Similar Triangles
Similar triangles are the triangles that have corresponding sides in proportion to each other and corresponding angles equal to each other. Similar triangles look the same but the sizes can be different. In general, similar triangles are different from congruent triangles. There are various methods by which we can find if two triangles are similar or not. Let us learn more about similar triangles and their properties along with a few solved examples.
What are Similar Triangles?
Similar triangles are the triangles that look similar to each other but their sizes might not be exactly the same. Two objects can be said similar if they have the same shape but might vary in size. That means similar shapes when magnified or demagnified superimpose each other. This property of similar shapes is referred to as "Similarity".
Similar Triangles Definition
Two triangles will be similar if the angles are equal (corresponding angles) and sides are in the same ratio or proportion(corresponding sides). Similar triangles may have different individual lengths of the sides of triangles but their angles must be equal and their corresponding ratio of the length of the sides must be the same. If two triangles are similar that means,
- All corresponding angle pairs of triangles are equal.
- All corresponding sides of triangles are proportional.
We use the "∼" symbol to represent the similarity. So, if two triangles are similar, we show it as △QPR ∼ △XYZ
Similar Triangles Examples
Similar triangles are triangles for which the corresponding angle pairs are equal. That means equiangular triangles are similar. Therefore, all equilateral triangles are examples of similar triangles. The following image shows similar triangles, but we must notice that their sizes are different.
Similar Triangles Formulas
In the previous section, we saw there are two conditions using which we can verify if the given set of triangles are similar or not. These conditions state that two triangles can be said similar if either their corresponding angles are equal or congruent or if their corresponding sides are in proportion. Therefore, two triangles △ABC and △EFG can be proved similar(△ABC ∼ △EFG) using either condition among the following set of similar triangles formulas,
Formula for Similar Triangles in Geometry:
- ∠A = ∠E, ∠B = ∠F and ∠C = ∠G
- AB/EF = BC/FG = AC/EG
Similar Triangles Theorems
We can find out or prove whether two triangles are similar or not using the similarity theorems. We use these similarity criteria when we do not have the measure of all the sides of the triangle or measure of all the angles of the triangle. These similar triangle theorems help us quickly find out whether two triangles are similar or not. There are three major types of similarity rules, as given below,
- AA (or AAA) or Angle-Angle Similarity Theorem
- SAS or Side-Angle-Side Similarity Theorem
- SSS or Side-Side-Side Similarity Theorem
Let us understand these similar triangles theorems with their proofs.
AA (or AAA) or Angle-Angle Similarity Criterion
AA similarity criterion states that if any two angles in a triangle are respectively equal to any two angles of another triangle, then they must be similar triangles. AA similarity rule is easily applied when we only know the measure of the angles and have no idea about the length of the sides of the triangle. In the image given below, if it is known that ∠B = ∠G, and ∠C = ∠F.
And we can say that by the AA similarity criterion, △ABC and △EGF are similar or △ABC ∼ △EGF.
⇒AB/EG = BC/GF = AC/EF and ∠A = ∠E.
Click here to understand AA Similarity Criterion in detail- AA similarity criterion
SAS or Side-Angle-Side Similarity Criterion
According to the SAS similarity theorem, if any two sides of the first triangle are in exact proportion to the two sides of the second triangle along with the angle formed by these two sides of the individual triangles are equal, then they must be similar triangles. This rule is generally applied when we only know the measure of two sides and the angle formed between those two sides in both the triangles respectively.
In the image given below, if it is known that AB/DE = AC/DF, and ∠A = ∠D
And we can say that by the SAS similarity criterion, △ABC and △DEF are similar or △ABC ∼ △DEF.
SSS or Side-Side-Side Similarity Criterion
According to the SSS similarity theorem, two triangles will the similar to each other if the corresponding ratio of all the sides of the two triangles are equal. This criterion is commonly used when we only have the measure of the sides of the triangle and have less information about the angles of the triangle.
In the image given below, if it is known that PQ/ED = PR/EF = QR/DF
And we can say that by the SSS similarity criterion, △PQR and △EDF are similar or △PQR ∼ △EDF.
Similar Triangles Properties
If two triangles are similar or proved similar by any of the above-stated criteria, then they possess few properties of the similar triangles. Properties of similar triangles are given below,
- Similar triangles have the same shape but different sizes.
- In similar triangles, corresponding angles are equal.
- Corresponding sides of similar triangles are in the same ratio.
- The ratio of area of similar triangles is the same as the ratio of the square of any pair of their corresponding sides.
How to Find Similar Triangles?
Two given triangles can be proved as similar triangles using the above-given theorems. We can follow the steps given below to check if the given triangles are similar or not,
- Step 1: Note down the given dimensions of the triangles (corresponding sides or corresponding angles).
- Step 2: Check if these dimensions follow any of the conditions for similar triangles theorems(AA, SSS, SAS).
- Step 3: The given triangles, if satisfy any of the similarity theorems, can be represented using the "∼" to denote similarity.
Let us understand these steps better using an example.
Example: Check if △ABC and △PQR are similar triangles or not using the given data: ∠A = 65°, ∠B = 70º and ∠P = 70°, ∠R = 45°.
Solution:
Using the given measurement of angles, we cannot conclude if the given triangles follow the AA similarity criterion or not. Let us find the measure of the third angle and evaluate.
We know, using angle sum property of a triangle, ∠C in △ABC = 180° - (∠A + ∠B) = 180° - 135° = 45°
Similarly, ∠Q in △PQR = 180° - (∠P + ∠R) = 180° - 115° = 65°
Therefore, we can conclude that in △ABC and △PQR, ∠A = ∠Q, ∠B = ∠P, and ∠C = R
⇒ △ABC ∼ △QPR
Difference Between Similar Triangles and Congruent Triangles
Similarity and congruency are two different properties of triangles. The following table helps in distinguishing similar triangles with congruent triangles:
| Similar Triangles | Congruent Triangles |
| Similar triangles have the same shape but may be different in size. They superimpose each other when magnified or demagnified. | Congruent triangles are the same in shape and size. They superimpose each other in their original shape. |
| They are represented using the symbol is ‘~’. For example, Similar triangles ABC and XYZ will be represented as, △ABC ∼ △XYZ | They are represented using the symbol is ‘≅’. For example, Congruent triangles ABC and XYZ will be represented as, △ABC ≅ △QPR |
| The ratio of all the corresponding sides is equal in similar triangles. This common ratio is also called as 'scale factor' in similar triangles. | The ratio of corresponding sides is equal to 1 for congruent triangles. |
☛Topics Related to Similar Triangles:
Important notes on Similar Triangles:
- The ratio of areas of similar triangles is equal to the square of the ratio of their corresponding sides.
- All congruent triangles are similar, but all similar triangles may not necessarily be congruent.
- Similar triangles are denoted using the ‘~’ symbol.
Examples on Similar Triangles
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Example 1:
Consider two similar triangles, ΔABC and ΔDEF:
AP and DQ are medians in the two triangles respectively. Show that
AP/BC = DQ/EF
Solution:
Since the two triangles are similar, they are equiangular.
This means that,
∠B=∠E
Also,
AB/DE = BC/EF→ (1)
⇒AB/DE = [(BC/2)/(EF/2)] = BP/EQ
Hence, by the SAS similarity criterion,
ΔABP∼ΔDEQ
Thus, the sides of these two triangles will be respectively proportional, and so:
AB/DE = AP/DQ
⇒AP/DQ = BC/EF . . . [From (1)]
⇒AP/BC = DQ/EF
Hence proved.
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Example 2: James is 140 in tall. He is standing 320 in away from a lamp post. His shadow from the light is 80 inches long. How high is the lamp post?
Solution:
Taking △ABD and △ECD, we can see that
∠B = ∠C = 90o, and ∠D = ∠D (common angle), hence by AA criterion △ABD is similar to △ECD.
Therefore,
AB/EC = BD/CD = AD/ED
Putting the given values
AB/140 = (320+80)/80
AB/140 = 5
AB = 700
Answer: The height of the pole is 700 in.
FAQs on Similar Triangles
What is Meant by Similar Triangles in Geometry?
In geometry, similar triangles are the triangles that are the same in shape, but may not be equal in size. All equilateral triangles are examples of similar triangles.
What Symbol Used for Similar Triangles?
Similar triangles can be expressed using the ‘~’'. This symbol means that the given two shapes have the same shape, but not necessarily the same size.
What is Similar Triangles Formula?
The formula used to check if two triangles are similar or not depends on the condition of similarity. For two triangles △PQR and △XYZ , similarity can be proved using either of the following conditions,
- ∠P = ∠X, ∠Q = ∠Y and ∠R = ∠Z
- PQ/XY = QR/YZ = PR/XZ
What are the 3 Similar Triangle Theorems?
The three similarity theorems used to check similarity in triangles are as given below,
- AA (or AAA) or Angle-Angle Similarity Theorem
- SAS or Side-Angle-Side Similarity Theorem
- SSS or Side-Side-Side Similarity Theorem
What are the Properties Similar Triangles?
The important properties of two similar triangles can be given as.
- The shape of two similar triangles is the same but their sizes might be different.
- Corresponding angles are equal in similar triangles.
- In similar triangles, corresponding sides are in the same ratio.
What are the Rules for Similar Triangles?
The rules or conditions used to check if the given set of triangles are similar or not as given as,
- All corresponding angle pairs of triangles should be the same.
- All corresponding sides of triangles are in the same proportion.
How Do You Know if Two Triangles are Similar?
Two triangles are similar if their corresponding angles are equal and their corresponding sides are in the same ratio. Similar triangles are the triangles that look the same but the sizes can be different.
How To Find the Proportion of Similar Triangles?
For similar triangles, the pairs of corresponding sides are in proportion. This relation/proportionality of corresponding sides can be used to find the length of the missing side of a figure, given a similar figure for which the corresponding measurements are known.
How Do You Find Missing Sides of Similar Triangles?
Missing sides of a similar triangle can find out by comparing the ratio of the consecutive corresponding sides of the triangle. We compare the ratios and find the length of the unknown side of the triangle.
Can Two Triangles be Similar and Congruent?
All the congruent triangles are also similar triangles but not all similar triangles are congruent triangles. So, two similar triangles can be congruent but not always. For two similar triangles to be congruent, they must have the same size, same shape, and the same measure of the corresponding angles.
Which Type of Triangles is Always Similar?
Equilateral triangles are always similar. Any two equilateral triangles are always similar irrespective of the length of the sides of the equilateral triangle. Two isosceles right triangles are also always similar.
Can Two Isosceles Triangles be Similar?
Two isosceles triangles can be similar if and only if their corresponding angles are equal and their corresponding sides are in the same ratio. Hence, it is not always true that isosceles triangles are similar.
How Do You Introduce Similar Triangles?
Similar triangles can be introduced as triangles that have the same shape but not necessarily the same size. Similar triangles are the triangles that look the same but the sizes can be different.
How To Find the Ratio of Area of Two Similar Triangles?
If two triangles are similar, then the ratio of the area of both triangles is proportional to the square of the ratio of their corresponding sides.If two similar triangles have two corresponding side lengths as a and b, then the ratio of their areas is a2:b2.