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. Subtraction of Decimals | Subtracting Decimals | Decimal Subtraction

    Apr 24, 25 03:25 PM

    Subtraction of Decimals
    We will discuss here about the subtraction of decimals. Decimals are subtracted in the same way as we subtract ordinary numbers. We arrange the digits in columns

    Read More

  2. How to Do Long Division? | Method | Steps | Examples | Worksheets |Ans

    Apr 24, 25 10:18 AM

    Long Division and Short Division Forms
    As we know that the division is to distribute a given value or quantity into groups having equal values. In long division, values at the individual place (Thousands, Hundreds, Tens, Ones) are dividend…

    Read More

  3. Division by Two-Digit Numbers | Knowledge of Estimation | Division

    Apr 24, 25 10:12 AM

    Divide 5-Digit by 2-Digit Number
    In division by two-digit numbers we will practice dividing two, three, four and five digits by two-digit numbers. Consider the following examples on division by two-digit numbers: Let us use our knowl…

    Read More

  4. Addition of Decimals | How to Add Decimals? | Adding Decimals|Addition

    Apr 24, 25 01:45 AM

    Addition of Decimals
    We will discuss here about the addition of decimals. Decimals are added in the same way as we add ordinary numbers. We arrange the digits in columns and then add as required. Let us consider some

    Read More

  5. Addition of Like Fractions | Examples | Videos | Worksheet | Fractions

    Apr 23, 25 09:23 AM

    Adding Like Fractions
    To add two or more like fractions we simplify add their numerators. The denominator remains same. Thus, to add the fractions with the same denominator, we simply add their numerators and write the com…

    Read More