Similar triangles are the triangles that have corresponding sides in proportion to each other and corresponding angles equal to each other. Similar triangles are the triangles that look the same but the sizes can be different. 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.
|1.||Introduction to Similar Triangles|
|2.||Similar Triangles Rules|
|3.||Similar Triangles Properties|
|5.||Practice Questions on Similar Triangles|
|6.||Frequently Asked Questions (FAQs)|
Introduction to Similar Triangles
Similar triangles are the triangles that look similar to each other but not exactly the same. 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.
We use the "∼" symbol to represent the similarity. So, if two triangles are similar, we show it as △QPR ∼ △XYZ
The following image shows similar triangles, but we must notice that their sizes are different.
Similar Triangles Rules
We can find out or prove whether two triangles are similar or not using the similarity rules. We use these rules when we do not have the measure of all the sides of the triangle or measure of all the angles of the triangle. These rules help us quickly find out whether two triangles are similar or not. There are three major types of similarity rules, which are explained below.
AA (or AAA) or Angle-Angle Similarity
AA similarity criterion tells us 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 ∼ △EFG.
SAS or Side-Angle-Side Similarity
According to the SAS similarity criterion, 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
According to the SSS similarity criterion, 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/ DE = 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.
Check out some interesting topics related to similar triangles.
Solved Examples on Similar Triangles
A pole of height 2 yards casts a shadow of length 4 yards. A tree casts a shadow of 24 yards.
Find the height of the tree, if it is known that the triangles formed by joining the tip of the tree and the shadow of the tree, are similar to the triangle formed by joining the tip of the pole with the tip of the shadow of the pole.
Here, we can see that △PQR is similar to △ABC
Since the corresponding sides of similar triangles are in the same ratio,
we get, PQ/AB = QR/BC = PR/AC
Putting the values, we get
PQ/2 = 24/4
On solving we get,
PQ = 12
Answer: The height of the tree is 12 yards.
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?
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.
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.
Frequently Asked Questions (FAQs)
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 b different.
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.