Similar Triangles - Conditions, Notation, Properties & Examples (2024)

In this tutorial, we will learn all about similar triangles – what similarity of triangles means, how to identify similar triangles, and also how the concept is useful.

Similar Triangles - Meaning

Two triangles are said to be similar if they have the exact same shape. They may or may not have the same size.

Similar Triangles - Conditions, Notation, Properties & Examples (1)

One way to think of similarity is – if one triangle can be turned into another by scaling it up or down (zooming in or out) and adjusting its orientation.

For example, if you flip the green triangle sideways and scale it up by 50% (1.5x), it will overlap perfectly with the yellow one.

How to Identify Similar Triangles

To determine whether two triangles are similar, we have three sets of conditions to check against. If we can prove that the two triangles fulfill any one set of conditions, we can be sure they are similar.

Here are the three criteria.

If you need help with hash marks and bands and terms like corresponding parts or the included angle, I have explained them in the footnotes.

Angle-Angle (AA)

By the AA condition, two triangles are similar if two angles in one triangle are the same as two angles in the other triangle.

Similar Triangles - Conditions, Notation, Properties & Examples (2)

Side-Side-Side (SSS)

By the SSS condition, two triangles are similar if the three sides in one triangle are proportional to the three sides in the other triangle.

Similar Triangles - Conditions, Notation, Properties & Examples (3)

Side-Angle-Side (SAS)

By the SAS condition, two triangles are similar if two sides in one triangle are proportional to two sides in the other triangle and the included angles (between the two sides in each triangle) are equal.

Similar Triangles - Conditions, Notation, Properties & Examples (4)

Similar Triangles - Notation

The symbol for similarity is ~. However, there’s more to it than just the symbol.

When we write the names of the two triangles, corresponding vertices must be in the same order/position.

For example, in the figure below vertices A, B, and C correspond to Q, P, and R respectively (corresponding angles are equal).

Similar Triangles - Conditions, Notation, Properties & Examples (5)

Remember, corresponding vertices have equal angles and sides opposite to corresponding vertices are proportional.

So the correct notation for their similarity would be △ ABC ~ △ QPR.

Properties of Similar Triangles

Here are the three most commonly used properties of similar triangles. We’ll use the triangles below as examples.

Similar Triangles - Conditions, Notation, Properties & Examples (6)

Property 1. Corresponding angles are equal.

m(A)=m(Y)m(B)=m(Z)m(C)=m(X)\begin{align*} m(\angle A) = m(\angle Y) \\[1em] m(\angle B) = m(\angle Z) \\[1em] m(\angle C) = m(\angle X) \end{align*}m(A)=m(Y)m(B)=m(Z)m(C)=m(X)

Property 2. The three pairs of corresponding sides are in proportion.

ABYZ=BCZX=ACYX\frac{AB}{YZ} = \frac{BC}{ZX} = \frac{AC}{YX}YZAB=ZXBC=YXAC

Property 3. The ratio of areas is equal to the square of the ratio of corresponding sides.

Area(ABC)Area(YZX)=(ABYZ)2\frac{\text{Area }(\triangle ABC)}{\text{Area }(\triangle YZX)} = \left ( \frac{AB}{YZ} \right )^2Area(YZX)Area(ABC)=(YZAB)2

How to Solve Similar Triangles

Alright, now that we know the criteria for similarity and the properties of similar triangles, it’s time to solve a couple of examples.

Example

Using the information provided in the figure below, prove that the two triangles are similar. And hence, find the length of PR.

Similar Triangles - Conditions, Notation, Properties & Examples (7)

Solution

To prove two triangles are similar, we need to see which of the three criteria would be the simplest or most convenient to apply.

Here, in △ ABC and △ PQR,

AYBZ\begin{align*} \angle A &\cong \angle Y \\[1em] \angle B &\cong \angle Z \end{align*}ABYZ

So, △ ABC ~ △ RQP by AA similarity. And we have proved the two triangles are similar.

Now, since corresponding sides of similar triangles are in proportion, we have –

RQAB=QPBC=RPAC\frac{RQ}{AB} = \frac{QP}{BC} = \frac{RP}{AC}ABRQ=BCQP=ACRP

Substituting the values from the question,

RQ15=912=RP20\frac{RQ}{15} = \frac{9}{12} = \frac{RP}{20}15RQ=129=20RP

Let’s rewrite this as two equations from which we can find the unknown sides – RQ and RP.

RQ15=912RQ=11.25\begin{align*} \frac{RQ}{15} &= \frac{9}{12} \\[1.3em] RQ &= 11.25 \end{align*}15RQRQ=129=11.25

912=RP20RP=15\begin{align*} \frac{9}{12} &= \frac{RP}{20} \\[1.3em] RP &= 15 \end{align*}129RP=20RP=15

That's it.

Example

Prove that triangles ABC and CDE are similar. Also, if the area of △ ABC is 69.28 in2, what is the area of △ CDE?

Similar Triangles - Conditions, Notation, Properties & Examples (8)

Solution

In this question, we know two sides of each triangle. So that gives us a clue into what we need to do.

Let’s find the ratio between the smaller sides (one from each triangle) and between the two larger sides.

CDBC=1020=2(1)CEAC=1632=2(2)\begin{align*} \frac{CD}{BC} &= \frac{10}{20} = 2 \hspace{0.25cm} \rule[0.1cm]{1cm}{0.1em} \hspace{0.15cm} (1) \\[1.3em] \frac{CE}{AC} &= \frac{16}{32} = 2 \hspace{0.25cm} \rule[0.1cm]{1cm}{0.1em} \hspace{0.15cm} (2) \end{align*}BCCDACCE=2010=2(1)=3216=2(2)

As expected, the two ratios are equal.

CDBC=CEAC\frac{CD}{BC} = \frac{CE}{AC}BCCD=ACCE

This means the two pairs of sides are in proportion. Also, the included angles are equal (they are vertically opposite angles).

ECDBCA\angle ECD \cong \angle BCAECDBCA

So, △ ABC ~ △ EDC by SAS similarity.

Now, we know that the ratio of areas of two similar triangles is equal to the square of the ratios of corresponding sides. So,

Area(EDC)Area(ABC)=(CDBC)2\frac{\text{Area }(\triangle EDC)}{\text{Area }(\triangle ABC)} = \left ( \frac{CD}{BC} \right )^2Area(ABC)Area(EDC)=(BCCD)2

Substituting the values from the question, we have –

Area(EDC)69.28=22Area(EDC)=4×69.28=277.12\begin{align*} \frac{\text{Area }(\triangle EDC)}{69.28} &= 2^2 \\[1.3em] \text{Area }(\triangle EDC) &= 4 \times 69.28 \\[1.3em] &= 277.12 \end{align*}69.28Area(EDC)Area(EDC)=22=4×69.28=277.12

So the area of CDE\hspace{0.2em} \triangle CDE \, \hspace{0.2em}CDE is 277.12in2\hspace{0.2em} \,277.12 \text{ in}^2 \hspace{0.2em}277.12in2.

Similarity and Congruency

Similarity and congruency are two closely related concepts. So let’s see what’s common and what’s different between them.

The idea is simple. Similarity requires two triangles (or any geometric figures) to have exactly the same shape. They may or may not have the same size. Congruency, on the other hand, requires them to have exactly the same shape and size.

So if two triangles are congruent, they must be similar too. But the converse is not true.

And that brings us to the end of this tutorial on similar triangles. Until next time.

Corresponding Parts

Corresponding parts (sides or angles) refer to parts in two triangles that have the same relative position - sort of matching parts.

Here's an example.

In the figure below, the second triangle is a scaled-up (1.5x) and flipped version of the first triangle. Also, if you imagine the transformation, vertex C in the first triangle turned into P in the second, and side BC turned into PR.

Similar Triangles - Conditions, Notation, Properties & Examples (9)

So which side in the second triangle corresponds to AB in the first? And which angle corresponds to B?

Well, AB is opposite C. And C and P are corresponding angles. So QR – the side opposite P – must correspond to AB.

Now, side BC corresponds to side PR. And C (one angle on BC) corresponds to P (one angle on PR). So B (the other angle on BC) and R (the other angle on PQ) must be corresponding parts.

In the figure below, the corresponding sides/angles are in the same color.

Similar Triangles - Conditions, Notation, Properties & Examples (10)

Included Side/Angle

For any two sides in a triangle (or a polygon), the included angle is the one formed between those sides – at their intersection. For example, here B is the included angle between AB and BC.

Similar Triangles - Conditions, Notation, Properties & Examples (11)

Similarly, the included side is the side that’s common to two angles. In the figure below, AC is the included side for angles A and C.

Similar Triangles - Conditions, Notation, Properties & Examples (12)

Hash Marks and Bands

When comparing two triangles or even two sides or angles within a triangle, hash marks and/or bands are used to show which sides or angles are equal. For example –

Similar Triangles - Conditions, Notation, Properties & Examples (13)
Similar Triangles - Conditions, Notation, Properties & Examples (2024)

FAQs

Similar Triangles - Conditions, Notation, Properties & Examples? ›

Similar triangles are triangles that have the same shape, but their sizes may vary. All equilateral triangles, squares of any side lengths are examples of similar objects. In other words, if two triangles are similar, then their corresponding angles are congruent and corresponding sides are in equal proportion.

What are the properties and conditions of similar triangles? ›

Similar triangles are triangles that have the same shape, but their sizes may vary. All equilateral triangles, squares of any side lengths are examples of similar objects. In other words, if two triangles are similar, then their corresponding angles are congruent and corresponding sides are in equal proportion.

What is the notation for 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 are the 3 triangle similarity conditions? ›

The three triangle similarity theorems to prove triangles similar are: Side-Angle-Side, or SAS. Side-Side-Side, or SSS. Angle-Angle, or AA.

What is the rule for similar triangles? ›

The Side-Side-Side (SSS) criterion for similarity of two triangles states that “If in two triangles, sides of one triangle are proportional to (i.e., in the same ratio of ) the sides of the other triangle, then their corresponding angles are equal and hence the two triangles are similar”.

What is the AA rule for similar triangles? ›

What does the AA similarity theorem state? The AA similarity theorem states that if two triangles of one triangle are congruent to two angles of a second triangle, then the two triangles are similar. Thus, corresponding angles in each triangle make the two triangles similar.

What is the formula for similar triangles in geometry? ›

Similar triangle formulas are the formulas that tell us whether two triangles are similar or not. For two triangles △ABC and △XYZ, the similar triangles formula are, ∠A = ∠X, ∠B = ∠Y and ∠C = ∠Z. AB/XY = BC/YZ = CA/ZX.

What are 4 characteristics of similar triangles? ›

Properties of Similar Triangle
  • They have a similar form but not the same size.
  • Each corresponding angle pair is equal.
  • Any pair of corresponding sides has the same ratio.
May 3, 2023

What is the symbol of congruence? ›

Notation. A symbol commonly used for congruence is an equals symbol with a tilde above it, ≅, corresponding to the Unicode character 'approximately equal to' (U+2245). In the UK, the three-bar equal sign ≡ (U+2261) is sometimes used.

What are the four rules for congruent triangles? ›

Congruent triangles
  • The three sides are equal (SSS: side, side, side)
  • Two angles are the same and a corresponding. side is the same (ASA: angle, side, angle)
  • Two sides are equal and the angle between the two sides is equal (SAS: side, angle, side)
  • A right angle, the hypotenuse.

What is the vocabulary of similar triangles? ›

Identifying Similar Triangles Vocabulary

Congruent Angles: Two angles are congruent if they have the same measure. Similar Triangles: Two triangles are similar if each corresponding angle is congruent and there is a common ratio between every pair of similar side lengths.

What are the 3 theorems that prove triangles are similar? ›

These three theorems, known as Angle-Angle (AA), Side-Angle-Side (SAS), and Side-Side-Side (SSS), are foolproof methods for determining similarity in triangles.

What are the properties of similar right triangles? ›

Similar Right triangles: Two right triangles are similar if the corresponding sides are proportional to each other, and the corresponding angles are congruent.

What are the conditions for congruence of triangles? ›

Congruent triangles
  • The three sides are equal (SSS: side, side, side)
  • Two angles are the same and a corresponding. side is the same (ASA: angle, side, angle)
  • Two sides are equal and the angle between the two sides is equal (SAS: side, angle, side)
  • A right angle, the hypotenuse.

What are the factors of similar triangles? ›

Scale Factor of Triangle

The triangles which are similar have same shape and measure of three angles are also same. The only thing which varies is their sides. However, the ratio of the sides of one triangle is equivalent to the ratio of sides of another triangle, which is called here the scale factor.

Top Articles
Latest Posts
Recommended Articles
Article information

Author: Melvina Ondricka

Last Updated:

Views: 6332

Rating: 4.8 / 5 (48 voted)

Reviews: 95% of readers found this page helpful

Author information

Name: Melvina Ondricka

Birthday: 2000-12-23

Address: Suite 382 139 Shaniqua Locks, Paulaborough, UT 90498

Phone: +636383657021

Job: Dynamic Government Specialist

Hobby: Kite flying, Watching movies, Knitting, Model building, Reading, Wood carving, Paintball

Introduction: My name is Melvina Ondricka, I am a helpful, fancy, friendly, innocent, outstanding, courageous, thoughtful person who loves writing and wants to share my knowledge and understanding with you.