Criteria for Congruency

Here we will learn different criteria for congruency of triangles.

I. SAS (Side-Angle-Side) Criterion:

If two triangles have two sides of one equal to two sides of the other, each to each, and the angles included by those sides are equal then the triangles are congruent.

Side-Angle-Side Congruency

Here in ∆KLM and ∆XYZ,

KL = XY, LM = YZ and ∠L = ∠Y

Therefore, ∆KLM ≅ ∆XYZ.

Note: It is necessary for the included angles to be equal for congruency. If in the above figure, ∠L ≠ ∠Y and ∠L = ∠X, the triangle may not be congruent.

II. AAS (Angle-Angle-Side) Criterion:

If two triangles have two angles of one equal to two angles of the other, each to each, and any side of the one equal to the corresponding side of the other, then the triangles are congruent.

Angle-Angle-Side Congruency

Here in ∆KLM and ∆XYZ,

∠L = ∠Y, ∠M = ∠Z and KM = XZ.

Therefore, ∆KLM ≅ ∆XYZ.

 

III. SSS (Side-Side-Side) Criterion:

If two triangles have three sides of one equal to three sides of the other, the triangles are congruent.

Here in ∆KLM and ∆XYZ,

KL = XY, LM = YZ and KM = XZ.

Therefore, ∆KLM ≅ ∆XYZ.

Side-Side-Side Congruency

IV: RHS (Right Angle-Hypotenuse-Side) Criterion:

If two right-angled triangles have their hypotenuses equal and one side of one equal to one side of the other, the triangles are congruent.

Right Angle-Hypotenuse-Side Congruency

Here, ∠L = ∠Y = 90°, KM = XZ and KL = XY.

Therefore, ∆KLM ≅ ∆XYZ.

Note: * Two triangles will be congruent only if they satisfy any one of the four criterion mentioned above.

** Two triangles may not be congruent if any three parts (elements) of one are equal to the corresponding parts of the other.

Examples:

(i) If two triangles have three angles of one equal to three angles of the other, they are said to be equiangular. But equiangular triangles need not be congruent.

Here, in the given figure, ∆KLM and ∆XYZ are equiangular but not congruent.

In short, if two triangles are congruent, they must be equiangular; but if they are equiangular, they may or may not be congruent.

Equiangular Triangles

(ii) If in two triangles, two sides and one angle of one are equal to the corresponding sides and corresponding angle of the other, the triangles need not be congruent.

Need Not be Congruent

In the adjoining figure, KL = XY, KM = XZ, ∠M = ∠Z.

But ∆KLM and ∆XYZ are not congruent. For congruency, two sides and the included angle of one must be equal to those of the other.

Note: The abbreviation CPCTC is generally used for ‘Corresponding parts of Congruent Triangles are Congruent’.




9th Grade Math

From Criteria for Congruency to HOME PAGE


New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.



Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



Share this page: What’s this?

Recent Articles

  1. Roman Numerals | System of Numbers | Symbol of Roman Numerals |Numbers

    Feb 22, 24 04:21 PM

    List of Roman Numerals Chart
    How to read and write roman numerals? Hundreds of year ago, the Romans had a system of numbers which had only seven symbols. Each symbol had a different value and there was no symbol for 0. The symbol…

    Read More

  2. Worksheet on Roman Numerals |Roman Numerals|Symbols for Roman Numerals

    Feb 22, 24 04:15 PM

    Roman Numbers Table
    Practice the worksheet on roman numerals or numbers. This sheet will encourage the students to practice about the symbols for roman numerals and their values. Write the number for the following: (a) V…

    Read More

  3. Roman Symbols | What are Roman Numbers? | Roman Numeration System

    Feb 22, 24 02:30 PM

    Roman Numbers
    Do we know from where Roman symbols came? In Rome, people wanted to use their own symbols to express various numbers. These symbols, used by Romans, are known as Roman symbols, Romans used only seven…

    Read More

  4. Place Value | Place, Place Value and Face Value | Grouping the Digits

    Feb 19, 24 11:57 PM

    Place-value of a Digit
    The place value of a digit in a number is the value it holds to be at the place in the number. We know about the place value and face value of a digit and we will learn about it in details. We know th…

    Read More

  5. Math Questions Answers | Solved Math Questions and Answers | Free Math

    Feb 19, 24 11:14 PM

    Math Questions Answers
    In math questions answers each questions are solved with explanation. The questions are based from different topics. Care has been taken to solve the questions in such a way that students

    Read More