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. Worksheet on Triangle | Homework on Triangle | Different types|Answers

    Jun 21, 24 02:19 AM

    Find the Number of Triangles
    In the worksheet on triangle we will solve 12 different types of questions. 1. Take three non - collinear points L, M, N. Join LM, MN and NL. What figure do you get? Name: (a)The side opposite to ∠L…

    Read More

  2. Worksheet on Circle |Homework on Circle |Questions on Circle |Problems

    Jun 21, 24 01:59 AM

    Circle
    In worksheet on circle we will solve 10 different types of question in circle. 1. The following figure shows a circle with centre O and some line segments drawn in it. Classify the line segments as ra…

    Read More

  3. Circle Math | Parts of a Circle | Terms Related to the Circle | Symbol

    Jun 21, 24 01:30 AM

    Circle using a Compass
    In circle math the terms related to the circle are discussed here. A circle is such a closed curve whose every point is equidistant from a fixed point called its centre. The symbol of circle is O. We…

    Read More

  4. Circle | Interior and Exterior of a Circle | Radius|Problems on Circle

    Jun 21, 24 01:00 AM

    Semi-circular Region
    A circle is the set of all those point in a plane whose distance from a fixed point remains constant. The fixed point is called the centre of the circle and the constant distance is known

    Read More

  5. Quadrilateral Worksheet |Different Types of Questions in Quadrilateral

    Jun 19, 24 09:49 AM

    In math practice test on quadrilateral worksheet we will practice different types of questions in quadrilateral. Students can practice the questions of quadrilateral worksheet before the examinations

    Read More