Identify equilateral, isosceles, right-angled and scalene triangles
Calculate missing angles in triangles
Solve multi-step problems combining triangle and straight-line rules
Apply to real-life contexts: roofing angles, structural supports
Real-World Applications
Construction: Roof trusses and rafters rely on triangle angle rules — builders calculate exact angles to cut timber correctly and ensure a stable structure.
Navigation: Triangulation uses triangle angle properties to determine an unknown position from two known points.
Engineering: Triangles are used in structural frames because they are rigid — knowing their angles is essential for calculating forces and ensuring stability.
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No matter the shape or size of a triangle — tall, flat, narrow or wide — the three interior angles always sum to exactly 180°. This is one of the most fundamental facts in geometry.
Example 1 — A triangle has angles 60° and 75°. Find the third angle.
Step 1
All angles in a triangle sum to 180°: \(60° + 75° + x = 180°\)
Step 2
Add the known angles: \(60 + 75 = 135\)
Step 3
Subtract from 180°: \(180 - 135 = \mathbf{45°}\)
Example 2 — A triangle has angles \(x\), \(2x\), and \(3x\). Find each angle.
Step 1
Set up: \(x + 2x + 3x = 180°\)
Step 2
Simplify: \(6x = 180°\)
Step 3
Solve: \(x = 30°\)
Step 4
Angles are: \(\mathbf{30°,\ 60°,\ 90°}\)
Tip: If you find one angle greater than 90°, the triangle can only have ONE such angle — two angles over 90° would already exceed 180°, making a triangle impossible.
✓ Quick Check 1 — Angles Sum to 180°
Question 1 of 10
Types of Triangles and Their Angle Properties
You met these four triangle types (equilateral, isosceles, right-angled, scalene) in Lesson 12.1 — 2D Shapes. Knowing the type instantly gives you extra information about missing angles — here's what each type tells you.
Example 1 — An isosceles triangle has a top angle of 40°. Find the base angles.
Step 1
In an isosceles triangle, the two base angles are equal. Let each = \(x\).
Step 2
Set up: \(x + x + 40° = 180°\)
Step 3
\(2x = 140°\)
Step 4
\(x = \mathbf{70°}\)
Example 2 — A right-angled triangle has one angle of 35°. Find the third angle.
Step 1
One angle = 90° (right angle), another = 35°.
Step 2
Third angle = \(180° - 90° - 35° = \mathbf{55°}\)
Equilateral triangles: ALL angles = 60°, ALL sides equal. Isosceles: exactly two equal angles (base angles), and two equal sides opposite those angles.
Common mistake: Assuming a triangle with two equal angles must be the only type with special properties — a right isosceles triangle (90°, 45°, 45°) is BOTH right-angled AND isosceles. These categories can overlap.
✓ Quick Check 2 — Types of Triangles
Question 1 of 10
Multi-Step Problems: Triangles on Lines and in Diagrams
Many exam questions combine the triangle angle sum with the straight-line rule (angles on a straight line = 180°). Strategy: find the interior angles first, then apply the straight-line rule for exterior angles.
Example 1 — Triangle on a straight line. Interior angles are 70° and 65°. The base extends past the 65° vertex — find the exterior angle there.
Step 1
Find the third interior angle: \(180° - 70° - 65° = 45°\)
Step 2
Exterior angle (on straight line, at the 65° vertex): \(180° - 65° = 115°\)
Shortcut
Exterior angle = sum of the two remote interior angles (the ones NOT at this vertex — here, 70° and 45°): \(70° + 45° = \mathbf{115°}\)
Example 2 — A triangular support has two equal angles. The third angle is 110°. Find the equal angles.
Step 1
Let each equal angle = \(x\). Set up: \(110° + x + x = 180°\)
Step 2
\(2x = 70°\)
Step 3
\(x = \mathbf{35°}\)
Key theorem: The exterior angle of a triangle equals the sum of the two non-adjacent (remote) interior angles. Shortcut: no need to find the interior angle at that vertex first!
Roofing check: A steeper roof has a LARGER base angle and a SMALLER apex angle. If the apex angle is 50°, the base angles are 65° each — not the other way round. Always label your diagram carefully before calculating.
✓ Quick Check 3 — Multi-Step Problems
Question 1 of 10
Practice: Mixed Triangle Angle Practice
Work out each missing angle, type your answer (numbers only), then click Check All.
#
Question
Answer
Result
1
Equilateral — all angles equal
2
Right-angled, one angle = 50°
3
Isosceles, apex = 100°, find each base angle
4
Angles: 55°, 75°, ?
5
Angles: x, x, 80° — find x
6
Right-angled, one angle = 72°
7
Isosceles, base angle = 65°, find apex angle
8
Angles: 3x, 2x, x — find x
9
Right-angled, one angle = 35°
10
Isosceles, apex = 40°, find each base angle
⚠ Things to Watch Out For
Using 360° instead of 180°: Angles in a TRIANGLE sum to 180°. Use 360° for angles in a quadrilateral or around a point.
Isosceles triangles — using the wrong pair of equal angles: In an isosceles triangle, the two BASE angles are equal (the angles opposite the two equal sides). Identify the equal sides first to find the equal angles.
Not identifying the triangle type first: Before calculating, identify if the triangle is equilateral (all 60°), isosceles (two equal angles), or right-angled (one angle is 90°). This reduces the unknowns.
Exterior angles: The exterior angle of a triangle equals the sum of the two NON-ADJACENT interior angles. Don't confuse this with interior angles.
Angles in more complex diagrams: When a triangle sits on a straight line, use angles-on-a-straight-line rules alongside the triangle angle sum rule.