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A Great Explanation of Similarity Statement in Geometry With Examples

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.
Buzzle Staff
Last Updated: May 5, 2018
Quick Tips to Remember
  1. Two similar triangles need not be congruent, but two congruent triangles are similar.
  2. 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.
  3. All equilateral triangles are similar.
The statement of similarity mentions that for two shapes to be similar, they must have the same angles and their sides must be in proportion. While writing a similarity statement in geometry, the reasons as to why the two shapes are similar, are explained. The concept is used to prove many theorems, as mentioned earlier.
It is also used to find the value of the unknown side of a geometric shape, while the values of the other sides are provided. In the paragraphs below, you will learn how to write similarity statements for different geometric figures.
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
Side scale 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
Similar Triangles
Similar Triangles
triangles
The figures above depict three similar triangles. In the triangles above,

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
Similar 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
For Rectangles
Similar 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
For Polygons
Similar 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
Inscribed Similar Right-angled Triangles
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.
In the above figure, assume that angle BAC = 30° and angle ACB = 60°. Then, according to the theorem, angle ABD = 60° and angle DBC = 30°.

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
This theorem can be used to solve for 'x' in the above figure. Since they are similar, their sides will be proportional as well. To begin with, separate the triangles and trace them individually. Then, by similarity theorem, consider any of the two inscribed triangles and the main right-angled triangle to find the value of the unknown. Let us consider triangle ABC and triangle BDC. Thus,

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.