Classification · Interior Angles · Isosceles & Equilateral · Exterior Angle · Solving for x
Section 1 of 6
🔺Types of Triangles
A triangle is a closed 2D shape with three straight sides — we classify by sides AND by angles
Classifying by SIDES
Equilateral
ALL three sides equal All angles = 60°
Isosceles
TWO sides equal Base angles equal
Scalene
NO sides equal No angles equal
Classifying by ANGLES
Acute-angled
ALL angles < 90°
Right-angled
ONE angle = 90° Longest side opposite the 90°
Obtuse-angled
ONE angle between 90° and 180°
⚠️ Think about it: Can a triangle have TWO right angles? No! \(90° + 90° = 180°\) already — there would be nothing left for the third angle. The same logic shows a triangle can't have two obtuse angles.
Check Your Understanding
A triangle has angles 30°, 30° and 120°. Classify it fully.
Two equal angles (30° and 30°) → isosceles. One angle of 120° (> 90°) → obtuse-angled.
Could 89°, 89° and 89° be the angles of a triangle?
Always test the sum first: \(89° \times 3 = 267° \neq 180°\)
📐Interior Angles of a Triangle
The single most important fact in triangle geometry
The sum of the interior angles of a triangle is 180°.int ∠s ∆
In \(\triangle ABC\): \(\hat{A} + \hat{B} + \hat{C} = 180°\)
Worked Example — with CAPS reasons
In \(\triangle ABC\), \(\hat{A} = 95°\) and \(\hat{B} = 50°\). Find \(\hat{C}\).
1
\(\hat{A} + \hat{B} + \hat{C} = 180°\) int ∠s ∆
2
\(95° + 50° + \hat{C} = 180°\)
3
\(\hat{C} = 180° - 145°\) = 35°
Writing Reasons — CAPS shorthand
Every statement in a geometry solution must have a reason. Use these standard abbreviations:
Full reason
Shorthand
Sum of interior angles of a triangle = 180°
int ∠s ∆
Angles opposite the equal sides of an isosceles ∆ are equal
∠s opp equal sides
Sides opposite the equal angles are equal
sides opp equal ∠s
Angles forming a straight line = 180°
∠s on a str line
Exterior angle of ∆ = sum of interior opposite angles
ext ∠ of ∆
Each angle of an equilateral ∆ = 60°
equilateral ∆
Check Your Understanding
In \(\triangle PQR\), \(\hat{Q} = 60°\) and \(\hat{R} = 45°\). Find \(\hat{P}\).
\(\hat{P} = 180° - 60° - 45°\) [int ∠s ∆]
A right-angled triangle has one angle of 32°. Find the third angle. (Type a number only)
\(180° - 90° - 32° = ?\) [int ∠s ∆]
⛰️Isosceles & Equilateral Triangles
Equal sides ↔ equal angles — the property works BOTH ways
Isosceles Triangle Properties
If two sides are equal, the angles opposite those sides are equal.∠s opp equal sides If two angles are equal, the sides opposite those angles are equal.sides opp equal ∠s
Worked Example 1
In \(\triangle PQR\), \(PQ = PR\) and \(\hat{Q} = 54°\). Find \(\hat{P}\). (from WMCS Worksheet 3.2)
1
\(\hat{R} = \hat{Q} = 54°\) ∠s opp equal sides
2
\(\hat{P} = 180° - 2(54°)\) int ∠s ∆
3
\(\hat{P} = 180° - 108°\) = 72°
Worked Example 2 — angle at the apex given
In \(\triangle ABC\), \(AB = AC\) and \(\hat{A} = 108°\). Find \(x\) (each base angle).
1
\(\hat{B} = \hat{C} = x\) ∠s opp equal sides
2
\(2x = 180° - 108° = 72°\) int ∠s ∆
3
\(x = \) 36°
Equilateral Triangle
All three sides equal → all three angles equal → each angle \(= 180° \div 3 = \) 60°equilateral ∆
💡 Interesting question (from WMCS): Can an isosceles triangle have an angle of 95°? Yes! If the 95° angle is at the apex, the base angles are \((180° - 95°) \div 2 = 42{,}5°\) each. The obtuse angle just can't be one of the two equal base angles — there can't be two angles of 95°.
Check Your Understanding
In an isosceles triangle the apex angle is 40°. What is each base angle?
\((180° - 40°) \div 2 = ?\)
In \(\triangle DEF\), \(DE = DF\) and \(\hat{E} = 65°\). Find \(\hat{D}\). (Type a number only)
\(\hat{F} = \hat{E} = 65°\) [∠s opp equal sides], then \(\hat{D} = 180° - 2(65°)\)
📏The Exterior Angle of a Triangle
Extend one side — the angle formed equals the sum of the two interior opposite angles
What IS an exterior angle?
The exterior angle is NOT just any angle outside the triangle. It is the angle between the extension of one side and the side next to it. The exterior angle and its neighbouring interior angle lie on a straight line, so they add to 180°.
The exterior angle of a triangle = the sum of the two interior opposite angles.ext ∠ of ∆
Worked Example 1 (from WMCS Worksheet 3.5)
The exterior angle at Q is 107° and one interior opposite angle is 75°. Find \(x\), the other interior opposite angle.
1
\(x + 75° = 107°\) ext ∠ of ∆
2
\(x = 107° - 75°\) = 32°
Worked Example 2 — combined with isosceles
The exterior angle of an isosceles triangle is 100°, adjacent to the apex. The extended side is one of the equal sides. Find the base angles.
1
Interior angle next to the exterior angle: \(180° - 100° = 80°\) ∠s on a str line
2
The two base angles are equal: \(2x = 180° - 80° = 100°\) int ∠s ∆ & ∠s opp equal sides
3
Each base angle \(x = \) 50°
⚠️ Two ways to find an exterior angle:
Method 1 — straight line: \(180° - \) (interior angle next to it)
Method 2 — shortcut: add the two interior opposite angles.
Both give the same answer — the shortcut saves a step!
Check Your Understanding
A triangle has interior angles 48° and 62° at two vertices. What is the exterior angle at the THIRD vertex?
Ext ∠ = sum of the two interior OPPOSITE angles = 48° + 62°
The exterior angle of a triangle is 118° and one of the interior opposite angles is 46°. Find the other interior opposite angle. (Type a number)
\(118° - 46° = ?\) [ext ∠ of ∆]
🧮Solving for x — Algebra Meets Geometry
When angles are expressions like 2x + 40°, set up an equation using a triangle fact
The Strategy
1. Identify the geometric fact that links the angles (int ∠s ∆, str line, ext ∠ of ∆, isosceles). 2. Write the equation. 3. Solve for \(x\). 4. Substitute back if the question asks for the actual angle sizes.
Worked Example 1 — interior angles with algebra
A triangle has angles \(x\), \(x + 20°\) and \(2x + 40°\). Find \(x\), then the largest angle.
1
\(x + (x + 20°) + (2x + 40°) = 180°\) int ∠s ∆
2
\(4x + 60° = 180°\)
3
\(4x = 120°\) so \(x = 30°\)
4
Largest angle: \(2(30°) + 40° = \) 100°
Worked Example 2 — straight line + triangle
Angle \(b\) and an angle of 130° form a straight line. The triangle containing \(b\) also has an angle of 30°. Find \(a\), the third angle. (From workbook)
1
\(b = 180° - 130° = 50°\) ∠s on a str line
2
\(a + 50° + 30° = 180°\) int ∠s ∆
3
\(a = \) 100°
Worked Example 3 — isosceles with algebra
An isosceles triangle has base angles of \(x\) each and an apex angle of \(x + 30°\). Find all three angles.
1
\(x + x + (x + 30°) = 180°\) int ∠s ∆
2
\(3x + 30° = 180°\) → \(3x = 150°\) → \(x = 50°\)
3
Angles: 50°, 50° and 80°
Check Your Understanding
A triangle has angles \(x\), \(2x\) and \(3x\). What is \(x\)?
\(x + 2x + 3x = 6x = 180°\)
A triangle has angles \(2x - 30°\), \(x + 10°\) and \(2x - 50°\). Find \(x\). (Type a number)
\(5x - 70° = 180°\) → \(5x = 250°\) (this is from your DBE workbook!)
🎯Practice Quiz
12 questions covering all triangle geometry topics