Triangles and their Congruence
Introduction to Triangles and their Congruence
Take three straight lines and enclose a closed area using those three straight lines, and you get a triangle. The three straight lines become the sides of the triangle, and the 3 points where the three pairs of sides meet are called the vertices of the triangle. Where do you see triangles normally? Think of nachos! Yep, they are triangles! Yum!
The Big Idea: What are triangles?
A triangle is a plane figure in two dimensions composed of three straight lines which enclose three angles, hence the name tri - meaning three, and angle.
Each of the angles inside the triangle are referred to as the three interior angles of a triangle. The angle sum rule of triangles states that the sum of the three interior angles of any triangle is always equal to 180°. If you extend all three sides of a triangle on both sides beyond the three vertices, then the angles that are formed on the outside of the triangle are called exterior angles.
Types of triangles
Every triangle has three angles, three sides and three vertices. Depending on the relative values of the sides and angles, there are six types of triangles. Let us try and understand these six types.
- A scalene triangle is a triangle which has all three sides unequal to each other, and all 3 angles unequal as well. There is no similarity in any of the dimensions. Theoretically, there are unlimited ways in which a scalene triangle can be constructed.
- An isosceles triangle is a triangle in which two sides are equal to each other. In all isosceles triangles, the two interior angles opposite to the equal sides are also equal to each other. If one of the two equal angles is known, it is easy to find the third unequal angle.
- When all three sides of a triangle are equal, the triangle is called an equilateral triangle. The three interior angles of an Equilateral triangle are also equal. Since the sum of 3 interior angles of every triangle is equal to 180°, therefore each of the three angles of an equilateral triangle is equal to 60°. An equilateral triangle can also be called an isosceles triangle in which the two equal angles are both equal to 60°.
- A triangle in which one of the three angles is equal to 90° is called a right-angled triangle. Because of the angle sum rule of a triangle, this means that the sum of the other two angles is also 90°. Additionally, a right-angled triangle has special names for each of the sides. The side opposite the right angle is called the hypotenuse. Both arms of the right angle are referred to as the base (which is parallel to the bottom of the page) and the altitude (or height).
- An acute angle is an angle of less than 90°. A triangle where all their interior angles are acute angles is called an acute triangle or acute-angled triangle. Depending on the three angles, all the above four types of triangles (scalene, isosceles, equilateral and right-angled triangle) can also be acute triangles.
- An obtuse angle is an angle greater than 90° and less than 180°. When a triangle has one obtuse angle, it is referred to as an obtuse triangle or an obtuse angled triangle. No triangle can have more than one obtuse angle. Also, a right-angled triangle and an isosceles triangle cannot be an obtuse angled triangle.
Congruence of triangles
There are some easy rules to determine if two triangles are congruent. These rules are represented in short by their acronyms, which you will find easy to remember if you keep in mind that A stands for angle, S stands for side and RHS is used for right-angled triangles. These are the five rules you need to keep in mind:
- SSS (side side side): All three sides of the first triangle are equal to the corresponding sides of the second triangle
- SAS (side angle side): Any two sides of the first triangle are equal to the corresponding sides of the second triangle, and the angle included between these two sides of the first triangle is equal to the corresponding angle of the second triangle
- ASA (angle side angle): Two angles of the first triangle are equal to the corresponding angles of the second triangle, and the side of the first triangle that joins these two angles is equal to the corresponding side of the second triangle
- AAS (angle angle side): Any two angles of the first triangle are equal to the corresponding angles of the second triangle, and a side of the first triangle which is not included within these two angles is equal to the corresponding side of the second triangle
The key to understanding most concepts of geometry is to be able to visualize what is happening. In order to do that with regards to the congruence of triangles, it’s best to imagine two congruent triangles as mirror images. If they fit perfectly together as mirror images, they are congruent.