Learn / Math / Geometry and Trigonometry
Lines, Angles, and Triangles
I worked these out from the practice questions. Each code says where to start reading, where to stop, what you can skip, and why it is safe.
1. Small and big angles
- Start:
- Look at the angle you know and the angle you want.
- Stop:
- Stop when you decide: both small or both big (equal), or one of each (add to 180°).
- Skip:
- Skip the names like corresponding or alternate interior.
- Why:
- There are only two angle sizes with parallel lines, so this one decision is enough.
Used in questions 3, 1
2. Triangle total of 180
- Start:
- Underline every angle that is given.
- Stop:
- Stop when you have two angles; subtract from 180°.
- Skip:
- Skip the side lengths for angle-only questions.
- Why:
- The 180° sum is always true, so it needs no extra information.
Used in questions 2, 4, 8
3. Exterior angle shortcut
- Start:
- Find the angle outside the triangle.
- Stop:
- Stop after setting it equal to the two far inside angles.
- Skip:
- Skip finding the near inside angle first.
- Why:
- It saves one step and avoids a subtraction error.
Used in question 4
4. Whole side, not the piece
- Start:
- In a parallel-line picture, find the full side by adding the pieces (AD + DB).
- Stop:
- Stop when your ratio uses small triangle over whole triangle.
- Skip:
- Do not use the leftover piece (DB) as a matching side.
- Why:
- The small triangle matches the whole big triangle. Using DB is the usual wrong answer.
Used in questions 7, 5
5. Third side window
- Start:
- Write the two given sides.
- Stop:
- Stop after calculating difference and sum.
- Skip:
- Skip drawing the triangle.
- Why:
- The answer must be strictly between the difference and the sum, so test each option against the window.
Used in question 6
6. Ratio total method
- Start:
- Add the ratio numbers.
- Stop:
- Stop when you know the value of one part.
- Skip:
- Do not guess the angles.
- Why:
- Once one part is known every angle is a simple multiple, and you can use a calculator for 180 ÷ 9.
Used in questions 8, 5
7. Names the same-side pair
- Start:
- Look at the two marked angles and ask: are they on the same side of the transversal, between the lines?
- Stop:
- Stop when you decide 'equal' or 'add to 180°'.
- Skip:
- Skip the long names (co-interior, corresponding) once you know the rule.
- Why:
- With parallel lines there are only two sizes of angles. Same-side angles between the lines add to 180°, as in question 11, while alternate ones are equal.
Used in questions 11, 3
8. Bowtie means similar triangles
- Start:
- Find the two triangles that touch at the crossing point and match each vertex to the one opposite it (A with D, B with C).
- Stop:
- Stop when you have one scale factor from a pair of known matching sides.
- Skip:
- Skip finding unknown sides one by one.
- Why:
- Perimeter scales by the plain factor, not its square, so 24 × 1.5 gives 36 in one step. Use the same factor on any matching side.
Used in questions 15, 13, 5
9. Test the triangle rule before you add
- Start:
- Write each side set you could use, for example 5-5-11 and 5-11-11.
- Stop:
- Stop at the first set where the two shorter sides do not add to more than the longest.
- Skip:
- Skip drawing the triangle.
- Why:
- A set that breaks the rule is not a triangle, so its perimeter is bait (21 in question 14).
Used in questions 14, 6