How to Know If Two Triangles Are Congruent: Key Criteria & Tips
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: Rules, Proofs & Easy Examples
1. What is the concept of 'Congruency of Triangles' in geometry?
In geometry, two triangles are said to be congruent if they are identical in both shape and size. This means that if you could place one triangle on top of the other, they would match up perfectly, point for point, side for side, and angle for angle. All corresponding sides and corresponding angles of congruent triangles are equal.
2. What are the main criteria used to prove that two triangles are congruent?
There are five primary criteria, or rules, to prove that two triangles are congruent. These rules provide shortcuts without needing to check all six corresponding parts (3 sides and 3 angles). The criteria are:
- SSS (Side-Side-Side): If all three sides of one triangle are equal to the corresponding three sides of another triangle.
- SAS (Side-Angle-Side): If two sides and the included angle (the angle between those two sides) of one triangle are equal to the corresponding parts of another triangle.
- ASA (Angle-Side-Angle): If two angles and the included side of one triangle are equal to the corresponding parts of another triangle.
- AAS (Angle-Angle-Side): If two angles and a non-included side of one triangle are equal to the corresponding parts of another triangle.
- RHS (Right Angle-Hypotenuse-Side): In two right-angled triangles, if the hypotenuse and one corresponding side are equal.
3. What is the difference between congruent triangles and similar triangles?
The key difference lies in size. Congruent triangles have the same shape and the exact same size; their corresponding sides and angles are equal. Similar triangles have the same shape but can have different sizes; their corresponding angles are equal, but their corresponding sides are only proportional. For example, a photograph and its enlargement are similar, but two identical prints from the same negative are congruent.
4. How is the concept of triangle congruency applied in real-world examples?
Congruency is a fundamental principle in engineering, architecture, and manufacturing. For instance, in construction, triangular trusses in bridges and roofs are made of congruent triangles to ensure stability and even weight distribution. In mass production, items like tiles, biscuits in a pack, or components of a machine are manufactured to be congruent for consistency and interchangeability.
5. What does CPCTC stand for, and why is it important?
CPCTC stands for 'Corresponding Parts of Congruent Triangles are Congruent'. This is not a rule to prove congruence, but rather a conclusion you can draw after you have proven two triangles are congruent using a criterion like SSS, SAS, or ASA. Its importance lies in allowing you to state that all the other corresponding parts (sides or angles) that you didn't use in your proof are also equal.
6. In the SAS congruence criterion, why must the angle be 'included' between the two sides?
The angle must be included because it 'locks' the shape and size of the triangle. If the angle were not between the two known sides (a situation known as SSA or Side-Side-Angle), the side opposite the angle could be positioned in two different ways, potentially creating two different triangles. The included angle provides a fixed, rigid structure, ensuring that only one unique triangle can be formed, which is essential for proving congruence.
7. Why isn't there an AAA (Angle-Angle-Angle) rule for proving congruence?
There is no AAA congruence rule because knowing that all three corresponding angles are equal is not enough to guarantee that the triangles are the same size. Two triangles can have the same angles but different side lengths—one being an enlarged version of the other. This condition proves that the triangles are similar, but not necessarily congruent. For congruence, you must know at least one corresponding side length.
8. Is SSA (Side-Side-Angle) a valid criterion for proving congruence? Why or why not?
No, SSA (Side-Side-Angle) is not a valid criterion for proving triangle congruence. This is because the given information can be ambiguous. Depending on the lengths of the sides and the measure of the angle, it might be possible to construct two different valid triangles. Since SSA does not guarantee a unique triangle, it cannot be used as a reliable rule for congruence, except in the specific case of right-angled triangles, which is covered by the RHS rule.
