# An Account of Similar and Congruent Triangles

- Apr 18, 2016
- Sudipto Das

Triangles that have the same shapes but with different sizes are known as similar triangles. Two triangles are said to be congruent if both of them have the exact same 3 sides and the exact same 3 angles. Let's go through the various attributes of these triangles in more details.

**Similar Triangles**

*Definition** *

Two triangles are said to be similar if both of them have identical shapes, but both of them can have different sizes.(Note: Both the triangles can be similar even if one of them is rotated. They are also similar if one is placed as a mirror image of another.) Go through the following picture that depicts 2 similar triangles.* *

*Similar Triangles*

*Image courtesy - *https://www.mathplanet.com/Oldsite/media/43915/similar_triangles.jpg* *

The triangles in the above picture have identical shapes but they aren't of the same size. They are similar triangles. The formal written notation through which we can represent their similarity is:

?ABC ~ ?DEF.

*Properties *

- Corresponding angles of similar triangles are congruent (means that the angles should be exactly same). Hence, according to the figure depicted above, ∠ABC = ∠DEF, ∠ACB = ∠DFE, ∠BAC = ∠EDF.
- Corresponding sides of similar triangles should be in identical proportion. According to the above figure,

*How to determine similarities of Triangles*

Each triangle has six forms of measurement values that are three sides and three angles. It's not necessary for you to know each and every values of the three sides and three angles of each triangle among the two triangles to determine the similarity of those two triangles. A few conditions comprising of 3 values each should be enough to determine that fact. These conditions are:

**AAA (Angle, Angle, Angle)**: All 3 pairs of their corresponding angles should be same.**SSS (Side, Side, Side)**in the same proportion: All 3 pairs of corresponding sides should be in the exact same proportion.**SAS (Side, Angle, Side)**: 2 pairs of sides should be in the same proportion, the included angle should also be equal.

If the two triangles satisfy at least one of the above three conditions, then the two triangles are similar triangles.

**Congruent Triangles **

*Definition *

Two triangles are said to be congruent if all of their corresponding sides as well as the interior angles are proved to be congruent. The size of the two triangles as well as their shapes should be same. (Note: If one triangle is the mirror image of another, the two triangles may be deemed congruent.) The following picture depicts 2 congruent triangles.

*Congruent Triangles*

*Image courtesy- *https://upload.wikimedia.org/wikipedia/en/6/65/Congruent_triangles.png* *

The two triangles in the above picture have same corresponding sides as well as same interior angles. The two triangles depicted above are congruent triangles. Mathematically, it can be represented as,

?ABC ≅ ?DEF.* *

An easier way to understand congruency of triangles is to imagine that the two triangles are made of cardboard. If you place one triangle on top of the other and you find that the two triangles fit perfectly, the two triangles can be deemed congruent.

*Properties*

- If 2 triangles are deemed congruent, each of the parts (side/angle) of one triangle should be congruent to the respective corresponding part of another triangle.

- Another point to be noted here is that if it's proved that the two triangles are congruent to one another, the angles as well as the sides of any one of the two triangles can be found from the other one. This point can be easily remembered through an acronym CPCTC that stands for “Corresponding Parts of Congruent Triangles are Congruent”.

*How to determine Congruency of Triangles*

Two triangles can be deemed congruent by the following conditions:

**SSS (Side, Side, Side):**The 3 corresponding sides should have equal length.**SAS (Side, Angle, Side):**A pair of sides that are corresponding to each other along with the included angle should be equal.**ASA (Angle, Side, Angle):**A pair of angles that are corresponding to each other along with the inclusive side should be equal.**AAS (Angle, Angle, Side):**A pair of angles that are corresponding to each other along with a non-inclusive side should be equal.**HL (Hypotenuse and leg of a right-angled triangle):**This is possible only in case of two right-angled triangles. Two right-angled triangles are said to be congruent if their hypotenuses and a leg are equal. You can go through an example here.

Congruency and similarity of triangles are an important part of the geometry curriculum. Students need to understand the underlying concepts clearly to solve certain type of problems based on the same topics. Apart from school lessons, there are after-school online mathematics tutorials that can help the students develop a clear concept of these type of topics. That would not only enhance their critical-thinking abilities but will also enhance their performance in mathematics.

### Sudipto Das

Sudipto writes technical and educational content periodically for wizert.com and backs it up with extensive research and relevant examples. He's an avid reader and a tech enthusiast at the same time with a little bit of “Arsenal Football Club” thrown in as well. He's got a B.Tech in Electronics and Instrumentation.

Follow him on twitter @SudiptoDas1993

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