What are the 3 shortcut methods to prove triangles are similar?

Similar Triangles (Definition, Proving, & Theorems)

Similarity in mathematics does non mean the aforementioned thing that similarity in everyday life does. Similar triangles are triangles with the same shape but different side measurements.

1. Similar Triangles Definition
• Corresponding Angles
• Proportion
• Included Angle
2. Proving Triangles Similar
3. Triangle Similarity Theorems
• AA Theorem
• SAS Theorem
• SSS Theorem

Similar Triangles Definition

Mint chocolate chip ice cream and chocolate bit water ice foam are like, but not the same. This is an everyday use of the word "like," just it not the manner we use information technology in mathematics.

In geometry, two shapes are similar if they are the same shape but unlike sizes. You could take 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 square with sides xiv cm would not be like because they are different shapes.

Similar triangles are easy to identify because you can apply 3 theorems specific to triangles. These 3 theorems, known as Angle - Bending (AA), Side - Angle - Side (SAS), and Side - Side - Side (SSS), are foolproof methods for determining similarity in triangles.

1. Angle - Angle (AA)
2. Side - Angle - Side (SAS)
3. Side - Side - Side (SSS)

Respective Angles

In geometry, correspondence means that a particular part on one polygon relates exactly to a similarly positioned function on another. Even if two triangles are oriented differently from each other, if y'all tin can rotate them to orient in the same way and meet that their angles are alike, yous can say those angles correspond.

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

Proportion

Similarity is related to proportion. Triangles are easy to evaluate for proportional changes that keep them similar. Their comparative sides are proportional to 1 another; their corresponding angles are identical.

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

Included Angle

The included angle refers to the bending between 2 pairs of respective sides. Yous cannot compare two sides of two triangles then leap over to an angle that is not between those two sides.

Proving Triangles Like

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

$△FOX$ is compared to $△H\mathrm{Eastward}N$. Detect that $\angle O$ on $△FOX$ corresponds to $\angle E$ on $△H\mathrm{Due\; east}\mathrm{North}$. Both $\angle O$ and $\angle E$ are included angles between sides $FO$ and $OX$ on $△FOX$, and sides $HE$ and $E\mathrm{Due\; north}$ on $△HE\mathrm{North}$.

Side $FO$ is congruent to side $HE$; side $OX$ is congruent to side $\mathrm{East}N$, and $\angle O$ and $\angle E$ are the included, congruent angles.

The ii equilateral triangles are the aforementioned except for their letters. They are the same size, and then they are identical triangles. If they both were equilateral triangles but side $EN$ was twice as long as side $HE$, they would be similar triangles.

Triangle Similarity Theorems

Angle-Angle (AA) Theorem

Bending-Angle (AA) says that two triangles are similar if they accept two pairs of corresponding angles that are congruent. The two triangles could get on to be more than similar; they could be identical. For AA, all you have to do is compare 2 pairs of corresponding angles.

Trying Bending-Angle

Hither are two scalene triangles $△JA\mathrm{Chiliad}$ and $△OUT$. We have already marked ii of each triangle's interior angles with the geometer's autograph for congruence: the little slash marks. A single slash for interior $\angle A$ and the same unmarried slash for interior $\angle U$ mean they are congruent. Notice $\angle M$ is coinciding to $\angle T$ because they each have two little slash marks.

Since $\angle A$ is congruent to $\angle U$, and $\angle M$ is congruent to $\angle T$, we now have two pairs of coinciding angles, so the AA Theorem says the two triangles are similar.

Tricks of the Trade

Scout for trickery from textbooks, online challenges, and mathematics teachers. Sometimes the triangles are not oriented in the same way when you await at them. You may accept to rotate one triangle to see if you can observe two pairs of respective angles.

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

Because each triangle has merely 3 interior angles, i each of the identified angles has to be congruent. Past subtracting each triangle'due south measured, identified angles from 180°, you can learn the measure out of the missing angle. And so you can compare any two corresponding angles for congruence.

Side-Angle-Side (SAS) Theorem

The second theorem requires an verbal society: a side, so the included bending, so the next side. The Side-Angle-Side (SAS) Theorem states if 2 sides of one triangle are proportional to two corresponding sides of some other triangle, and their corresponding included angles are congruent, the 2 triangles are like.

Trying Side-Angle-Side

Here are two triangles, side by side and oriented in the same way. $△RAP$ and $△EMO$ both have identified sides measuring 37 inches on $△RAP$ and 111 inches on $△EMO$, and besides sides 17 on $△RAP$ and 51 inches on $△EMO$. Notice that the angle betwixt the identified, measured sides is the same on both triangles: $47°$.

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

Side-Side-Side (SSS) Theorem

The last theorem is Side-Side-Side, or SSS. This theorem states that if two triangles accept proportional sides, they are similar. This might seem like a large leap that ignores their angles, but remember nearly it: the but way to construct a triangle with sides proportional to another triangle's sides is to copy the angles.

Trying Side-Side-Side

Here are 2 triangles, $△FLO$ and $△HIT$. Discover we have not identified the interior angles. The sides of $△FLO$ measure xv, 20 and 25 cms in length. The sides of $△HIT$ measure 30, twoscore and fifty cms in length.

You need to set up ratios of respective sides and evaluate them:

$\frac{15}{\mathrm{thirty}}=\frac{\mathrm{ane}}{\mathrm{two}}$

$\frac{20}{40}=\frac{1}{\mathrm{ii}}$

$\frac{25}{50}=\frac{1}{2}$

They all are the same ratio when simplified. They all are $\frac{\mathrm{i}}{\mathrm{two}}$. Then 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 accept studied this lesson, you are able to ascertain and identify like figures, and you can describe the requirements for triangles to be like (they must either have two congruent pairs of corresponding angles, ii proportional respective sides with the included corresponding angle congruent, or all corresponding sides proportional).

You lot also can apply the 3 triangle similarity theorems, known equally Bending - Angle (AA), Side - Angle - Side (SAS) or Side - Side - Side (SSS), to make up one's mind if two triangles are like.

Next Lesson:

Triangle Congruence Postulates

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