Criteria of Similarity between Triangles

We will discuss here about the different criteria of similarity between triangles with the figures.

1. SAS criterion of similarity:

If two triangles have an angle of one equal to an angle of the other and the sides including them are proportional, the triangles are similar.

SAS Criterion of Similarity

In ∆XYZ and ∆PQR, if ∠Y = ∠Q and \(\frac{XY}{PQ}\) = \(\frac{YZ}{QR}\) then ∆XYZ ∼ ∆PQR.

Similarly, if ∠X = ∠P and \(\frac{XY}{PQ}\) = \(\frac{XZ}{PR}\) then ∆XYZ ∼ ∆PQR.

Also, if ∠Z = ∠R and \(\frac{XY}{PR}\) = \(\frac{YZ}{QR}\) then ∆XYZ ∼ ∆PQR.

2. AA criterion of similarity:

If two triangles have two angles of one equal to two angles of the other, the triangles are similar.

AA Criterion of Similarity Image

In ∆XYZ, if ∠X = ∠P and ∠Y then ∆XYZ ∆PQR.

If in two triangles, two angles of one are equal to two angles of the ther, then the third angle of the first triangle is also equal to the third angle of the other because the sum of the three angles in a triangle is 180°.

Thus, similar triangles are equiangular.


3. SSS criterion of similarity:

If in two triangles, three sides of one are proportional to the three sides of the other, the triangles are similar.

SSS Criterion of Similarity

In ∆XYZ and ∆PQR, \(\frac{XY}{PQ}\) = \(\frac{YZ}{QR}\) = \(\frac{ZX}{RP}\) then ∆XYZ ∼ ∆PQR.


Theorem on Similarity between Triangles

If ∆XYZ is similar to ∆PQR and XM, PN are corresponding medians of the triangles respectively, show that \(\frac{XY}{PQ}\) = \(\frac{XM}{PN}\).

Theorem on Similarity between Triangles

Solution:

In ∆XYM and ∆PQN,

∠Y = ∠Q and \(\frac{XY}{PQ}\) = \(\frac{YM}{QN}\), (since, ∆XYZ ∼ ∆PQR and YM = \(\frac{1}{2}\)YZ, QN = \(\frac{1}{2}\)QR)

Therefore, ∆XYM ∼ ∆PQN

Therefore, \(\frac{XY}{PQ}\) = \(\frac{XM}{PN}\) (Proved)





9th Grade Math

From Criteria of Similarity between Triangles 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. Comparison of Numbers | Compare Numbers Rules | Examples of Comparison

    May 18, 24 02:59 PM

    Rules for Comparison of Numbers
    Rule I: We know that a number with more digits is always greater than the number with less number of digits. Rule II: When the two numbers have the same number of digits, we start comparing the digits…

    Read More

  2. Numbers | Notation | Numeration | Numeral | Estimation | Examples

    May 12, 24 06:28 PM

    Numbers are used for calculating and counting. These counting numbers 1, 2, 3, 4, 5, .......... are called natural numbers. In order to describe the number of elements in a collection with no objects

    Read More

  3. Face Value and Place Value|Difference Between Place Value & Face Value

    May 12, 24 06:23 PM

    Face Value and Place Value
    What is the difference between face value and place value of digits? Before we proceed to face value and place value let us recall the expanded form of a number. The face value of a digit is the digit…

    Read More

  4. Patterns in Numbers | Patterns in Maths |Math Patterns|Series Patterns

    May 12, 24 06:09 PM

    Complete the Series Patterns
    We see so many patterns around us in our daily life. We know that a pattern is an arrangement of objects, colors, or numbers placed in a certain order. Some patterns neither grow nor reduce but only r…

    Read More

  5. Worksheet on Bar Graphs | Bar Graphs or Column Graphs | Graphing Bar

    May 12, 24 04:59 PM

    Bar Graph Worksheet
    In math worksheet on bar graphs students can practice the questions on how to make and read bar graphs or column graphs. Test your knowledge by practicing this graphing worksheet where we will

    Read More