# Multi-Step Angle Puzzles

> Chain together the triangle, straight line, point, isosceles and parallel line rules to find a hidden angle.

From Selective School Maths on WhizzNotes, free study notes for NSW students: https://whizznotes.com/selective-school-maths/angle-puzzles/. Updated 3 November 2026.
Year 11 and Year 12 boxes go deeper than the Year 10 text around them.

### Story time: The detective's notebook
Detective Hana arrives at a puzzle, not a crime. A diagram shows lots of lines, a few angles and one question mark.

She does not guess. She writes down every angle she can find, one at a time, and next to each one she writes a **reason**: "straight line", "triangle", "isosceles". Each new angle leads to the next, like clues in a mystery. Finally she reaches the question mark.

Hard angle questions are just several easy steps in a row. The trick is knowing which rule to use next.

**In short:** A hard angle puzzle is really several small steps. Mark everything you know first, then look for a triangle, a straight line or a point where only one angle is missing, and give a reason for each angle you find.

## Before you start

You need every rule from this unit. Here they are in one place:

| Rule | Fact |
|---|---|
| Straight line | angles add to $180^\circ$ |
| Around a point | angles add to $360^\circ$ |
| Vertically opposite | equal |
| Triangle | angles add to $180^\circ$ |
| Exterior angle of a triangle | equals the two opposite inside angles added |
| Isosceles triangle | base angles are equal |
| Equilateral triangle | every angle is $60^\circ$ |
| Quadrilateral | angles add to $360^\circ$ |
| Parallel lines | corresponding and alternate angles equal, co-interior add to $180^\circ$ |

## The detective method

### Key idea: Four steps
1. **Mark** every angle you are given, and every equal side or parallel line.
2. **Look for** a triangle or straight line where you already know two of the three angles.
3. **Write** the new angle on the diagram, with its reason.
4. **Repeat** until you reach the angle you want.

### Exam tip: Tick marks mean isosceles
Whenever two sides of a triangle have tick marks, write "isosceles" straight away and mark the two equal base angles. It is almost always part of the solution.

## Worked puzzles

*Diagram: Line from A through B to C. Point D above makes triangle BCD. Angle ABD is 128 degrees, angle at C is 68 degrees, and angle f is at B inside the triangle*
*A, B and C are on one straight line. Find angle f, then angle D.*

### Worked example: Straight line, then triangle
Angle $ABD = 128^\circ$ and $A$, $B$, $C$ are on a straight line.

Step 1: $f = 180 - 128 = 52^\circ$ (straight line).

Step 2: In triangle $BCD$: angle $D = 180 - 52 - 68 = 60^\circ$ (triangle).

### Worked example: An isosceles chain
In triangle $PQR$, $PQ = PR$ and angle $P = 38^\circ$. Point $S$ is on $PR$ with $QS = QR$. Find angle $SQR$.

Step 1: $PQR$ is isosceles, so angles $Q$ and $R$ are equal: $\frac{180 - 38}{2} = 71^\circ$ each.

Step 2: Triangle $QSR$ is isosceles with $QS = QR$, so angle $QSR = $ angle $QRS = 71^\circ$.

Step 3: Angle $SQR = 180 - 71 - 71 = 38^\circ$.

### Worked example: Around a point, then a triangle
At a point $P$, three angles meet: $110^\circ$, a reflex angle of $212^\circ$, and angle $a$. Angle $a$ is also one corner of a triangle whose other corners are $85^\circ$ and $b$.

Step 1: $a = 360 - 110 - 212 = 38^\circ$ (around a point).

Step 2: $b = 180 - 38 - 85 = 57^\circ$ (triangle).

### Worked example: Two triangles on a line
Two triangles stand on a straight line and share a point $X$ on the line. The left triangle has angles $70^\circ$ and $80^\circ$, so its angle at $X$ is $30^\circ$. The right triangle has a $70^\circ$ angle and a right angle, so its angle at $X$ is $20^\circ$.

The angle $x$ between the two triangles at $X$ completes the straight line: $x = 180 - 30 - 20 = 130^\circ$.

### Watch out: Do not assume
A diagram "not to scale" may look like it has a right angle or equal sides when it does not. Only use what is marked or written.

### Challenge: Selective level: parallelogram plus triangle
$ABCD$ is a parallelogram with angle $A = 72^\circ$. A triangle is drawn on side $BC$ with angle $y$ at $B$, where $y$ is half of the parallelogram's angle at $B$.

Angle $B$ of the parallelogram $= 180 - 72 = 108^\circ$ (co-interior). So $y = 54^\circ$.

If the triangle on $BC$ is isosceles with its two equal sides meeting at $B$, the other two angles are $\frac{180 - 54}{2} = 63^\circ$ each.

**Mind map**

```text
Angle puzzles
  Method
    Mark what you know
    Find a shape with one gap
    Write angle and reason
    Repeat
  Favourite clues
    Tick marks: isosceles
    Arrows: parallel
    Straight line: 180°
  Traps
    Not to scale
    Reflex vs inside angle
```

## Quick summary

- Big angle problems are several small steps.
- Mark everything you know first, including equal sides and parallel lines.
- Find a triangle, straight line or point with only one angle missing.
- Write each new angle with its reason.
- Tick marks almost always mean "use the isosceles rule".
- Never trust how a diagram looks if it says "not to scale".

## Quiz

Try these questions from the chapter quiz.

1. Points $A$, $B$, $C$ are on a straight line. Angle $ABD = 128^\circ$. What is angle $DBC$?
   - A. $38^\circ$
   - B. $52^\circ$
   - C. $62^\circ$
   - D. $232^\circ$
   - Answer: B. $52^\circ$. Straight line: $180 - 128 = 52$.

2. A triangle has angles $52^\circ$ and $68^\circ$. What is the third?
   - A. $60^\circ$
   - B. $50^\circ$
   - C. $70^\circ$
   - D. $120^\circ$
   - Answer: A. $60^\circ$. $180 - 52 - 68 = 60$.

3. Two sides of a triangle have tick marks. The angle between them is $50^\circ$. What is each of the other angles?
   - A. $50^\circ$
   - B. $130^\circ$
   - C. $65^\circ$
   - D. $40^\circ$
   - Answer: C. $65^\circ$. Isosceles: $\frac{180 - 50}{2} = 65$.

The whole chapter quiz is on the page, for signed-in students: https://whizznotes.com/selective-school-maths/angle-puzzles/#quiz