Congruence of Triangles

As a plane enclosed figures with 3-sides, segments - “triangles” are of different types based upon their sides and angles. The common variants are isosceles, equilateral, scalene, etc. So, what are congruent triangles? Triangles are said to be in congruence when every corresponding side and interior angles are congruent (of the same length). Though the triangles will have the same shape and size, one will appear as a mirror image of the other. In simple terms, any object when laid over its other counterpart appears to be the same figure or Xerox copies of each other are congruent.

Imagine the Congruent Triangles

(ImagewillbeUploadedSoon)

(ImagewillbeUploadedSoon)

In the simple case above, the two triangles ABC and DEF are congruent as each of their corresponding sides are equal, and all corresponding interior angles have the same measure. The angle at “B” measures the same (in degrees) as the angle at “E”, while the side “BA” is the same length as the side “ED” etc. What’s amazing is that no matter how you keep flipping it, the other triangle i.e “DEF” will rotate to remain in congruence to triangle “ABC” and vice-e-versa.

Easiest Way to Find if the Triangle is Congruent

Every triangle is typically represented by 6 measures i.e. 3 sides & three angles. But the fact is you need not know all of them to prove that two triangles are congruent with each other. In a similar vein, different various groups of three will do the needful. Thus, the Triangles will be congruent based on certain properties that are as follows.

Different Rules of Congruent Triangle

1. Sides and Angles

The property is based on making a triangle congruent depending on how many sides and angles of equal measures make a congruent pair. Under this criterion of congruence— when two equal sides and one equal angle forms the two similar sides, it will result in triangles appearing similar.

(ImagewillbeUploadedSoon)

(ImagewillbeUploadedSoon)

2. SAS (Side-Angle-Side)

By this property a triangle declares congruence with each other - If two sides and the involved interior angle of one triangle is equivalent to the sides and involved angle of the other triangle. This rule is a self-evident truth and does not need any validation to support the principle.

(ImagewillbeUploadedSoon)

(ImagewillbeUploadedSoon)

3. ASA (Angle Side Angle)

By this rule, two triangles are congruent to each other - If two angles and the involved side of one triangle is equivalent to the two angles and the included side of the other triangle. When we have proved the two triangles in congruence through this benchmark, the remaining two sides and the third angle will also be equal.

(ImagewillbeUploadedSoon)

(ImagewillbeUploadedSoon)

4. SSS (Side - Side - Side)

By this rule, two triangles are said to be congruent to each - If all the three sides of one triangle are of the same length as all the three sides of the other triangle. Now that all three corresponding sides are of the same length, you can be confident the triangles are congruent. And since we can be sure the triangles are congruent, this suggests that the three angles of one triangle are equal to the angles of the other triangle respectively.

(ImagewillbeUploadedSoon)

(ImagewillbeUploadedSoon)

5. AAS (Angle-Angle Side) 

By this rule, two triangles are congruent to each other - If one pair of corresponding sides and either of the two pairs of angles is equivalent to each other. The criterion of this principle is the Angle sum property of triangles that suggests that the sum of 3 angles in a triangle is 180°. Thus, if two triangles are of the same measure, automatically the 3rd side is also equal, therefore forming triangles ideally congruent.

(ImagewillbeUploadedSoon)

(ImagewillbeUploadedSoon)

6. RHS (Right Angle Hypotenuse)

By this rule of congruence, in two triangles at right angles - If the hypotenuse and one side of a triangle measure the same as the hypotenuse and one side of the other triangle, then the pair of two triangles are congruent with each other.

(ImagewillbeUploadedSoon)

(ImagewillbeUploadedSoon)

Rules that do not Apply to Make Congruent Triangle

1. AAA (Angle Angle Angle)

By this rule, if all the corresponding angles of a triangle measure equal, the triangles will become about the same shape, but not necessarily the same size. It will be a case of Two triangles of the same shape, but one is bigger than the other.

(ImagewillbeUploadedSoon)

(ImagewillbeUploadedSoon)

2. SSA (Side Side Angle) Does not Work.

Given two sides and a non-involved angle, it is likely to form two different triangles that convince the values but are certainly not adequate to show congruence.

(ImagewillbeUploadedSoon)

(ImagewillbeUploadedSoon)

Fun Facts

1. A surprising phenomenon of congruent triangles as well as other congruent shapes is that they can be reflected, flipped or converted, and still remain congruent.

2. Congruent triangles cannot be expanded or contracted, and still, be congruent.

3. Two bangles of the same shape and size are congruent with each other.

4. Imagine all the pawns on a chessboard and they are congruent.

5. Do you know cigarettes in packing are in congruence to each other?

Overview of Congruence of Triangles

1. If the three sides of a triangle and the three angles of two angles have equal orientation then it is known to be congruent triangles. 

2. CPCT expands to Corresponding Parts of Congruent Triangles. 

3. CPCT Theorem states that if two triangles bear similar angles and sides and are congruent to each other then the angles and the sides of the triangles are congruent to each other as well. 

4. If all the three sides of a triangle are corresponding to the sides of the second triangle then it is said that the triangles are congruent to each other by SSS Rule. 

5. If any two angles along with the side in between the angles are equivalent to the corresponding to the two angles and the side between them then the triangles are congruent to each other by ASA Rule. 

6. In AAS Rule, both the angles along with the side that is not included are equivalent to the triangle in the same manner. 

7. In RHS Rule, the hypotenuse and the side of the right-angled triangle are equivalent to the side and hypotenuse of another triangle.  

8. The perimeter and areas of congruent triangles are equal. 

9. There is no AAA criteria rule for the congruent rule because one triangle might be an enlarged copy of another triangle even though they might have equal angles. 

10. Two equilateral triangles might have equal angles but the sides of those triangles might be different hence, there is no congruent triangle in the case of triangles with equilateral triangles. 

Become a Pro in Congruent Triangles

• Remember the rules

Students must pay closer attention to the rules of Congruent triangles in order to solve the sums more effectively and within less time. Calculating the angles and sides can be time-consuming so a student must be well versed with the rules and criteria. 

• Practice religiously 

Math is a subject that requires constant practice and so does this chapter. In order to solve the sums like an expert, a student must practice the sums regularly to stay in touch with the concepts and rules of congruent triangles. 

• Solve questions from RD Sharma’s book.

A student should solve the questions from the textbook in order to prepare for the exam. Practicing sums of Congruent triangles from RD Sharma can give an idea of possible questions to the students. 

• Practice solved examples

The solved examples establish the fundamental knowledge and idea about Congruent triangles for the students. It can be helpful in understanding the basic concepts of the chapter including the rules of congruent triangles. 

• Attend classes on Congruent Triangles  

Paying attention to the teacher and the classes can help a student understand the chapter in the best possible way. A student must be willing to listen and learn carefully about Congruent triangles in order to solve the sums and calculate the angles and sides of the triangle. 

• Make notes during class

A student must make notes simultaneously during the classes so that the notes can help later during revision sessions. It is very important to note the rules and the formulas in order to calculate the sums correctly. 

• Stay in touch with Congruent triangles by the help of sample papers

A student must be willing to solve the sample papers after solving the examples and questions from the textbook in order to truly master the chapter. Congruent triangles can have various possible questions for exams so it is always better to be prepared. 

• Test knowledge with quizzes 

After solving questions from the books, students may test their knowledge through quizzes which will help them clock their time taken for solving one sum. It is a great way to challenge their skills on a lighter note. 

• Participate in online interactive classes 

Students can participate in online classes in order to clear their doubts as well as for the revision of this chapter. It will surely help the students understand the Congruence of triangles in more efficient ways. It will give them room to make notes and answer the questions to test their skills. 

Students can access the online materials such as notes, practice papers, sample questions, solutions to RD Sharma book along with puzzles and quizzes on Congruent Triangles from Vedantu’s official website or mobile application. The solutions can be downloaded in PDF formats as well. If a student is stuck at any point in Congruent Triangles then the student can access the online live classes and doubt clearing sessions in order to progress in the chapter.