Mathematics

Angles &
Triangles

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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
🎓

Module Complete!

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