This Year 7 angles and parallel lines lesson is written for pupils aged 11-12 and follows the National Curriculum. In short, it shows you how to calculate angles in triangles, on lines and with parallel lines. Before you start, it helps to revise ratio and proportion and perimeter, area and volume, because this lesson builds on them.
Year 7 angles and parallel lines lesson: what you will learn
Angles are measured in degrees. In this chapter you will use angle facts to find missing angles, in triangles, on lines and with parallel lines, and explain each step with a reason.
Starter activity
Look at a clock face. At 3 o'clock the hands make a right angle. At what other time on the hour is the angle between the hands exactly 90°? What angle do the hands make at 2 o'clock? Explain your method using a full turn of 360° and the 12 divisions of the clock.
Measuring and naming angles
Definition: angle types
An acute angle is less than \(90^{\circ}\). A right angle is exactly \(90^{\circ}\). An obtuse angle is between \(90^{\circ}\) and \(180^{\circ}\). A reflex angle is between \(180^{\circ}\) and \(360^{\circ}\).
We name an angle with three letters. \(\angle ABC\) is the angle at B between the lines BA and BC. Measure with a protractor, with its centre on the vertex and one arm on the zero line.
Angle facts
Rule: basic angle facts
Angles on a straight line add up to \(180^{\circ}\). Angles around a point add up to \(360^{\circ}\). Vertically opposite angles are equal.
Example 1
Angles on a straight line are \(112^{\circ}\) and \(x\). Find \(x\).
\(x=180-112=68^{\circ}\) (angles on a straight line add up to \(180^{\circ}\)).
Example 2
Angles at a point are \(90^{\circ}\), \(130^{\circ}\) and \(x\). Find \(x\).
\(x=360-90-130=140^{\circ}\) (angles at a point add up to \(360^{\circ}\)).
Triangles
Rule: angles in a triangle
The angles in a triangle add up to \(180^{\circ}\). An equilateral triangle has three angles of \(60^{\circ}\). An isosceles triangle has two equal sides and the angles opposite them are equal (the base angles). A right-angled triangle has one angle of \(90^{\circ}\).
Example 3
Two angles of a triangle are \(48^{\circ}\) and \(67^{\circ}\). Find the third.
\(180-48-67=65^{\circ}\).
Example 4
An isosceles triangle has an apex angle of \(40^{\circ}\). Find the base angles.
The base angles are equal, so each is \((180-40)\div2=70^{\circ}\).
Quadrilaterals
Rule: angles in a quadrilateral
The angles in a quadrilateral add up to \(360^{\circ}\), because any quadrilateral can be split into two triangles.
Example 5
Three angles of a quadrilateral are \(95^{\circ}\), \(110^{\circ}\) and \(70^{\circ}\). Find the fourth.
\(95+110+70=275\), so the fourth angle is \(360-275=85^{\circ}\).
Parallel lines
Definition: parallel lines and a transversal
Parallel lines stay the same distance apart and never meet. We mark them with arrows. A line that crosses them is a transversal.Key fact: parallel line angles
Corresponding angles are equal (they make an F shape). Alternate angles are equal (they make a Z shape). Co-interior angles add up to \(180^{\circ}\) (they make a C or U shape).Example 6
Two parallel lines are crossed by a transversal and one angle is \(64^{\circ}\). Find the corresponding, alternate and co-interior angles.
The corresponding angle is \(64^{\circ}\). The alternate angle is \(64^{\circ}\). The co-interior angle is \(180-64=116^{\circ}\).
Method: finding angles with parallel lines
- Look for the F, Z or C shape that links the angle you know with the angle you want.
- Use the matching rule (equal, equal or add to \(180^{\circ}\)).
- Write the reason beside the angle.
Using algebra with angles
Example 7
Angles on a straight line are \(2x\) and \(x+30\). Find \(x\) and both angles.
\(2x+x+30=180\), so \(3x=150\) and \(x=50\). The angles are \(2\times50=100^{\circ}\) and \(50+30=80^{\circ}\). Check: \(100+80=180\).
A multi-step problem
Triangle ABC has a line drawn through C parallel to AB. The alternate angles are equal, and the angles on a straight line add up to \(180^{\circ}\). This proves that the angles in a triangle add up to \(180^{\circ}\).
Writing the reason beside each angle earns marks in exams and helps you spot mistakes.
Example 8
In a triangle ABC, angle A is \(50^{\circ}\) and angle B is \(60^{\circ}\). The line CD extends BC beyond C. Find the exterior angle at C.
Angle ACB is \(180-50-60=70^{\circ}\). The exterior angle is on a straight line with it, so \(180-70=110^{\circ}\). This equals \(50+60\), the sum of the two opposite interior angles.
Common mistakes
- Mixing up \(180^{\circ}\) (straight line, triangle) and \(360^{\circ}\) (point, quadrilateral).
- Measuring a diagram that says “not drawn accurately”.
- Confusing alternate and co-interior angles.
- Giving an answer without a reason.
Watch out
Do not assume lines are parallel unless the diagram marks them with arrows. Co-interior angles add up to \(180^{\circ}\); they are not equal.
Worked example in context
A ladder leans against a wall making an angle of 70° with the ground. What angle does it make with the wall?
The wall is vertical so the triangle is right-angled: \(180-90-70=20\). The ladder makes an angle of 20° with the wall.
Common mistakes
- Using 360° for angles on a straight line instead of 180°.
- Confusing alternate and co-interior angles.
- Not giving a reason, such as "angles in a triangle add to 180°".
Did you know?
Ancient Babylonians divided a circle into 360 parts about 4 000 years ago, probably because their number system was based on 60 and because a year has about 360 days. That is why we still use 360 degrees today.
Giving reasons in angle problems
In angle questions the number is only half of the answer. Examiners also want to know why the angle has that size, so every step of your working should end with a short reason written in words. A good reason names the angle fact you used, for example “angles on a straight line add up to 180°” or “alternate angles are equal”. Reasons such as “because it looks like it” or “I measured it” do not score marks, because diagrams are usually not drawn accurately.
Method: solving a multi-step angle problem
- Copy or sketch the diagram and mark every angle you are given.
- Look for the angle facts that link a known angle to an unknown one: straight lines, points, triangles, quadrilaterals, vertically opposite angles and parallel lines.
- Work out one new angle at a time, write it on the diagram and write the reason next to your calculation.
- Keep going until you reach the angle you were asked for, then check that the angles in every triangle add up to 180°.
Example 1
In triangle PQR, angle P is 64° and angle Q is 58°. The side QR is extended beyond R to a point S. Find angle PRS.
First we find the third angle of the triangle. The angles in a triangle add up to 180°, so angle PRQ is \(180-64-58=58°\). Next we use the straight line QRS: angles on a straight line add up to 180°, so angle PRS is \(180-58=122°\). Notice that 122° is exactly \(64+58\). This is the exterior angle fact: an exterior angle of a triangle equals the sum of the two opposite interior angles.
Example 2
Two parallel lines are crossed by a transversal. One angle is \(2x+15\) and its co-interior angle is \(3x-5\). Find x.
Co-interior angles lie between the parallel lines on the same side of the transversal, and they add up to 180°. So we form the equation \(2x+15+3x-5=180\). Collecting like terms gives \(5x+10=180\), then \(5x=170\) and \(x=34\). The two angles are \(2\times34+15=83°\) and \(3\times34-5=97°\). They add up to 180°, which confirms the answer.
Example 3
Three angles of a quadrilateral are equal and the fourth angle is 87°. Find each of the equal angles.
The angles in a quadrilateral add up to 360°. If we call each equal angle a, then \(3a+87=360\). Subtracting 87 gives \(3a=273\), so \(a=91°\). Check: \(91+91+91+87=360\).
Note
Recognising angle pairs is easier if you look for letters. Corresponding angles sit in an F shape, alternate angles sit in a Z shape and co-interior angles sit in a C (or U) shape. Use the letters to spot the pairs, but in your written reason always use the proper names: corresponding, alternate and co-interior.
Watch out
Parallel line facts only work when the lines really are parallel. Look for the arrow marks on the lines or the word “parallel” in the question. If the lines are not parallel, corresponding angles are not equal and co-interior angles do not add up to 180°. In that case you can only use the facts about straight lines, points and triangles.
Did you know?
The idea of splitting a full turn into 360 degrees comes from the ancient Babylonians, who counted in sixties. The number 360 is very convenient because it can be divided exactly by 2, 3, 4, 5, 6, 8, 9, 10, 12 and many other numbers, so halves, thirds, quarters and sixths of a turn are all whole numbers of degrees.
Otto’s tip
Look for the letters! Z means alternate (equal), F means corresponding (equal), and C means co-interior (add to \(180^{\circ}\)).
How to use this Year 7 angles and parallel lines lesson
First, read the key points so that you know where the Year 7 angles and parallel lines lesson is going. Then work through each definition and worked example, and cover the solution before you read it. After that, practise with the angles and parallel lines worksheet and check every question with the worksheet answers. When you feel ready, test yourself with the multiple choice questions or the quick quiz. Finally, sit the angles and parallel lines test and mark it with the test answers.
Also, the starters are a good warm-up, so you can keep the key facts fresh. Moreover, all the pages of this chapter work together, so you can move between them at any time. For example, many pupils read the lesson on Monday, practise on Wednesday and test themselves on Friday. In short, this Year 7 angles and parallel lines lesson gives you everything you need for the chapter.
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