Hello, and welcome to today's mathematics lesson. Today, we are going to explore Chapter 17: Triangles. By the end of this lesson, you will understand what makes a triangle, how to classify different types of triangles, and some fascinating properties that triangles possess — especially the isosceles triangle.
Let us begin with the basics. A triangle is a closed figure with three sides. We denote a triangle using the Greek letter delta, written as Δ. So, triangle ABC is written as ΔABC.
Every triangle has three vertices — these are the points where two sides meet. In triangle ABC, the vertices are A, B, and C. Each vertex has an opposite side: A is opposite side BC, B is opposite side AC, and C is opposite side AB.
A triangle has three interior angles. These are the angles that lie inside the triangle. Here is a fundamental truth about triangles: the sum of the three interior angles is always 180 degrees. We write this as ∠A + ∠B + ∠C = 180°.
Now, let us talk about exterior angles. When you extend any side of a triangle, you create an exterior angle. Imagine extending side BC of triangle ABC to a point D. The angle ∠ACD formed outside is an exterior angle.
At each vertex, two exterior angles can be formed, and they are equal because they are vertically opposite angles. Also, an exterior angle and its adjacent interior angle always add up to 180 degrees.
Here is a powerful property you must remember. The exterior angle of a triangle equals the sum of its two interior opposite angles.
Let me explain. In triangle ABC, if we extend BC to D, then exterior angle ∠ACD equals the sum of angles ∠BAC and ∠ABC. These two angles are called the interior opposite angles.
Let us see this in action. Suppose an exterior angle measures 130 degrees, and the two interior opposite angles are 7x and 6x. Using our property: 130 equals 7x plus 6x, which gives 13x equals 130, so x equals 10. Therefore, the two interior angles are 70 degrees and 60 degrees.
Now, let us classify triangles. We can do this in two ways: by their angles, or by their sides.
By angles, we have three types. First, an acute-angled triangle, where every angle is less than 90 degrees. Second, a right-angled triangle, where one angle is exactly 90 degrees. In a right-angled triangle, the side opposite the right angle is called the hypotenuse, and it is always the longest side. The other two angles are acute and add up to 90 degrees. Third, an obtuse-angled triangle, where one angle is greater than 90 degrees.
By sides, we also have three types. A scalene triangle has all sides unequal, and therefore all angles are unequal too. An isosceles triangle has at least two equal sides. The angle between the equal sides is called the vertical angle, and the third side is called the base. The angles opposite the equal sides are called base angles, and they are equal.
Here is the crucial property of isosceles triangles. The angles opposite to equal sides are equal. Conversely, if two angles of a triangle are equal, then the sides opposite to them are also equal.
An equilateral triangle has all three sides equal. Since all sides are equal, all angles are equal too, and each angle measures exactly 60 degrees. An equilateral triangle is also called equiangular. Note that every equilateral triangle is isosceles, but not every isosceles triangle is equilateral.
There is also a special case: the isosceles right-angled triangle. This has one right angle and two equal sides. Since the angles must sum to 180 degrees, the two base angles are each 45 degrees.
Let us work through an example. In an isosceles triangle, the vertical angle is 40 degrees. What are the base angles? Let each base angle be x. Then x plus x plus 40 equals 180. This gives 2x equals 140, so x equals 70 degrees. Each base angle is 70 degrees.
Another example: one base angle of an isosceles triangle is 65 degrees. Since base angles are equal, the other base angle is also 65 degrees. The vertical angle is 180 minus 130, which equals 50 degrees.
One more: if a base angle is double the vertical angle, let the vertical angle be x. Then each base angle is 2x. So x plus 2x plus 2x equals 180. This gives 5x equals 180, so x equals 36 degrees. The vertical angle is 36 degrees, and each base angle is 72 degrees.
Let us recap the key takeaways from today's lesson.
First, a triangle has three sides, three vertices, and three interior angles that always sum to 180 degrees. Second, an exterior angle equals the sum of the two interior opposite angles. Third, triangles can be classified by angles as acute, right, or obtuse; and by sides as scalene, isosceles, or equilateral. Fourth, in an isosceles triangle, angles opposite equal sides are equal, and conversely, equal angles have equal opposite sides. Fifth, an equilateral triangle has all angles equal to 60 degrees. Sixth, an isosceles right-angled triangle has angles of 90, 45, and 45 degrees.
Triangles are everywhere in mathematics and in the world around you. Understanding their properties opens doors to geometry, trigonometry, and beyond. Keep practicing, stay curious, and I will see you in the next lesson.