figure, ask: do the two angles share an arm and a vertex (adjacent)? Do they add to 90° (complementary) or 180° (supplementary)? Are they across the crossing point from each other (vertically opposite)? Sometimes a pair has more than one name: two adjacent angles on a straight line are both adjacent and supplementary.

Summary of angle pairs
Pair	How to spot it	Relationship
Adjacent	share a vertex and an arm	none by itself
Adjacent on a line	adjacent, forming a straight line	sum = 180°
Vertically opposite	opposite each other at a crossing	equal
Complementary	any two angles	sum = 90°
Supplementary	any two angles	sum = 180°

Worked examples

Try each one yourself first, then open the solution and compare every step.

Example 1

Angles on a straight line

On a straight line, three angles meet at a point. Two of them are 40° and 65°. Find the third angle

x

Show the solution

Step 1Angles on a straight line add up to 180°.
Step 2 $40 + 65 + x = 180$ .
Step 3 $105 + x = 180$ , so $x = 75$ .
Step 4Check: 40° + 65° + 75° = 180° ✓.

Answer: $x = 75°$

Example 2

Angles at a point

Four angles meet at a point: 100°, 120°, 90° and

y

. Find

y

Show the solution

Step 1Angles at a point add up to 360°.
Step 2 $100 + 120 + 90 + y = 360$ .
Step 3 $310 + y = 360$ , so $y = 50$ .
Step 4Check: 100 + 120 + 90 + 50 = 360 ✓.

Answer: $y = 50°$

Example 3

Two crossing roads

Two straight roads cross at Kwame Nkrumah Circle. One of the angles between them is 72°. Find the other three angles.

▶ Vertically opposite angles are equal; neighbouring angles add up to 180°.

Show the solution

Step 1The angle opposite 72° is vertically opposite, so it is 72°.
Step 2The angles next to 72° are on a straight line: 180° − 72° = 108°.
Step 3The angle opposite the 108° angle is also 108°.
Step 4Check: 72 + 108 + 72 + 108 = 360 ✓.

Answer: 72°, 108°, 72°, 108°

Example 4

An equation from vertically opposite angles

Two straight lines cross. Two vertically opposite angles are

(2 x + 10) °

and

(3 x - 20) °

. Find

x

and the size of each of the four angles.

Show the solution

Step 1Vertically opposite angles are equal: $2 x + 10 = 3 x - 20$ .
Step 2Subtract $2 x$ from both sides: $10 = x - 20$ .
Step 3Add 20: $x = 30$ .
Step 4Each of the pair is $2 (30) + 10 = 70°$ . Check: $3 (30) - 20 = 70°$ ✓.
Step 5The other two angles are $180° - 70° = 110°$ each.

Answer: $x = 30$ ; angles 70°, 110°, 70°, 110°

Example 5

Adjacent angles on a line, with algebra

Two adjacent angles on a straight line are

(3 y + 10) °

and

(7 y - 30) °

. Find

y

and the angle vertically opposite the larger one.

Show the solution

Step 1They lie on a straight line: $(3 y + 10) + (7 y - 30) = 180$ .
Step 2 $10 y - 20 = 180$ , so $10 y = 200$ and $y = 20$ .
Step 3The angles are $3 (20) + 10 = 70°$ and $7 (20) - 30 = 110°$ . Check: 70 + 110 = 180 ✓.
Step 4The larger angle is 110°, so the angle vertically opposite it is also 110°.

Answer: $y = 20$ ; the opposite angle is 110°

Summary

Adjacent angles share a vertex and an arm and do not overlap.
Angles on a straight line add up to 180° (they are supplementary).
Angles at a point add up to 360°.
Vertically opposite angles, across a crossing point, are equal.
Always give a reason for each step: 'angles on a straight line', 'angles at a point' or 'vertically opposite angles'.

Scissors at the tailor's shop

1. Adjacent angles and angles on a straight line

2. Angles at a point

3. Vertically opposite angles

4. Using the facts together

Worked examples

Angles on a straight line

Angles at a point

Two crossing roads

An equation from vertically opposite angles

Adjacent angles on a line, with algebra

Summary

Practice: 15 questions