What is Congruency of Triangles Definition Rules and Solved Examples
Congruent Triangles
Congruent Triangles are triangles that have an equivalent size and shape. This means that the corresponding sides are equal and therefore the corresponding angles are equal. In this article, we are going to discuss the congruence of triangles class 7 cbse.
It can be told whether two triangles are congruent without testing all the sides and all the angles of the two triangles. In this lesson, we'll consider the four rules to prove triangle congruence. It’s called the SSS rule, SAS rule, ASA rule, and AAS rule. In congruence of triangles class 7, we'll consider a symbol used for right triangles called the Hypotenuse Leg Rule.
Properties of Congruent Triangles
Now we know about the congruence of triangles class 7 CBSE. Let’s discuss the properties. If two triangles are congruent, then each part of the Triangulum (side or angle) is congruent to the corresponding part within the other triangle. This is the truth value of the concept; once you've proven two triangles are congruent, you'll find the angles or sides of 1 of them from the opposite .
To remember this important idea, we usually find it very helpful to use the acronym CPCTC, which is the short form for "Corresponding Parts of Congruent Triangles are Congruent".
In addition to sides and angles, all other properties of Triangulum are equivalent also, like area, perimeter, location of centers, circles, etc.
How Will You Know That a Triangle is Congruent?
We can prove the congruence of triangles for class 7 CBSE using a few ways. A triangle is basically defined by six measures (three sides, three angles). But you do not get to know all of them to point out that two triangles are congruent. Various groups of three will do. Triangles are congruent if:
SSS (side side side)
All three corresponding sides are equal in length.SAS (side angle side)
A pair of corresponding sides and therefore the included angle are equal.ASA (angle side angle)
A pair of corresponding angles and therefore the included side are equal.AAS (angle angle side)
A pair of the corresponding angles and a non-included side is equal.HL (hypotenuse leg of a right triangle)
Two right angled triangles are congruent only if the hypotenuse and one leg are the same.
CPCT Rules in Maths
The full sort of CPCT is corresponding parts of congruence of triangles class 7 CBSE. Congruency are often predicted without actually measuring the edges and angles of a triangle. Different rules of congruency are as follows.
SSS (Side-Side-Side)
SAS (Side-Angle-Side)
ASA (Angle-Side-Angle)
AAS (Angle-Angle-Side)
RHS (Right angle-Hypotenuse-Side)
Let Us Learn All Three Conditions of Congruence of Triangles Class 7 CBSE in Detail.
SSS (Side-Side-Side)
If all three sides of one triangle are equivalent to the corresponding three sides of the second triangle, then the two triangles are said to be congruent by SSS rule.
SAS (Side-Angle-Side)
If any two sides and the angle included between the sides of one triangle are equivalent to the corresponding two sides and the angle between the sides of the second triangle, then the two other triangles are known to be congruent by SAS rule.
ASA (Angle-Side- Angle)
If any two angles and sides included between the angles of 1 triangle are like the corresponding two angles and side included between the angles of the second triangle, then the 2 triangles are said to be congruent by ASA rule.
AAS (Angle-Angle-Side)
AAS stands for Angle-angle-side. When two angles and a non-included side of a triangle are adequate to the corresponding angles and sides of another triangle, then the triangles are said to be congruent.
Students many times get confused for AAS with ASA congruency. But remember that AAS is for the non-included side, whereas ASA is for included sides of the triangles.
RHS (Right Angle-Hypotenuse-Side)
If the hypotenuse and a side of a right-angled triangle are equivalent to the hypotenuse and a side of the second right-angled triangle, then the two right triangles are said to be congruent by RHS rule.
Information You Need to Check Whether the Triangles Are Congruent or Not
Let us draw a congruent triangle for ΔABC. You can do so if you've got the subsequent information:
The lengths of all of the three sides of ΔABC OR
The length of two sides are therefore the angle between them OR
The measure of the two angles and length of their side included by them.
Important Observations: Two angles are said to be congruent whenever they have the same measurement. Instead of denoting congruence by ≅ you can also denote it by = since they are equal in measure.
FAQs on Congruency of Triangles Concepts and Criteria Explained
1. What is congruency of triangles?
The congruency of triangles means that two triangles are exactly the same in shape and size, so all their corresponding sides and angles are equal. If triangle ABC is congruent to triangle DEF, we write it as ΔABC ≅ ΔDEF. This means:
- AB = DE
- BC = EF
- CA = FD
- ∠A = ∠D, ∠B = ∠E, ∠C = ∠F
2. What are the conditions for congruency of triangles?
The main congruence conditions for triangles are SSS, SAS, ASA, AAS, and RHS. These rules prove two triangles are congruent if specific corresponding parts are equal:
- SSS (Side–Side–Side): All three sides are equal.
- SAS (Side–Angle–Side): Two sides and the included angle are equal.
- ASA (Angle–Side–Angle): Two angles and the included side are equal.
- AAS (Angle–Angle–Side): Two angles and a non-included side are equal.
- RHS (Right angle–Hypotenuse–Side): In right triangles, hypotenuse and one side are equal.
3. What is the SSS congruence rule?
The SSS congruence rule states that if all three sides of one triangle are equal to the three corresponding sides of another triangle, then the triangles are congruent. For example:
- If AB = DE = 5 cm
- BC = EF = 6 cm
- CA = FD = 7 cm
4. What is the SAS congruence rule?
The SAS congruence rule states that if two sides and the included angle of one triangle are equal to those of another triangle, the triangles are congruent. The angle must be between the two equal sides. For example:
- AB = DE = 4 cm
- BC = EF = 6 cm
- ∠B = ∠E = 60°
5. What is the ASA congruence rule?
The ASA congruence rule states that if two angles and the included side of one triangle are equal to those of another triangle, then the triangles are congruent. Example:
- ∠A = ∠D = 50°
- ∠B = ∠E = 60°
- AB = DE = 5 cm
6. What is the RHS congruence rule in right triangles?
The RHS (Right angle–Hypotenuse–Side) rule states that two right triangles are congruent if their hypotenuse and one corresponding side are equal. This rule applies only to right-angled triangles. Example:
- ∠A = ∠D = 90°
- Hypotenuse BC = EF = 10 cm
- Side AB = DE = 6 cm
7. What is the difference between congruent and similar triangles?
The main difference is that congruent triangles are equal in both shape and size, while similar triangles have the same shape but not necessarily the same size. In congruent triangles:
- All corresponding sides are equal.
- All corresponding angles are equal.
- Corresponding angles are equal.
- Corresponding sides are proportional (not necessarily equal).
8. How do you prove two triangles are congruent?
To prove two triangles are congruent, you must show that they satisfy one of the valid triangle congruence criteria such as SSS, SAS, ASA, AAS, or RHS. Follow these steps:
- Identify equal sides and angles from the diagram or given data.
- Match corresponding parts carefully.
- State the appropriate congruence rule.
- Write the conclusion, e.g., ΔABC ≅ ΔDEF by SAS.
9. Is AAA a valid congruence rule for triangles?
No, AAA (Angle–Angle–Angle) is not a valid congruence rule because it only guarantees similarity, not equal size. If all three angles of two triangles are equal, the triangles may differ in side lengths. Therefore, AAA proves similar triangles, not congruent triangles.
10. Why is congruency of triangles important in geometry?
The congruency of triangles is important because it helps prove equality of sides and angles in geometric figures. It is used to:
- Prove properties of isosceles and equilateral triangles.
- Show diagonals bisect each other in parallelograms.
- Solve geometric proofs and constructions.
- Find unknown lengths and angles accurately.
