30 Most Important Triangle Concepts Every Student Should Know | Part-1

This video will be divided into two different parts, and this is the first part. Let us begin. 1. Definition of a Triangle A triangle is a shape that has three sides, three angles, and three corners. It is one of the simplest shapes in geometry. These are the corners, also called vertices, or triangles, and these three lines are three sides of the triangle. Now, inside every triangle, there are three angles, one at each corner. 2. Sum of Angles in a Triangle A special rule about triangles is that if you add up all the angles inside a triangle, the total will always be 180 degrees. This works for every triangle, no matter how big or small, skinny or fat. It's like a magic rule of triangles. For example, if this angle is 40 degrees, and this is 60 degrees, then this angle, let us call it x, plus 40 plus 60, will be equal to 180. This means x equals 180 degrees, minus 40 degrees, minus 60 degrees, or 80 degrees. 3. Types of Triangles by Sides Triangles can be grouped in two ways, by their sides or by their angles. If a triangle has all three sides of the same length, it is called an equilateral triangle, and all its angles are 60 degrees. If only two sides are the same length, it is called an isosceles triangle, and the angles opposite these sides are equal. If all three sides are different, it is called a scalene triangle, and all the angles are different. 4. Types of Triangles by Angles Triangles can also be grouped based on their angles. If all three angles are less than 90 degrees, it is called an acute triangle. If one angle is exactly 90 degrees, it is called a right triangle. If one angle is more than 90 degrees, it is called an obtuse triangle. 5. Exterior Angle Theorem Here's a fun one. Draw a triangle and pick one corner. Now, extend one side past that corner to make a new angle outside the triangle. That's the exterior angle. This outside angle has a neat trick. Its size equals the sum of the two inside angles that aren't next to it. For example, if your triangle has angles 30 degrees, 60 degrees, and 90 degrees, and you extend the side at 90 degrees, the exterior angle there is 30 plus 60, which is 90 degrees, which we can also visually observe. 6. Triangle Inequality Theorem Can any three lines make a triangle? Nope. The triangle inequality theorem says that the sum of any two sides of a triangle must be greater than the third side. Imagine you have sticks of lengths 3, 4, and 8. Try making a triangle. 3 plus 4 is 7, but that's less than 8, so they will never make a triangle, no matter how hard you try. But if you have 3, 4, and 5, then 3 plus 4 is 7, more than 5. 3 plus 5 is 8, more than 4. And 4 plus 5 is 9, more than 3. That works. Now, let me know in the comments. Can sticks of length 5, 7, and 13 make a triangle? Every right answer will get my hug. 7. Perimeter of a Triangle The perimeter of a triangle is simply the sum of all three sides. If a triangle has sides 4, 5, and 6, the perimeter is 4 plus 5 plus 6, which is 15. Think of it like measuring a fence around a triangular garden. You walk along each side and add up the steps. Base and Height Every triangle has a base, which is nothing but any side that you would like to pick, and a height for that base. The height is a straight line from the opposite corner down to the base, hitting it at a 90-degree angle like this. This height helps us find the area later. Here's the fun part. For an isosceles triangle, if these two sides are of equal length, and if we choose this as the base of the triangle, the height will split this base in half. 9. Area of a Triangle Area is the amount of space inside a shape or surface. Imagine a square floor, and you want to cover it with small square tiles. If each tile takes up one unit of space, then the total number of tiles needed to cover the floor is the area of the square. The area of a triangle is found by multiplying half of the base with the height. So consider this acute angled triangle. If this side is the base, and then this is the height, an area will be this. Then consider this right angled triangle. If this side is the base, and then this side itself becomes the height, an area will be this. Then consider this obtuse angled triangle. If this side is the base, and then this is the height, an area will be this. Yes, height can also go outside of the triangle in case of obtuse angled triangle. Cool! 10. Congruence of Triangles If two triangles have the same shape and size, like copy-paste or like twins, they are called congruent. Triangles can be congruent in different ways. If all three sides are the same, it is called side-side-side congruence. If two sides and the angle between them are the same, it is called side-angle-side congruence. If two angles and the side between them are the same, it is called angle-side-angle congruence. If two angles and a side that is not between them are the same, it is called angle-angle-side congruence. And if If two right triangles have the same hypotenuse and one leg, it is called right hypotenuse side congruence. Now if this is an isosceles triangle and this is the height of this triangle, then tell me in comments whether both these triangles are congruent or not. Also let me know which of these rules you have used to prove the same. 11. Similarity of triangles. Similar triangles look alike, but can be different in size, like a big and small version of the same photograph. There are three main ways to check similarity. First, the angle-angle rule. If two angles of one triangle are the same as two angles of another, the third angle must also be the same, making the triangles similar. Second, the side-side-side rule. If all three sides of one triangle are in the same ratio as the sides of another, they are similar. Third, the side-angle-side rule. If two sides of one triangle are in the same ratio as two sides of another, and the angle between them is the same, then the triangles are similar. For example, if one triangle has sides 3 and 4, with an angle of 60 degrees between them, and another triangle has sides 6 and 8, with the same 60-degree angle, they are similar because 6 is twice of 3, and 8 is twice of 4. Wausam. 12. Pythagorean theorem. This is the most widely known, most loved, and most useful rule in all of geometry. It applies to right-angled triangles, and says that the square of the longest side, the hypotenuse, is equal to the sum of the squares of the other two sides. 13. Median theorem. A median is a line drawn from any corner of a triangle to the middle of the opposite side. So if this is the center point of this side of the triangle, which divides it into half, then this line is called the median line. Here's the cool part. Each median splits the triangle's total area into two equal parts. Which means, suppose this triangle has a total area of 30 square centimeters. If you draw this median, it will split the triangle into two triangles, this and this, and each triangle will have an area of 15 square centimeters, even if their shapes look different. 14. Centroid theorem. There is another cool part about medians. Now, if you draw all three medians in a triangle, all three medians of a triangle will always intersect at a single point. This special point is called the centroid of the triangle. The centroid is the triangle's center of balance, meaning if you cut out the triangle from a solid sheet and tried to balance it on the centroid, it would stay level. Consider a triangle with vertices at 0,0, 6,0, and 3,6. The midpoints of its sides are this will be 3,0, this will be 6 plus 3 or 9 over 2 or 4.5,6 plus 0 divided by 2 or 3, and similarly, this will be 1.5,3. If you draw medians from each vertex to the opposite midpoint, all three will meet at a single point, the centroid, which is located at 3,2. This is because the centroid is given by this formula, where x1, x2, and x3 are the x-values of the corners of this triangle, and y1, y2, and y3 are the y-values of the corners of this triangle. So, put it here to get the centroid as 3,2. Another interesting fact is that the centroid always divides each median into two parts, with the longer part being twice the length of the shorter part. Suppose a median in a triangle is 9 centimeters long. The centroid will split it into 3 centimeters, which will be from the centroid to the midpoint of the opposite side, and 6 centimeters, which will be from the centroid to the vertex. The ratio of the two parts is always 1 to 2, no matter what the triangle looks like. 15. Angle Bisector Theorem It says that if you draw a line inside a triangle that cuts an angle into two equal parts, this line is called an angle bisector, then it divides the opposite side into two segments that are proportional to the other two sides of the triangle. Imagine a triangle with sides 4 centimeters, 6 centimeters, and an unknown third side. If we draw an angle bisector from one of the angles to the opposite side, it will split that side into two parts in the same ratio as the other two sides. Suppose the opposite side is 5 centimeters long, since the other two sides are in the ratio 4 to 6, which simplifies to 2 to 3. This first segment is 2 divided by 2 plus 3 times 5, which is 2 centimeters. The second segment is 3 divided by 2 plus 3 times 5, giving 3 centimeters. 16. Properties of Equilateral Triangles An equilateral triangle is super special because all three sides are exactly the same length, and all three angles are always 60 degrees. One amazing fact about an equilateral triangle is that the height, median, and angle bisector from any corner are actually the same line. Phew, I am tired for now. Rest of them will be covered in Part 2. If you enjoy my videos and want to support my channel, consider becoming a Patreon, as it helps me create more awesome content for you. Link is in the description. Also you can support my channel by joining our community and becoming a member. So good!