# Geometry: Congruency Explained

Congruency simply refers to two or more shapes that have the same shape and size, that is, the length and angles are of the same magnitude. Triangles are usually the shape that are used when examining the concept of congruency. Therefore, you will learn about the concept of congruent triangles.

Two triangles are considered congruent if the following criteria are satisfied:

- The length of all three three sides of one triangle must be the same length of all three sides of the other triangle. This is called the “SSS” condition, which is Side-Side-Side.
- Two angles and one of the lengths of one triangle must be the same magnitude of the corresponding two angles of the other triangle. This is the “ASA”, which is the Angle-Side-Angle.
- An angle that is between two sides of a triangle is equal to the corresponding angle in the other triangle with respect to the same sides. This is the SAS rule. Side-Angle-Side.
- Two right angled triangle that the hypotenuse and one other side have the same magnitude.

Consider the following triangles:

Therefore the hypotenuse of both triangles are equal (a = b) and one of the lengths are equal (c = d).

**Example 1**

Prove that the following triangles are congruent.

In order for us to prove that they are congruent we need to remember the rule. It states that in order for two right angled triangles to be congruent, the hypotenuse along with one other side of both triangles must be equal. From the question, we can see that both triangles have the same corresponding side the same length which is 4. Since we know that the hypotenuse of one of the tangle is 5, we need to check of the hypotenuse of the other triangle is also 5. In order for us to do that we must find the unknown ‘a’. To solve for ‘a’ we need to recall knowledge of triangles in terms of finding missing sides.

Recall the Pythagoras theorem: The square of the hypotenuse is equal to the sum of the squares of the sides adjacent to the right angle:

## a^{2} = b^{2} + c^{2}

Therefore from the question:

a^{2 }= 4^{2} + 3^{2}

a^{2} = 16 + 9

a^{2} = 25.

Since we have a^{2} and we need to find ‘a’ without the exponent. Therefore, we need to find the square root of both sides to get ‘a’.

√a^{2} = √25

Therefore a = 5.

Since both triangles have hypotenuse of the same length and another corresponding side of the same length then both triangles are congruent.

Sometimes the questions may come in worded form and you will have to decipher the information correctly with your knowledge of the concepts of triangles to solve the problem. Sometimes you might have to use the sine and cosine rule to find either an angle or a side. You may also note if a question requires you prove if two shapes are congruent and you cannot prove it, that is perfectly fine, just show that it cannot proven and state the same.

**Example 2**

A triangle has dimensions 3, 5, 3 respectively and an angle at the base of magnitude 33.5ᵒ. Another triangle has dimensions 3, 5, and ‘a’ and an angle not at the base of 113ᵒ. State the rule of congruency that applies and prove that the triangle is congruent.

To solve this problem you may need to draw a representation of both triangles to get an idea of what you are working with. Note that you do not have do that, but it is much easier to do it that way. From the question you can see that the first triangle is isosceles triangle because it has two sides of the same length. Therefore:

You might be wondering how we came by Sine 33.5ᵒ since no angle is there. From our knowledge of triangle, all angles in a triangle adds up to 180, and in an isosceles triangle the angles at the base are equal, therefore:

2*x* + 113 = 180

2*x* = 180 – 113

2*x* = 67

*x* = 67/2

*x* = 33.5ᵒ

So each of the angles at the base equals to 33.5ᵒ

5/0.9205 = a/0.5519

0.9205(a) = 5(0.5519)

0.9205a = 2.7597

a = 2.7597/0.9205

a = 2.998 which is approximately 3.

Since all corresponding sides are equal (SSS), thus the shapes are congruent. You can run the SAS test in your own time.

Congruency is also important when you do topics such as Transformation. If a shape is reflected, rotated and translated then it should produce a congruent shape. In Transformation, there is an object and an image. The image is produced after some transformation acts on an object. There are different rules that apply to each concept but the idea is that the image should be congruent to the object. If an object is rotated, it simply moves from one position to another, and therefore it doesn’t change the length of the side nor the size of the angles, the same holds true for reflection, because while the shape might look different the magnitude of the lengths and angles.

In the diagram above, the two triangles are congruent because the conditions named above:

- SSS (a = d, b = f, and c = e)
- ASA (A = D, C = E and a = d)
- SAS (C = E, b = f and a = d)

The right angled rule:

Now that we have the diagrams the next thing is to see what rule of congruency can be applied. We can either check for side-side-side (SSS), because we have two corresponding sides that are the same. Or we could check for the side-angle-side (SAS) because we can check if the corresponding angles of both triangles are the same.

Let us check for (SSS): We can check by finding side ‘a’. To do that we can use the sine rule:

You might be wondering how we came by Sine 33.5ᵒ since no angle is there. From our knowledge of triangle, all angles in a triangle adds up to 180, and in an isosceles triangle the angles at the base are equal, therefore:

2*x* + 113 = 180

2*x* = 180 – 113

2*x* = 67

*x* = 67/2

*x* = 33.5ᵒ

So each of the angles at the base equals to 33.5ᵒ

5/0.9205 = a/0.5519

0.9205(a) = 5(0.5519)

0.9205a = 2.7597

a = 2.7597/0.9205

a = 2.998 which is approximately 3.

Since all corresponding sides are equal (SSS), thus the shapes are congruent. You can run the SAS test in your own time.

Congruency is also important when you do topics such as Transformation. If a shape is reflected, rotated and translated then it should produce a congruent shape. In Transformation, there is an object and an image. The image is produced after some transformation acts on an object. There are different rules that apply to each concept but the idea is that the image should be congruent to the object. If an object is rotated, it simply moves from one position to another, and therefore it doesn’t change the length of the side nor the size of the angles, the same holds true for reflection, because while the shape might look different the magnitude of the lengths and angles.