Similar triangles (Definition, Proving, & Theorems)

Similarity in mathematics does not median the very same thing the similarity in everyday life does. Comparable triangles are triangles v the exact same shape however different side measurements.

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Similar triangle Definition

Mint cacao chip ice cream cream and also chocolate chip ice cream cream space similar, but not the same. This is an daily use of the word "similar," yet it no the method we usage it in mathematics.

In geometry, two shapes are similar if they space the same shape however different sizes. You might have a square with sides 21 cm and also a square v sides 14 cm; they would be similar. An it is intended triangle v sides 21 cm and also a square through sides 14 cm would certainly not be similar because they are various shapes.

Similar triangles are easy to identify because you can use three theorems specific to triangles. These 3 theorems, well-known as Angle - edge (AA), Side - angle - next (SAS), and also Side - side - next (SSS), room foolproof techniques for identify similarity in triangles.

Angle - edge (AA)Side - edge - side (SAS)Side - side - next (SSS)

Corresponding Angles

In geometry, correspondence means that a particular component on one polygon relates precisely to a similarly positioned component on another. Also if two triangles are oriented in different way from each other, if you deserve to rotate them come orient in the same way and see that your angles are alike, you have the right to say those angles correspond.

The three theorems for similarity in triangles depend upon corresponding parts. Friend look in ~ one edge of one triangle and compare it to the same-position angle of the other triangle.



Similarity is related to proportion. Triangles are straightforward to evaluate for proportional alters that keep them similar. Their comparative sides are proportional come one another; their corresponding angles room identical.

You can create ratios to compare the lengths that the two triangles" sides. If the ratios are congruent, the equivalent sides are comparable to every other.

Included Angle

The included angle describes the angle in between two bag of corresponding sides. You can not compare 2 sides of 2 triangles and also then leap over to one angle the is not between those two sides.

Proving triangle Similar

Here are two congruent triangles. Come make your life easy, us made them both equilateral triangles.


△FOX is compared to △HEN. An alert that ∠O on △FOX coincides to ∠E top top △HEN. Both ∠O and ∠E space included angles in between sides FO and also OX ~ above △FOX, and sides HE and EN on △HEN.

Side FO is congruent to next HE; next OX is congruent to side EN, and also ∠O and also ∠E are the included, congruent angles.

The 2 equilateral triangles are the same except for their letters. They room the very same size, so they room identical triangles. If they both were equilateral triangles however side EN was twice as long as side HE, they would certainly be similar triangles.

Triangle Similarity Theorems


Angle-Angle (AA) Theorem

Angle-Angle (AA) says that 2 triangles are similar if they have actually two bag of matching angles that room congruent. The 2 triangles can go on to it is in more 보다 similar; they might be identical. For AA, all you have to do is compare 2 pairs of corresponding angles.

Trying Angle-Angle

Here are two scalene triangle △JAM and △OUT. We have actually already significant two of each triangle"s interior angles v the geometer"s shorthand because that congruence: the small slash marks. A single slash for interior ∠A and the same single slash for interior ∠U median they room congruent. Notice ∠M is congruent come ∠T due to the fact that they each have two tiny slash marks.

Since ∠A is congruent come ∠U, and ∠M is congruent to ∠T, we now have actually two pairs of congruent angles, so the AA Theorem claims the 2 triangles are similar.


Tricks that the Trade

Watch because that trickery native textbooks, digital challenges, and mathematics teachers. Occasionally the triangles room not oriented in the same method when girlfriend look at them. Girlfriend may have to rotate one triangle to see if friend can uncover two pairs of corresponding angles.

Another challenge: 2 angles room measured and also identified top top one triangle, yet two different angles room measured and also identified on the various other one.

Because every triangle has actually only three interior angles, one each of the identified angles needs to be congruent. By subtracting each triangle"s measured, established angles from 180°, you deserve to learn the measure up of the missing angle. Then you can compare any two matching angles because that congruence.

Side-Angle-Side (SAS) Theorem

The second theorem requires an accurate order: a side, then the consisted of angle, climate the following side. The Side-Angle-Side (SAS) Theorem claims if 2 sides that one triangle are proportional come two equivalent sides of another triangle, and also their corresponding consisted of angles space congruent, the two triangles space similar.

Trying Side-Angle-Side

Here room two triangles, side by side and also oriented in the very same way. △RAP and also △EMO both have established sides measure 37 inch on △RAP and also 111 customs on △EMO, and also sides 17 top top △RAP and 51 customs on △EMO. Notification that the angle between the identified, measured political parties is the same on both triangles: 47°.


Is the ratio 37/111 the exact same as the ratio 17/51? Yes; the 2 ratios room proportional, since they each leveling to 1/3. V their had angle the same, these two triangles are similar.

Side-Side-Side (SSS) Theorem

The last theorem is Side-Side-Side, or SSS. This theorem claims that if 2 triangles have actually proportional sides, they are similar. This might seem prefer a large leap that ignores their angles, however think about it: the only method to construct a triangle v sides proportional to an additional triangle"s sides is come copy the angles.

Trying Side-Side-Side

Here room two triangles, △FLO and △HIT. Notification we have actually not established the interior angles. The sides of △FLO measure up 15, 20 and also 25 cms in length. The sides of △HIT measure 30, 40 and 50 cms in length.


You require to set up ratios of equivalent sides and also evaluate them:

1530 = 12

2040 = 12

2550 = 12

They all space the very same ratio as soon as simplified. Lock all room 12. So even without understanding the inner angles, we recognize these 2 triangles space similar, because their sides are proportional to each other.

Lesson Summary

Now the you have actually studied this lesson, you are able to define and also identify similar figures, and you can explain the demands for triangle to be similar (they have to either have two congruent bag of equivalent angles, 2 proportional equivalent sides v the included corresponding edge congruent, or all matching sides proportional).

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You additionally can apply the three triangle similarity theorems, well-known as angle - angle (AA), next - edge - side (SAS) or side - next - next (SSS), to determine if 2 triangles room similar.

Next Lesson:

Triangle Congruence Postulates

What friend learned:

After studying this lesson and the video, girlfriend learned to:

Define and identify similar figures, consisting of trianglesExplain and apply 3 triangle similarity theorems, known as angle - edge (AA), side - angle - next (SAS) or side - next - next (SSS)Apply the three theorems to determine if 2 triangles being compared are similar