• Home
• /
• Blog
• /
• Similarity of Triangles Criteria – SSS, SAS, AA

# Similarity of Triangles Criteria – SSS, SAS, AA

December 14, 2022

This post is also available in: हिन्दी (Hindi)

In general two objects are similar if they look alike. In layman’s words, objects with the same shape, whether they have the same size or not are usually called similar. In mathematics, it is quite different. In mathematics, two objects are similar when their shapes are the same but their sizes are different.

Let’s understand what is the similarity of triangles and the criteria for similarity of triangles with examples.

Is your child struggling with Maths?
We can help!
Country
• Afghanistan 93
• Albania 355
• Algeria 213
• American Samoa 1-684
• Andorra 376
• Angola 244
• Anguilla 1-264
• Antarctica 672
• Antigua & Barbuda 1-268
• Argentina 54
• Armenia 374
• Aruba 297
• Australia 61
• Austria 43
• Azerbaijan 994
• Bahamas 1-242
• Bahrain 973
• Belarus 375
• Belgium 32
• Belize 501
• Benin 229
• Bermuda 1-441
• Bhutan 975
• Bolivia 591
• Bosnia 387
• Botswana 267
• Bouvet Island 47
• Brazil 55
• British Indian Ocean Territory 246
• British Virgin Islands 1-284
• Brunei 673
• Bulgaria 359
• Burkina Faso 226
• Burundi 257
• Cambodia 855
• Cameroon 237
• Cape Verde 238
• Caribbean Netherlands 599
• Cayman Islands 1-345
• Central African Republic 236
• Chile 56
• China 86
• Christmas Island 61
• Cocos (Keeling) Islands 61
• Colombia 57
• Comoros 269
• Congo - Brazzaville 242
• Congo - Kinshasa 243
• Cook Islands 682
• Costa Rica 506
• Croatia 385
• Cuba 53
• Cyprus 357
• Czech Republic 420
• Denmark 45
• Djibouti 253
• Dominica 1-767
• Egypt 20
• Equatorial Guinea 240
• Eritrea 291
• Estonia 372
• Ethiopia 251
• Falkland Islands 500
• Faroe Islands 298
• Fiji 679
• Finland 358
• France 33
• French Guiana 594
• French Polynesia 689
• French Southern Territories 262
• Gabon 241
• Gambia 220
• Georgia 995
• Germany 49
• Ghana 233
• Gibraltar 350
• Greece 30
• Greenland 299
• Guam 1-671
• Guatemala 502
• Guernsey 44
• Guinea 224
• Guinea-Bissau 245
• Guyana 592
• Haiti 509
• Heard & McDonald Islands 672
• Honduras 504
• Hong Kong 852
• Hungary 36
• Iceland 354
• India 91
• Indonesia 62
• Iran 98
• Iraq 964
• Ireland 353
• Isle of Man 44
• Israel 972
• Italy 39
• Jamaica 1-876
• Japan 81
• Jersey 44
• Jordan 962
• Kazakhstan 7
• Kenya 254
• Kiribati 686
• Kuwait 965
• Kyrgyzstan 996
• Laos 856
• Latvia 371
• Lebanon 961
• Lesotho 266
• Liberia 231
• Libya 218
• Liechtenstein 423
• Lithuania 370
• Luxembourg 352
• Macau 853
• Macedonia 389
• Malawi 265
• Malaysia 60
• Maldives 960
• Mali 223
• Malta 356
• Marshall Islands 692
• Martinique 596
• Mauritania 222
• Mauritius 230
• Mayotte 262
• Mexico 52
• Micronesia 691
• Moldova 373
• Monaco 377
• Mongolia 976
• Montenegro 382
• Montserrat 1-664
• Morocco 212
• Mozambique 258
• Myanmar 95
• Namibia 264
• Nauru 674
• Nepal 977
• Netherlands 31
• New Caledonia 687
• New Zealand 64
• Nicaragua 505
• Niger 227
• Nigeria 234
• Niue 683
• Norfolk Island 672
• North Korea 850
• Northern Mariana Islands 1-670
• Norway 47
• Oman 968
• Pakistan 92
• Palau 680
• Palestine 970
• Panama 507
• Papua New Guinea 675
• Paraguay 595
• Peru 51
• Philippines 63
• Pitcairn Islands 870
• Poland 48
• Portugal 351
• Puerto Rico 1
• Qatar 974
• Romania 40
• Russia 7
• Rwanda 250
• Samoa 685
• San Marino 378
• Saudi Arabia 966
• Senegal 221
• Serbia 381 p
• Seychelles 248
• Sierra Leone 232
• Singapore 65
• Slovakia 421
• Slovenia 386
• Solomon Islands 677
• Somalia 252
• South Africa 27
• South Georgia & South Sandwich Islands 500
• South Korea 82
• South Sudan 211
• Spain 34
• Sri Lanka 94
• Sudan 249
• Suriname 597
• Svalbard & Jan Mayen 47
• Swaziland 268
• Sweden 46
• Switzerland 41
• Syria 963
• Sao Tome and Principe 239
• Taiwan 886
• Tajikistan 992
• Tanzania 255
• Thailand 66
• Timor-Leste 670
• Togo 228
• Tokelau 690
• Tonga 676
• Tunisia 216
• Turkey 90
• Turkmenistan 993
• Turks & Caicos Islands 1-649
• Tuvalu 688
• U.S. Outlying Islands
• U.S. Virgin Islands 1-340
• UK 44
• US 1
• Uganda 256
• Ukraine 380
• United Arab Emirates 971
• Uruguay 598
• Uzbekistan 998
• Vanuatu 678
• Vatican City 39-06
• Venezuela 58
• Vietnam 84
• Wallis & Futuna 681
• Western Sahara 212
• Yemen 967
• Zambia 260
• Zimbabwe 263
• Less Than 6 Years
• 6 To 10 Years
• 11 To 16 Years
• Greater Than 16 Years

## What are Similar Triangles?

Two triangles will be similar if the corresponding angles are equal and the corresponding sides are in the same ratio or proportion. 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.

In the above figure, two triangles $\triangle \text{ABC}$ and $\triangle \text{XYZ}$ are similar. We use the ‘$\sim$’ symbol to represent the similarity. So, if two triangles are similar, we show it as $\triangle \text{ABC} \sim \triangle \text{XYZ}$.

## Criteria for Similarity of Triangles – Similar Triangles Formula

There are two conditions using which we can verify if the given set of triangles is similar or not. According to these conditions, 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 $\triangle \text{ABC}$ and $\triangle \text{XYZ}$ can be proved similar $\left( \triangle \text{ABC} \sim \triangle \text{XYZ} \right)$ using either condition among the following set of similar triangles formulas

• AA (Angle Angle): If any two of the angles of the triangles are equal, then the triangles are said to be similar.
• SAS (Side Angle Side): If two triangles have two pairs of sides in the same ratio and the included angles are also equal, then the triangles are similar.
• SSS (Side Side Side): If two triangles have three pairs of sides in the same ratio, then the triangles are similar.

### 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 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 above figure, if it is known that $\angle \text{A} = \angle \text{D}$, and $\angle \text{B} = \angle \text{E}$.

So we can say that by the AA similarity criterion, $\triangle \text{ABC}$ and $\triangle \text{DEF}$ are similar or $\triangle \text{ABC} \sim \triangle \text{ABC}$.

This means, $\frac{\text{AB}}{\text{DE}} = \frac{\text{BC}}{\text{EF}} = \frac{\text{CA}}{\text{FD}}$ and $\angle \text{C} = \angle \text{F}$.

### SAS or Side-Angle-Side Similarity Criterion

SAS similarity criterion states that 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 above figure if it is known that $\frac{\text{AB}}{\text{DE}} = \frac{\text{CA}}{\text{FD}}$, and $\angle \text{A} = \angle \text{D}$.

And we can say that by the SAS similarity criterion, $\triangle \text{ABC}$ and $\triangle \text{DEF}$ are similar or $\triangle \text{ABC} \sim \triangle \text{DEF}$.

### SSS or Side-Side-Side Similarity Criterion

SSS similarity criterion states that two triangles will the similar to each other if the corresponding ratio of all the sides of the two triangles is 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 above figure if it is known that $\frac{\text{AB}}{\text{DE}} = \frac{\text{BC}}{\text{EF}} = \frac{\text{CA}}{\text{FD}}$, then we can say that by the SSS similarity criterion, $\triangle \text{ABC}$ and $\triangle \text{DEF}$ are similar or $\triangle \text{ABC} \sim \triangle \text{DEF}$.

## Difference Between Similar Triangles and Congruent Triangles

Similarity and congruency are two different properties of triangles. The following are the differences between similar triangles and congruent triangles.

## Key Takeaways

If two triangles are similar or proved similar by any of the above-stated criteria, then they possess few properties of similar triangles. The following are the properties of similar triangles

• 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 the area of similar triangles is the same as the ratio of the square of any pair of their corresponding sides.

## Practice Problems

1. What is the meaning of the similarity of figures in math?
2. What does the similarity of triangles mean?
3. Explain the AA similarity criterion.
4. Explain the SAS similarity criterion.
5. Explain the SSS similarity criterion.

## FAQs

### What is meant by similar triangles in Geometry?

In geometry, similar triangles are triangles that are the same in shape, but may not be equal in size. For example, all equilateral triangles are similar triangles as each triangle has three angles of $60^{\circ}$ each but the length of the sides are different.

### What are the 3 similar triangle Theorems?

The three theorems that are used to prove the similarity of triangles are
a) AA (or AAA) or Angle-Angle Similarity Theorem
b) SAS or Side-Angle-Side Similarity Theorem
c) SSS or Side-Side-Side Similarity Theorem

### What are the properties of similar triangles?

The important properties of two similar triangles are
a) The shape of two similar triangles is the same but their sizes might be different.
b) Corresponding angles are equal in similar triangles.
c) In similar triangles, the corresponding sides are in the same ratio.

## Conclusion

In geometry, similar triangles are triangles that are the same in shape, but may not be equal in size. When two triangles are similar, then their corresponding angles are equal and the corresponding sides are in the same ratio. The three rules that are used to prove the similarity of two triangles are AA, SAS, and SSS similarity rules.