# A Great Explanation of Similarity Statement in Geometry With Examples

The concept of similarity is fairly important in geometry and helps prove many theorems and corollaries. The ScienceStruck article provides an explanation of similarity statement in geometry with examples.

ScienceStruck Staff

Last Updated: May 5, 2018

Quick Tips to Remember

- Two similar triangles need not be congruent, but two congruent triangles are similar.
- If an acute angle of a right-angled triangle is congruent to an acute angle of another right-angled triangle, then the triangles are similar.
- All equilateral triangles are similar.

How to Write a Similarity Statement

Step I

To begin with, identify the similar shapes. Then, draw them on paper. The figures you will be provided will be in different orientations, so, even if they are similar, they might appear different. Do not get swayed.

Step II

Draw the shapes such that equal angles line up similar to each other, i.e., you will either be given the values of the angles, or the congruent angles will be marked already. Thus, you can identify the angle and start drawing them accordingly. For example, if, in triangles ABC and PQR, angle ABC is congruent to angle PQR, trace these angles on paper first. Name the vertices correctly.

Step III

Next, move on to the next set of congruent triangles, and label them accordingly. Repeat the same with the third set of congruent angles.

Step IV

Now that you are done with understanding the similarity, write down the similar angles. Mention that angle ABC is congruent to angle PQR, angle BCA is congruent to angle QRC, and so on.

Step V

Calculate the side lengths and verify that they are in proportion. Now, write the similarity statement. You have to write triangle ABC ~ triangle PQR. The '~' sign is a congruence sign in geometry.

Similarity Statement and Ratio

- In similar shapes, the sides are in proportion. This ratio of two corresponding side lengths is called scale factor. This must be mentioned while writing the similarity statement.
- In the above figure, for instance, AB/PQ = BC/QR = AC/PR = 1/2.
- In similar triangles, the ratio of their areas is equal to the square of the ratio of their sides.
- The scale factor is used to find out the value of the unknown side in geometrical problems. It is especially useful in case of polygons.

Examples of Similarity Statements

For Three Triangles

angle A = angle P = angle X

angle B = angle Q = angle Y

angle C = angle R = angle Z

Similarity Statement: triangle ABC ~ triangle PQR ~ triangle XYZ

For a Quadrilateral

angle A = angle P

angle B = angle Q

angle C = angle R

angle D = angle S

AB/PQ = BC/QR = CD/RS = AD/PS = ½

Similarity Statement: quadrilateral ABCD ~ quadrilateral PQRS

angle B = angle Q

angle C = angle R

angle D = angle S

AB/PQ = BC/QR = CD/RS = AD/PS = ½

Similarity Statement: quadrilateral ABCD ~ quadrilateral PQRS

For Rectangles

angle A = angle P

angle B = angle Q

angle C = angle R

angle D = angle S

AB/PQ = BC/QR = CD/RS = AD/PS = 2/1

Similarity Statement: rectangle ABCD ~ rectangle PQRS

angle B = angle Q

angle C = angle R

angle D = angle S

AB/PQ = BC/QR = CD/RS = AD/PS = 2/1

Similarity Statement: rectangle ABCD ~ rectangle PQRS

For Polygons

angle A = angle P

angle B = angle Q

angle C = angle R

angle D = angle S

angle E = angle T

angle F = angle U

AB/PQ = BC/QR = CD/RS = DE/ST = EF/TU = ½

Similarity Statement: hexagon ABCDEF ~ hexagon PQRSTU

angle B = angle Q

angle C = angle R

angle D = angle S

angle E = angle T

angle F = angle U

AB/PQ = BC/QR = CD/RS = DE/ST = EF/TU = ½

Similarity Statement: hexagon ABCDEF ~ hexagon PQRSTU

Inscribed Similar Right-angled Triangles

Theorem: If an altitude is drawn from the right angle of any right-angled triangle, then the two triangles so formed are similar to the original triangle, and all three triangles are similar to each other. |

Thus,

angle ABC = angle ADB = angle BDC

angle BAC = angle DAB = angle DBC

angle ACB = angle ABD = angle BCD

Similarity Statement: triangle ABC ~ triangle ADB ~ triangle BDC

AB/BD = AC/BC

3/x = 5/4

5x = 12

x = 2.4

Similar geometric shapes follow the same order and direction. Whatever set of vertices you choose while tracing, the others will connect accordingly.