Chapter Overview

A triangle is the simplest polygon. Three line segments join three non-collinear points to form a closed shape.

1. Parts of a Triangle

A triangle has:

• 3 vertices;
• 3 sides;
• 3 interior angles.

A triangle with vertices $A$, $B$ and $C$ is written as:

2. Types of Triangles by Sides

3. Triangle Inequality

For sides $a$, $b$, $c$:

In practice, if the sides are arranged from smallest to largest, checking whether the two smaller sides add to more than the longest side is enough.

4. Constructing a Triangle from Three Sides

If three side lengths satisfy the triangle inequality:

1. draw the longest side as a base;
2. from one endpoint, draw an arc with radius equal to the second side;
3. from the other endpoint, draw an arc with radius equal to the third side;
4. join the intersection point of the arcs to the endpoints.

5. Constructing with Sides and Angles

A unique triangle can also be constructed when suitable measurements are given, such as:

• two sides and the included angle;
• two angles and the included side.

6. Angle Sum Property

7. Exterior Angle

An exterior angle equals the sum of the two remote interior angles.

8. Altitude of a Triangle

A triangle has three altitudes, one from each vertex.

9. Types of Triangles by Angles

10. Special Triangle Facts

For an equilateral triangle:

so every angle is:

In an isosceles triangle, the angles opposite the equal sides are equal.

11. Triangle Inequality in a Stronger Form

For three positive lengths to form a triangle, the sum of any two sides must be greater than the third side.

For sides $a$, $b$ and $c$:

A quick test is to arrange the lengths from smallest to largest. It is enough to check whether the sum of the two smaller lengths is greater than the largest.

12. Angle Sum and Exterior Angle Together

The three interior angles of every triangle add to $180^\circ$.

An exterior angle forms a linear pair with the adjacent interior angle. This leads to an important result: an exterior angle of a triangle equals the sum of the two opposite interior angles.

13. Altitudes Can Look Different

An altitude is a perpendicular segment from a vertex to the opposite side or to the line containing the opposite side.

• In an acute triangle, all three altitudes lie inside the triangle.
• In a right triangle, two sides themselves act as altitudes.
• In an obtuse triangle, some altitudes meet extensions of opposite sides and lie partly outside the triangle.

14. Special Triangles

In an isosceles triangle, equal sides have equal opposite angles. Conversely, if two angles of a triangle are equal, the sides opposite them are equal.

In an equilateral triangle, all three sides are equal. Therefore all three angles are equal. Since their total is $180^\circ$:

So every angle of an equilateral triangle is $60^\circ$.

15. Construction Thinking

A triangle construction is possible only if the given information determines a triangle and satisfies the required conditions. When constructing with ruler and compass, do not measure the final intersection by eye. The arcs themselves locate the point that satisfies the given lengths.

16. Worked Understanding of Triangles

Naming Sides and Angles

In $\triangle ABC$, the side opposite $\angle A$ is $BC$, the side opposite $\angle B$ is $AC$, and the side opposite $\angle C$ is $AB$. This opposite relationship becomes important when working with equal sides and equal angles.

In an isosceles triangle, if:

then the opposite angles satisfy:

Testing Three Lengths Quickly

For sides 6 cm, 9 cm and 11 cm, check the two smaller sides:

so a triangle can be formed.

For 3 cm, 5 cm and 8 cm:

The sum must be greater than, not equal to, the third side. So these lengths form a straight arrangement, not a triangle.

Finding Angles in Isosceles Triangles

Suppose an isosceles triangle has vertex angle $40^\circ$. The two base angles are equal. Their total is:

So each base angle is:

Right Triangles

A right triangle contains one $90^\circ$ angle. Therefore the other two angles together must add to:

So the two remaining angles are always acute.

Exterior Angles as a Shortcut

Suppose two remote interior angles are $38^\circ$ and $67^\circ$. The exterior angle is:

You could also find the third interior angle first and use a linear pair, but the exterior-angle property is quicker.

Construction Accuracy

When constructing a triangle from three sides, draw one side first. Use the other two lengths as compass radii from the endpoints. Their arc intersection gives the third vertex. If the arcs do not meet, the three lengths fail the triangle inequality.

Important Terms

Common Mistakes

Exam Focus

Quick Revision

Practice Questions

Additional Practice

Chapter Summary