Similar Triangles are triangles with the same shape but can have variable sizes. Similar triangles have corresponding sides in proportion to each other and corresponding angles equal to each other. Similar triangles are different from congruent triangles. Two congruent figures are always similar, but two similar figures need not be congruent.
Two triangles are considered similar when their corresponding angles match and their sides are proportional. This means that similar triangles have the same shape, although their sizes may differ. On the other hand, triangles are defined as congruent when they not only share the same shape but also have corresponding sides that are identical in length.
Now, let's learn more about similar triangles and their properties with solved examples and others in detail in this article.
What are Similar Triangles?
Similar triangles are triangles that look similar to each other, but their sizes might be different. Similar objects are of the same shape but different sizes. This implies similar shapes, when magnified or demagnified, should superimpose over each other. This property of similar shapes is known as "Similarity".
There are three similar triangle theorems:
- AA (or AAA) or Angle-Angle Similarity Theorem
- SAS or Side-Angle-Side Similarity Theorem
- SSS or Side-Side-Side Similarity Theorem
Similar Triangles Definition
Two triangles are called similar triangles if their corresponding angles are equal and the corresponding sides are in the same proportion. The corresponding angles of two similar triangles must be equal. Similar triangles can have different respective lengths of the sides of the triangle, but the ratio of lengths of corresponding sides must be the same.
When two triangles are similar it implies that:
- All pairs of corresponding angles in the triangles are equal.
- All pairs of corresponding sides of the triangle are proportional.
The symbol "∼" is used to represent the similarity between similar triangles. So, when two triangles are similar, we write it as △ABC ∼ △DEF.
Similar Triangles Examples
Various examples of the similar triangles are:
- If we take two triangles that have sides in the ratio then they are the similar triangles.
- The Flagpoles and their Shadows represent similar triangles.
The triangles shown in the image below are similar and we represent them as, △ABC ∼ △PQR.
Basic Proportionality Theorem (Thales Theorem)
Basic Proportionality Theorem, also known as Thales' Theorem, is a fundamental concept in geometry that relates to the similarity of triangles. It states that if a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally. In simpler terms, if a line parallel to one side of a triangle intersects the other two sides, it divides those sides proportionally.
Mathematically, if a line DE is drawn parallel to one side of triangle ABC, intersecting sides AB and AC at points D and E respectively, then according to the Basic Proportionality Theorem:
BD/DA = CE/EA
This theorem is a consequence of the similarity of triangles formed by the parallel line and the sides of the original triangle. Specifically, triangles ADE and ABC, as well as triangles ADC and AEB, are similar due to corresponding angles being equal. Consequently, the ratios of corresponding sides in similar triangles are equal, leading to the proportionality relationship described by the Basic Proportionality Theorem.
Basic Proportionality Theorem is widely used in geometry and trigonometry to solve various problems involving parallel lines and triangles. It serves as a foundational principle for understanding the properties of similar triangles and the relationships between their corresponding sides and angles. Additionally, it forms the basis for more advanced concepts in geometry, such as the Parallel Lines Theorem and applications in various geometric constructions and proofs.
Similar Triangles Criteria
If two triangles are similar they must meet one of the following rules,
- Two pairs of corresponding angles are equal. (AA Rule)
- Three pairs of corresponding sides are proportional. (SSS Rule)
- Two pairs of corresponding sides are proportional and the corresponding angles between them are equal. (SAS Rule)
Read in Detail: Criteria for Similar Triangles
In the last section, we studied two conditions using which we can verify whether the given triangles are similar or not. The conditions are when two triangles are similar; their corresponding angles are equal, or the corresponding sides are in proportion. Using either condition, we can prove △PQR and △XYZ are similar from the following set of similar triangle formulas.
In △PQR and △XYZ if,
- ∠P = ∠X , ∠Q = ∠Y, ∠R = ∠Z
- PQ/XY = QR/YZ = RP/ZX
The above two triangles are similar, i.e., △PQR ∼ △XYZ.
Similar Triangle Rules
The similarity theorems help us to find whether the two triangles are similar or not. When we do not have the measure of angles or the sides of the triangles, we use the similarity theorems.
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
Angle-Angle (AA) or AAA Similarity Theorem
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.
Side-Angle-Side or SAS Similarity Theorem
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 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.
Side-Side-Side or SSS Similarity Theorem
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
Similar triangles have various properties which are widely used for solving various geometrical problems. Some of the common properties of similar triangle:
- The shape of similar triangles is fixed but their sizes may be different.
- Corresponding angles of similar triangles are equal.
- Corresponding sides of similar triangles are in common ratios.
- The ratio of the area of similar triangles is equal to the square of the ratio of their corresponding side.
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.
This can be understood better with the help of the following 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°.
Using 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 it.
We know, using the 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
Area of Similar Triangles - Theorem
Similar Triangle Area Theorem states that for two similar triangles ratio of area of the triangles is proportional to the square of the ratio of their corresponding sides. Suppose we are given two similar triangles, ΔABC and ΔPQR then
According to Similar Triangle Theorem:
(Area of ΔABC)/(Area of ΔPQR) = (AB/PQ)2 = (BC/QR)2 = (CA/RP)2
Difference Between Similar Triangles and Congruent Triangles
Similar triangles and congruent triangles are two types of triangles that are widely used in geometry for solving various problems. Each type of triangle has different properties and the basic difference between them is discussed in the table below.
Similar Triangles | Congruent Triangles |
|---|
| Similar triangles are triangles that have equal corresponding angles. | Congruent triangles are triangles that have equal corresponding angles and equal corresponding sides. |
| Similar triangles have the same shape but their sizes may or may not be the same | Congruent triangles have the same size and the same area. |
| Similar triangles are not superimposed images of each other until magnified or demagnified. | Congruent triangles are superimposed images of each other if arrange in the proper orientation. |
| Similar triangles are represented with the ‘~’ symbol. | Congruent triangles are represented with the ‘≅’ symbol. |
| Their corresponding sides are in the ratio. | Their corresponding sides are equal. |
Applications of Similar Triangles
Various applications of the similar triangle that we see in the real life are,
- Shadow and Height of various objects are calculated using the concept of similar triangles.
- Map Scaling uses the concept of the similar triangle.
- Photographic devices uses the similar triangle properties to capture various images.
- Model Making uses the concept of similar triangles.
- Navigation and Trigonometry also uses the similar triangle approach to solve various problems, etc.
Important Notes on Similar Triangles:
- Ratio of areas of similar triangles is equal to square of ratio of their corresponding sides.
- All congruent triangles are similar, but all similar triangles may not necessarily be congruent.
- This ‘~’ symbol is used to denote similar triangles.
Solved Questions on Similar Triangles
Question 1: In the given figure 1, DE || BC. If AD = 2.5 cm, DB = 3 cm, and AE = 3.75 cm. Find AC?
Solution:
In △ABC, DE || BC
AD/DB = AE/EC (By Thales' Theorem)
2.5/3 = 3.75/x, where EC = x cm
(3 × 3.75)/2.5 = 9/2 = 4.5 cm
EC = 4.5 cm
Hence, AC = (AE + EC) = 3.75 + 4.5 = 8.25 cm.
Question 2: In Figure 1 DE || BC. If AD = 1.7 cm, AB = 6.8 cm, and AC = 9 cm. Find AE?
Solution:
Let AE = x cm.
In △ABC, DE || BC
By Thales Theorem we have,
AD/AB = AE/AC
1.7/6.8 = x/9
x = (1.7×9)/6.8 = 2.25 cm
AE = 2.25 cm
Hence AE = 2.25 cm
Question 3: Prove that a line drawn through the midpoint of one side of a triangle (figure 1) parallel to another side bisects the third side.
Solution:
Given a ΔΑΒC in which D is the midpoint of AB and DE || BC, meeting AC at E.
TO PROVE AE = EC.
Proof: Since DE || BC, by Thales' theorem, we have:
AE/AD = EC/DB =1 (AD = DB, given)
AE/EC = 1
AE = EC
Question 4: In the given Figure 2, AD/DB = AE/EC and ∠ADE = ∠ACB. Prove that ABC is an isosceles triangle.
Solution:
We have AD/DB = AE/EC DE || BC [by the converse of Thales' theorem]
∠ADE = ∠ABC (corresponding ∠s)
But, ∠ADE = ∠ACB (given).
Hence, ∠ABC = ∠ACB.
So, AB = AC [sides opposite to equal angles].
Hence, △ABC is an isosceles triangle.
Question 5: If D and E are points on the sides AB and AC respectively of △ABC (figure 2) such that AB = 5.6 cm, AD = 1.4 cm, AC = 7.2 cm, and AE = 1.8 cm, show that DE || BC.
Solution:
Given, AB = 5.6 cm, AD = 1.4 cm, AC = 7.2 cm and AE = 1.8 cm
AD/AB = 1.4/5.6 = 1/4 and AE/AC = 1.8/7.2 = 1/4
AD/AB = AE/AC
Hence, by converse of Thales Theorem, DE || BC.
Question 6: Prove that the line segment joining the midpoints of any two sides of a triangle (figure 2) is parallel to the third side.
Solution:
In △ABC in which D and E are the midpoints of AB and AC respectively.
Since D and E are the midpoints of AB and AC respectively, we have :
AD = DB and AE = EC.
AD/DB = AE/EC (each equal to 1)
Hence, by converse of Thales Theorem, DE || BC
Important Maths Related Links:
Practice Questions Similar Triangles
Q1. In two similar triangle △ABC and △ADE, if DE || BC and AD = 3 cm, AB = 8 cm, and AC = 6 cm. Find AE.
Q2. In two similar triangle △ABC and △PQR, if QR || BC and PQ = 2 cm, AB = 12 cm, and AC = 9 cm. Find PR.
Q3. In two similar triangles ΔABC and ΔAPQ, the length of the sides are given as AP = 9 cm , PB = 12 cm and BC = 24 cm. Find the ratio of the areas of ΔABC and ΔAPQ.
Q4. In two similar triangles ΔABC and ΔAPQ, the length of the sides are given as AP = 3 cm , PB = 4 cm and BC = 8 cm. Find the ratio of the areas of ΔABC and ΔAPQ.
Summary - Similar Triangles
Similar triangles are geometric figures that share the same shape but differ in size, characterized by equal corresponding angles and proportional corresponding sides. Key theorems like Angle-Angle (AA), Side-Angle-Side (SAS), and Side-Side-Side (SSS) establish criteria for triangle similarity.
These principles are foundational in fields such as engineering, computer graphics, and architecture due to their ability to maintain shape integrity under scaling. Thales' Theorem, or the Basic Proportionality Theorem, illustrates how a line parallel to one side of a triangle divides the other two proportionally, further demonstrating the concept of similarity in triangles.
Similar triangles are crucial for practical applications ranging from calculating heights and distances in navigation to optimizing designs in technology and construction, demonstrating their wide-reaching relevance in both academic and real-world contexts.
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