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




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.



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.




Share this page: What’s this?

Recent Articles

  1. Conversion of Rupees and Paise | How to convert rupees into paise?

    Feb 12, 25 02:45 AM

    Conversion of Rupees and Paise
    We will discuss about the conversion of rupees and paise (i.e. from rupees into paise and from paise into rupees). How to convert rupees into paise? First we need to remove the point and then remove

    Read More

  2. Writing Money in Words and Figure Worksheet With Answers | All Grades

    Feb 12, 25 02:30 AM

    In writing money in words and figure worksheet we will get different types of questions on expressing money in words and figures, write the amount in short form (in figures) and write the amount in lo…

    Read More

  3. Writing Money in Words and Figure | Rules for Writing Money in Words

    Feb 12, 25 01:59 AM

    Rules for writing money in words and figure: 1. Abbreviation used for a rupee is Re. and for 1-rupee it is Re. 1 2. Rupees is written in short, as Rs., as 5-rupees is written as Rs. 5

    Read More

  4. Worksheet on Money | Conversion of Money from Rupees to Paisa

    Feb 11, 25 09:39 AM

    Amounts in Figures
    Practice the questions given in the worksheet on money. This sheet provides different types of questions where students need to express the amount of money in short form and long form

    Read More

  5. Worksheet on Measurement | Problems on Measurement | Homework |Answers

    Feb 10, 25 11:56 PM

    Measurement Worksheet
    In worksheet on measurement we will solve different types of questions on measurement of length, conversion of length, addition and subtraction of length, word problems on addition of length, word pro…

    Read More