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Year 9 Mathematics

Triangles

Choose the chapter you need. Each chapter opens with its learning page before the practice questions.

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01

Angles in a Triangle

Use the 180-degree angle sum to calculate missing interior angles.

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02

Triangles by Sides

Recognise equilateral, isosceles and scalene triangles from equal-side markings.

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03

Triangles by Angles

Classify acute, right and obtuse triangles using their largest angle.

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04

Supplementary Angles

Find missing angles on a straight line by subtracting from 180 degrees.

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05

Co-Interior Angles

Identify C-angles, learn the co-interior law and calculate missing angles.

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06

Parallel Line Calculations

Combine straight-line and co-interior rules across multi-step diagrams.

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07

Names of Angles

Learn corresponding, alternate, co-interior and vertically opposite angle names.

Reference chapter
08

Geometry Test Practice: Q1-2 Skills

Ten original missing-angle diagrams using triangle sums, isosceles triangles, exterior angles and parallel lines.

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09

Geometry Test Practice: Q3-4 Skills

Ten original questions on triangle equations and angle relationships across parallel lines.

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10

Triangle Congruence Tests

Learn SSS, SAS, AAS and RHS, then identify the evidence in 40 original diagrams.

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11

Reorient and Match Triangles

Match six not-to-scale triangles by their labelled evidence, then classify each pair as AAA, ASA, SSS or RHS.

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12

Isosceles and Supplementary Angles

A 16-question class on equal base angles, straight-line angles and two-step triangle problems.

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13

Similar Triangles

Use a scale factor to enlarge or shrink a triangle, then deduce the multiplier and find missing sides.

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A

Geometry Practice Test A

Seven original test-style questions using every major scaffold from angle sums through indirect measurement.

Practice test
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B

Geometry Practice Test B

A fresh seven-question version covering angle chains, equations, congruence, similarity and scale factors.

Practice test
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C

Geometry Practice Test C

A third fully original version with reoriented diagrams and independent measurements and answers.

Practice test
→
Test Practice · Questions 1-2 Skills

Find Missing Angles

Read the markings before calculating. The drawing may not be to scale, so use the angle rules rather than guessing from its appearance.

Triangle sumThe three interior angles of every triangle add to 180 degrees.
Isosceles triangleEqual side marks mean the angles opposite those sides are equal.
Straight lineAdjacent angles on a straight line add to 180 degrees.
Parallel linesTransfer a known angle using corresponding or alternate angles, then use the triangle sum.
Working order: mark any angle you can transfer or calculate first, then find the final missing angle.
Test Practice · Questions 3-4 Skills

Equations and Parallel Lines

These questions use the same mathematical skills as Questions 3 and 4, with fresh values, wording and diagrams.

Triangle equationWrite all three interior angles in one equation and set their sum equal to 180 degrees.
Solve for xCollect the x terms, collect the number terms, then isolate x.
Equal angle rulesCorresponding and alternate angles are equal when the lines are parallel.
Supplementary rulesCo-interior angles and angles on a straight line add to 180 degrees.
Working order: first decide whether the marked angles are equal or add to 180 degrees. Then write the equation before calculating x.
Triangle Congruence · Identification Unit

SSS, SAS, AAS and RHS

Congruent triangles have exactly the same size and shape. Read the matching marks on each pair of triangles, then identify which congruence test proves they are congruent.

SSS · Side, Side, SideAll three pairs of corresponding sides are equal. Look for three different sets of matching side marks.
SAS · Side, Angle, SideTwo corresponding sides and the angle between those sides are equal. The marked angle must be included between the marked sides.
AAS · Angle, Angle, SideTwo corresponding angles and one corresponding side are equal. In this unit, the marked side is not between the two marked angles.
RHS · Right angle, Hypotenuse, SideBoth are right triangles, with equal hypotenuses and one other pair of equal sides.
Do not use AAA or SSA: AAA proves only that triangles are similar, while SSA does not guarantee one unique triangle.
Practice balance: 10 SSS diagrams · 10 SAS diagrams · 10 AAS diagrams · 10 RHS diagrams.
Geometry Test Practice · Question 5 Skill

Reorient and Match Triangles

Each page contains six equilateral-looking triangles. The drawings are deliberately not to scale: a triangle may look equilateral while its labels show completely different angles and sides.

1 · Ignore appearanceTrust the written angles, side lengths and right-angle squares. Do not assume an angle is 60 degrees.
2 · Derive missing anglesIf only two angles are written, subtract both from 180 degrees to determine the third angle.
3 · Match the pairSelect two triangles carrying the same complete set of labelled evidence.
4 · Reorient and nameMentally turn or flip the evidence, then classify the match as AAA, ASA, SSS or RHS.
Important: AAA establishes similarity only. ASA, SSS and RHS can prove congruence.
Ten pages: every page contains six triangles forming three matching pairs.
Year 9 Triangles · Chapter 12

Isosceles and Supplementary Angles

This class builds one skill at a time. Every question must be correct before the next question becomes available.

Part A · 7 questionsUse the equal-side marks on an isosceles triangle. Equal sides are opposite equal angles.
Part B · 4 questionsAngles on a straight line total 180°. Calculate the missing supplementary angle directly.
Part C · 5 questionsFirst subtract the exterior angle from 180°. Then use the three interior angles of the triangle.
Part A rule: equal sides in an isosceles triangle produce equal angles opposite those sides.
Part B rule: missing angle = 180° - known angle.
Part C order: find the interior angle beside the exterior angle first, then subtract both known interior angles from 180°.
Year 9 Triangles · Chapter 13

Similar Triangles

Similar triangles have the same shape but a different size. Their matching angles are equal, and their matching sides are all multiplied by the same number — the scale factor.

Part A · 10 questionsApply a given scale factor. Multiply each side of the original triangle by the multiplier to find the matching side of the similar triangle.
Part B · 8 questionsDeduce the scale factor from one pair of matching sides, then use it to find the missing side. Enter the scale factor first, then x.
Scale factor = new side ÷ original side.
New side = original side × scale factor.
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Section 1 · Learn

Angles in a Triangle

Every triangle has exactly three interior angles. No matter the shape — tall, flat, lopsided — those three angles will always add up to exactly 180°. This is the most important rule in all of triangle geometry.

a° b° c°
a° + b° + c° = 180° Missing angle = 180° − (sum of the other two)
Why 180°? A straight line measures 180°. If you tear the three corners off any triangle and line them up along a straight edge, they fit together perfectly — every time. The angles sum to exactly one straight line.
✦   The Secret Weapon   ✦
180° − b° − c° = a°
Know two angles in a triangle? Subtract them both from 180°. What's left is your missing angle. Every time. No exceptions. No triangle can resist this.
a° = the angle you're solving for
b°, c° = the two angles you already know

Worked Examples

See the secret weapon in action. Press an example.

▼ Select an example
Section 1: Angles in a Triangle
Section 1a · Learn

Triangles by Sides

To classify a triangle by its sides, you just need to count how many sides are equal. There are only three possibilities: 3, 2, or 0. That's the entire system.

The rule: Count the equal sides → that tells you the type.
▼ Press a triangle type to explore it
Remember: A triangle can belong to more than one group at once. An equilateral triangle is also isosceles (two — actually all three — sides equal). A right-angled triangle can be isosceles too (the 45°–45°–90° triangle).
1a: Triangles by Sides
Section 1b · Learn

Triangles by Angles

Triangles are also classified by their largest angle. Is it less than, equal to, or greater than 90°? Press each type below to explore.

Key fact: The three angles of any triangle always add up to 180° — no exceptions.
▼ Press a triangle type to explore it
1b: Triangles by Angles
Section 2 of 4 · Learn

Supplementary Angles

180°
The Supplementary Angle Sum
★   A Foundation of All Geometry   ★

What Are Supplementary Angles?

A straight line is always exactly 180°. No exceptions. When a ray splits a straight line into two angles, those two angles must add up to 180°. They are called supplementary angles.

The straight line rule: any two angles sitting on a straight line sum to 180°.

Formula: x = 180° − (known angle)

The Rule — Three Ways

A straight line is 180°.  If one angle is 60°, the other is 120°.  (180 − 60 = 120)
A straight line is 180°.  If one angle is 135°, the other is 45°.  (180 − 135 = 45)
A straight line is 180°.  If one angle is 90°, the other is 90°.  (180 − 90 = 90 — it is its own supplement)

You will use this every single time you work with parallel lines. Straight line = 180°. Subtract to find x.

Section 2: Supplementary Angles
Section 3a · Learn

Co-Interior Angles

When a straight line (called a transversal) crosses two parallel lines, it creates angle pairs at each crossing. One special pair is called co-interior angles.

Co-interior angles are the two angles that sit between the parallel lines, on the same side of the transversal. They form a C-shape.

How to spot them

Look at where the transversal crosses each parallel line. On one side of the transversal, there is one angle below the top line and one angle above the bottom line. Those two angles — trapped between the parallel lines on the same side — are the co-interior pair.

In the diagram, the angles are labelled A to H. The co-interior pair on the right side of the transversal is C and F. The co-interior pair on the left side is D and E.

3a: What are co-interior angles?
Section 3a · Practice

Find the Co-Interior Angles

Click the two angle regions that are co-interior — they sit between the parallel lines on the same side of the transversal.

3a: Identify co-interior angles
Section 3b · Learn

The Co-Interior Law

Now you can identify them. Here is what makes co-interior angles powerful.

⚠   THE LAW — MUST BE REMEMBERED

Co-interior angles always add up to 180°

co-interior angle₁ + co-interior angle₂ = 180°

Just like angles on a straight line — but stretched across to the other parallel line.

Using the law

If you know one co-interior angle, subtract from 180° to find the other. Same rule as supplementary angles.

Co-interior angle = 65° → Other = 180° − 65° = 115°
Co-interior angle = 110° → Other = 180° − 110° = 70°
Co-interior angle = 48° → Other = 180° − 48° = 132°
3b: The Co-Interior Law
Section 4 of 4 · Learn

Parallel Line Calculations

① Straight line → angles sum to 180°   ② Co-interior → angles sum to 180°
Section 4: Parallel Line Calculations
Unlocked · Final Section · Vocabulary

Names of Angles with Parallel Lines

You've already been calculating using these angle relationships throughout this module. Now it's time to learn the proper names. These names appear in exam questions, textbooks, and assessment criteria — you need to recognise them instantly.

Notice that you already know the rules. The names are just the formal language for what you've been doing.

Summary:
Co-interior (C-angles)  → sum to 180° (supplementary)
Alternate interior (Z)  → equal
Corresponding (F-angles) → equal
Vertically opposite    → equal
Alternate exterior     → equal
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Module Complete!

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