Sss Sas Asa Aas Worksheet

Similar Triangles (Definition, Proving, & Theorems)

Similarity in mathematics does not mean the same thing that similarity in everyday life does. Similar triangles are triangles with the same shape only different side measurements.

Similar Triangles Definition

Respective Angles
Proportion
Included Angle

Proving Triangles Like
Triangle Similarity Theorems

AA Theorem
SAS Theorem
SSS Theorem

Similar Triangles Definition

Mint chocolate chip ice cream and chocolate chip ice cream are similar, just not the same. This is an everyday use of the discussion "similar," but it not the fashion we utilize information technology in mathematics.

In geometry, two shapes are like if they are the aforementioned shape but dissimilar sizes. Yous could have a square with sides 21 cm and a square with sides 14 cm; they would be similar. An equilateral triangle with sides 21 cm and a foursquare with sides 14 cm would not be similar considering they are different shapes.

Similar triangles are easy to identify because you can apply three theorems specific to triangles. These iii theorems, known as Angle - Bending (AA), Side - Angle - Side (SAS), and Side - Side - Side (SSS), are foolproof methods for determining similarity in triangles.

Angle - Angle (AA)
Side - Bending - Side (SAS)
Side - Side - Side (SSS)

Corresponding Angles

In geometry, correspondence ways that a item part on one polygon relates exactly to a similarly positioned part on another. Fifty-fifty if two triangles are oriented differently from each other, if y'all can rotate them to orient in the same fashion and come across that their angles are alike, you can say those angles correspond.

The iii theorems for similarity in triangles depend upon corresponding parts. You look at one angle of 1 triangle and compare it to the aforementioned-position bending of the other triangle.

$Similar Triangles Corresponding Angles$

Proportion

Similarity is related to proportion. Triangles are easy to evaluate for proportional changes that keep them similar. Their comparative sides are proportional to one some other; their corresponding angles are identical.

You can establish ratios to compare the lengths of the two triangles' sides. If the ratios are congruent, the corresponding sides are like to each other.

Included Bending

The included angle refers to the angle betwixt two pairs of corresponding sides. You cannot compare ii sides of 2 triangles and then leap over to an bending that is not between those two sides.

Proving Triangles Like

Hither are two congruent triangles. To make your life like shooting fish in a barrel, we made them both equilateral triangles.

$Proving Triangles Similar$

$△ F O X$ is compared to $△ H Eastward N$ . Notice that $∠ O$ on $△ F O Ten$ corresponds to $∠ E$ on $△ H Eastward Northward$ . Both $∠ O$ and $∠ E$ are included angles between sides $F O$ and $O X$ on $△ F O X$ , and sides $H E$ and $Eastward N$ on $△ H E Northward$ .

Side $F O$ is coinciding to side $H E$ ; side $O X$ is congruent to side $Due east N$ , and $∠ O$ and $∠ E$ are the included, congruent angles.

The two equilateral triangles are the aforementioned except for their letters. They are the same size, so they are identical triangles. If they both were equilateral triangles but side $Due east N$ was twice equally long as side $H E$ , they would exist like triangles.

Triangle Similarity Theorems

$Triangle Similarity Theorems$

Angle-Angle (AA) Theorem

Angle-Angle (AA) says that two triangles are similar if they take two pairs of corresponding angles that are coinciding. The two triangles could go on to be more than similar; they could be identical. For AA, all you accept to exercise is compare ii pairs of corresponding angles.

Trying Angle-Angle

Here are two scalene triangles $△ J A M$ and $△ O U T$ . We have already marked ii of each triangle'south interior angles with the geometer's shorthand for congruence: the footling slash marks. A unmarried slash for interior $∠ A$ and the aforementioned single slash for interior $∠ U$ mean they are congruent. Notice $∠ Thousand$ is coinciding to $∠ T$ because they each accept two picayune slash marks.

Since $∠ A$ is coinciding to $∠ U$ , and $∠ G$ is congruent to $∠ T$ , nosotros now have two pairs of coinciding angles, so the AA Theorem says the two triangles are similar.

$Triangle Similarity - AA Theorem (Angle Angle)$

Tricks of the Merchandise

Watch for trickery from textbooks, online challenges, and mathematics teachers. Sometimes the triangles are non oriented in the same way when you look at them. You lot may take to rotate one triangle to come across if yous tin observe two pairs of corresponding angles.

Another challenge: two angles are measured and identified on ane triangle, but 2 different angles are measured and identified on the other one.

Because each triangle has merely three interior angles, one each of the identified angles has to be congruent. Past subtracting each triangle'south measured, identified angles from 180°, you can learn the measure out of the missing angle. Then you can compare any 2 corresponding angles for congruence.

Side-Bending-Side (SAS) Theorem

The second theorem requires an verbal order: a side, then the included angle, so the next side. The Side-Angle-Side (SAS) Theorem states if two sides of one triangle are proportional to two respective sides of some other triangle, and their corresponding included angles are congruent, the ii triangles are similar.

Trying Side-Angle-Side

Hither are two triangles, side by side and oriented in the same fashion. $△ R A P$ and $△ East M O$ both accept identified sides measuring 37 inches on $△ R A P$ and 111 inches on $△ East One thousand O$ , and besides sides 17 on $△ R A P$ and 51 inches on $△ East M O$ . Notice that the angle between the identified, measured sides is the same on both triangles: $47 °$ .

$Triangle Similarity - SAS Theorem (Side Angle Side)$

Is the ratio $37 / 111$ the same as the ratio $17 / 51$ ? Yes; the ii ratios are proportional, since they each simplify to $1 / 3$ . With their included bending the same, these ii triangles are similar.

Side-Side-Side (SSS) Theorem

The last theorem is Side-Side-Side, or SSS. This theorem states that if two triangles have proportional sides, they are like. This might seem like a big bound that ignores their angles, merely think about it: the only style to construct a triangle with sides proportional to some other triangle'southward sides is to copy the angles.

Trying Side-Side-Side

Here are ii triangles, $△ F Fifty O$ and $△ H I T$ . Find we have not identified the interior angles. The sides of $△ F L O$ measure 15, twenty and 25 cms in length. The sides of $△ H I T$ measure 30, forty and 50 cms in length.

$Triangle Similarity - SSS Theorem (Side Side Side)$

You need to gear up up ratios of corresponding sides and evaluate them:

$\frac{15}{xxx} = \frac{1}{two}$

$\frac{20}{twoscore} = \frac{ane}{2}$

$\frac{25}{l} = \frac{i}{2}$

They all are the same ratio when simplified. They all are $\frac{1}{ii}$ . So even without knowing the interior angles, we know these two triangles are similar, because their sides are proportional to each other.

Lesson Summary

Now that you lot take studied this lesson, you are able to define and identify similar figures, and you can describe the requirements for triangles to be similar (they must either take 2 coinciding pairs of respective angles, two proportional corresponding sides with the included corresponding angle congruent, or all corresponding sides proportional).

You likewise can apply the iii triangle similarity theorems, known as Bending - Angle (AA), Side - Bending - Side (SAS) or Side - Side - Side (SSS), to determine if two triangles are like.

Next Lesson:

Triangle Congruence Postulates