THE TRIANGLE AND ITS PROPERTIES

A triangle is a simple closed curve made of three line segments. It has:

• 3 Vertices: Points where sides meet (e.g., A, B, C).
• 3 Sides: Line segments forming the triangle (e.g., AB, BC, CA).
• 3 Angles: Formed by the intersection of sides (e.g., ∠BAC, ∠ABC, ∠BCA).

Classification of Triangles

Triangles can be classified based on two criteria:

1. Based on Sides:

• Scalene Triangle: All three sides have different lengths. Consequently, all three angles are also different.
• Isosceles Triangle: Any two sides are of equal length. The angles opposite to these equal sides are also equal (called base angles).
• Equilateral Triangle: All three sides are of equal length. Consequently, all three angles are also equal, each measuring 60°.

2. Based on Angles:

• Acute-angled Triangle: All three angles are acute (each angle < 90°).
• Right-angled Triangle: One angle is a right angle (exactly 90°). The other two angles are acute and their sum is 90°.
• Obtuse-angled Triangle: One angle is an obtuse angle (one angle > 90°). The other two angles are acute.

Key Points:

• The side opposite to a vertex is the side that does not include that vertex. E.g., for vertex A, side BC is opposite.
• The angle opposite to a side is the angle formed by the other two sides. E.g., for side AB, angle C is opposite.

A median of a triangle is a line segment that connects a vertex of the triangle to the mid-point of the opposite side.

• Every triangle has exactly three medians, one from each vertex.
• For $\triangle ABC$, if D is the mid-point of BC, then AD is a median.
• Medians always lie entirely within the interior of the triangle.
• The point where all three medians intersect is called the centroid of the triangle. (You'll learn more about this in higher classes).

Example: In $\triangle PQR$, if S is the mid-point of QR, then PS is a median.

An altitude of a triangle is a line segment from a vertex that is perpendicular to the line containing the opposite side.

• It represents the 'height' of the triangle from that vertex to its base.
• Every triangle has exactly three altitudes, one from each vertex.
• For $\triangle ABC$, if AL is perpendicular to BC (or its extension), then AL is an altitude.
• The point where all three altitudes intersect is called the orthocentre of the triangle.

Location of Altitudes:

• Acute-angled triangle: All three altitudes lie inside the triangle.
• Right-angled triangle: Two altitudes are the legs of the triangle themselves. The third altitude lies inside.
• Obtuse-angled triangle: Two altitudes lie outside the triangle (they meet the extended opposite side). The third altitude lies inside.

Important Note: An altitude does not necessarily bisect the opposite side, nor does it necessarily pass through the midpoint of the opposite side. It only has to be perpendicular.

When a side of a triangle is produced (extended), the angle formed outside the triangle is called an exterior angle.

• At each vertex, there are two exterior angles, which are vertically opposite and thus equal.
• An exterior angle and its adjacent interior angle form a linear pair, so their sum is 180°.
• The other two interior angles (not adjacent to the exterior angle) are called interior opposite angles or remote interior angles.

Exterior Angle Property:

The measure of an exterior angle of a triangle is equal to the sum of its two interior opposite angles.

For $\triangle ABC$, if side BC is extended to D, forming exterior angle $\angle ACD$: $$ \angle ACD = \angle BAC + \angle ABC $$ Or, in simpler terms: $$\text{Exterior Angle} = \text{Sum of Interior Opposite Angles}$$

This is a fundamental property of all triangles.

Angle Sum Property:

The sum of the measures of the three interior angles of any triangle is always 180°.

For $\triangle ABC$ with angles $\angle A$, $\angle B$, and $\angle C$: $$ \angle A + \angle B + \angle C = 180° $$

Proof using Exterior Angle Property:

1. Consider $\triangle ABC$. Extend side BC to D, forming exterior angle $\angle ACD$.
2. By Exterior Angle Property: $\angle ACD = \angle A + \angle B$.
3. Angles $\angle ACD$ and $\angle ACB$ (or $\angle C$) form a linear pair on the straight line BD.
4. Therefore, $\angle ACD + \angle ACB = 180°$.
5. Substitute (2) into (4): $(\angle A + \angle B) + \angle ACB = 180°$.
6. Hence, $\angle A + \angle B + \angle C = 180°$.

Implications:

• A triangle cannot have two right angles (90° + 90° = 180°, leaving 0° for the third angle).
• A triangle cannot have two obtuse angles (sum would be > 180°).
• A triangle must have at least two acute angles.
• If all three angles are equal (equilateral triangle), each angle is $180°/3 = 60°$.
• If two angles are equal (isosceles triangle), the third angle can be found by $180° - 2 \times (\text{equal angle})$ or $180° - (\text{unequal angle})$.

These are special types of triangles with specific properties related to their sides and angles.

Equilateral Triangle:

• Definition: A triangle in which all three sides are of equal length.
• Properties:
• All three angles are equal.
• Each angle measures 60° (since $180°/3 = 60°$).
• It is also an equiangular triangle.
• Medians, altitudes, and angle bisectors from each vertex are coincident (they are the same line segment).

Isosceles Triangle:

• Definition: A triangle in which at least two sides are of equal length.
• Properties:
• The angles opposite to the equal sides are equal. These are called base angles.
• The side that is not equal to the other two is called the base.
• The vertex where the two equal sides meet is called the vertex angle.
• The median and altitude drawn from the vertex angle to the base are the same line segment and also bisect the vertex angle.

Example: If $\triangle XYZ$ has XY = XZ, then $\angle Y = \angle Z$. XY and XZ are equal sides, YZ is the base, and $\angle X$ is the vertex angle.

This property helps determine if a set of three given lengths can actually form a triangle.

Triangle Inequality Theorem:

The sum of the lengths of any two sides of a triangle is always greater than the length of the third side.

For any $\triangle ABC$ with sides a, b, c:

1. $$ a + b > c $$
2. $$ b + c > a $$
3. $$ c + a > b $$

Difference Property:

The difference between the lengths of any two sides of a triangle is always smaller than the length of the third side.

For any $\triangle ABC$ with sides a, b, c:

1. $$ |a - b| < c $$
2. $$ |b - c| < a $$
3. $$ |c - a| < b $$

Practical Application: If you are given three lengths, you must check all three conditions of the sum property. If even one condition fails, a triangle cannot be formed.

The Pythagoras property (also known as the Pythagorean Theorem) applies only to right-angled triangles.

Terminology for Right-angled Triangles:

• Hypotenuse: The side opposite the right angle. It is always the longest side of a right-angled triangle.
• Legs: The other two sides that form the right angle.

Pythagoras Property:

In a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides (legs).

For a right-angled $\triangle ABC$, with the right angle at B, and sides AB, BC, and hypotenuse AC: $$ (\text{Hypotenuse})^2 = (\text{Leg 1})^2 + (\text{Leg 2})^2 $$ $$ AC^2 = AB^2 + BC^2 $$

Converse of Pythagoras Property:

If, in any triangle, the square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle. This means the triangle must be a right-angled triangle.

Applications:

• Finding the length of an unknown side in a right-angled triangle if the other two sides are known.
• Determining if a given triangle is a right-angled triangle.

Pythagorean Triplets: Sets of three positive integers a, b, and c, such that $a^2 + b^2 = c^2$. Common examples: (3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25).