Angle Facts (OCR GCSE Maths): Revision Notes

Example Question: Given that one angle on a straight line is 120°, find the other angle.

Step-by-Step Solution:

1. Let the unknown angle be $x.$

2. Since angles on a straight line add up to 180°, you can set up the equation:

3. Subtract 120° from both sides:

4. Calculate $x$:

So, the other angle is 60°.

Example Question: Given four angles around a point: 90°, 85°, 100°, and $x$. Find $x$.

Step-by-Step Solution:

5. Set up the equation using the fact that the sum of angles around a point is 360°:

6. Add the known angles:

7. Subtract 275° from both sides:

8. Calculate $x$:

So, the missing angle is 85°.

Example Question: In a triangle, two angles are 65° and 75°. Find the third angle.

Step-by-Step Solution:

9. Let the unknown angle be $x$.

10. Set up the equation using the fact that the sum of angles in a triangle is 180°:

11. Add the known angles:

12. Subtract 140° from both sides:

13. Calculate $x$:

So, the third angle is 40°.

Example Question: A quadrilateral has three angles measuring 85°, 95°, and 100°. Find the measure of the fourth angle.

Step-by-Step Solution:

14. Let the unknown angle be $x$.

15. Set up the equation using the fact that the sum of angles in a quadrilateral is 360°:

16. Add the known angles:

17. Subtract 280° from both sides:

18. Calculate $x$:

So, the fourth angle is 80°.

Example Question: Two intersecting lines form an angle of 110°. Find the measures of the three remaining angles.

Step-by-Step Solution:

19. Identify the angle that is opposite the given angle of 110°. By the property of opposite angles, this angle is also 110°.

20. The other two angles are opposite to each other and must both add up with 110° to equal 180° (since they are on a straight line).

21. Set up the equation:

22. Subtract 110° from both sides:

23. The other two angles are 70° each.

So, the remaining angles are 110°, 70°, and 70°.

Example Question: In the diagram below, if one of the corresponding angles is 75°, what are the measures of the other corresponding angles?

Step-by-Step Solution:

24. Identify the corresponding angles using the F shape.
25. Since corresponding angles are equal, all corresponding angles will be 75°. So, all corresponding angles in the diagram are 75°.

Example Question: In the diagram, if one of the alternate angles is 60°, what are the measures of the other alternate angles?

Step-by-Step Solution:

26. Identify the alternate angles using the Z shape.
27. Since alternate angles are equal, the other alternate angle will also be 60°. So, all alternate angles in the diagram are 60°.

Example Problem: If you have a diagram where $x$ and $y$ are interior angles, and you know that $x=$ 70°, find the value of $y$.

Step-by-Step Solution:

28. Identify the interior angles using the C shape.

29. Use the key fact:

30. Substitute the given value:

31. Solve for $y$:

32. $y=$ 110°

So, the value of $y$ is 110°.

Tips for Answering Angle Questions

1. Always Write Down the Name of Each Angle Fact You Use:

• Clearly state which angle fact (like corresponding, alternate, or interior angles) you've used to arrive at your answer. This shows your understanding and makes your working clear.

2. Check for Parallel Lines:

• Remember, lines are only parallel if they have the little arrows to indicate it. Never assume lines are parallel unless marked.

3. Use Alphabetical Order:

• If you are given multiple angles to find and aren't sure where to start, it's often a good idea to work in alphabetical order. This can help keep your working organised.

4. Multiple Methods:

• There are often various ways to solve angle problems. If you get stuck, consider if there's another angle fact or method that could help you find the solution.

Example 1: Using Angles on a Straight Line and Opposite Angles

Problem: Given that angle $b$ is 56°, find the values of $a$, $c$, and $d$.

Solution:

5. Finding $a$:

• $a=180°−56°=$ 124°
• Fact 1: Angles on a straight line add up to 180°.

6. Finding $c$:

• $c=$ 56° (opposite angle to $b$)
• Fact 5: Opposite angles are equal.

7. Finding $d$:

• $d=360°−56°−124°−56°=$ 124°
• Fact 2: Angles around a point add up to 360°.

Example 2: Using Opposite Angles and Corresponding Angles

Problem: Given that angle $d$ is 118°, find the values of $e$, $f$, and $g$.

Solution:

8. Finding $e$:

• $e=$ 118° (opposite angle to $d$)
• Fact 5: Opposite angles are equal.

9. Finding $f$:

• $f=$ 118° (corresponding angle to $d$)
• Fact 6: Corresponding angles are equal.

10. Finding $g$:

• $g=180°−118°=$ 62°
• Fact 1: Angles on a straight line add up to 180°.

Example 3: Using Angles in a Triangle and Quadrilateral

Problem: Given that angle $p$ is 51°, find the values of $q$, $r$, and $s$.

Solution:

11. Finding $q$:

• $q=180°−51°−61°=$ 68°
• Fact 3: Angles in a triangle add up to 180°.

12. Finding $r$:

• $r=360°−51°−68°−61°=$ 180°
• Fact 4: Angles in a quadrilateral add up to 360°.

13. Finding $s$:

• $s=180°−68°=$ 112°
• Fact 1: Angles on a straight line add up to 180°.

Example 4: Using Interior Angles and Angles in a Triangle

Problem: Given that angle $r$ is 30°, find the values of $s$ and $t$.

Solution:

14. Finding $s$:

• $s=180°−30°=$ 150°
• Fact 3: Angles in a triangle add up to 180°.

15. Finding $t$:

• $t=180°−106°−35°=$ 39°
• Fact 8: Interior angles add up to 180°.