How to Identify Similar Triangles
Similar triangles have the same characteristics as similar figures but can be identified much more easily. Learn the shortcuts for identifying similar triangles here and test your ability with a quiz.
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The Language of Similarity
Similar figures are figures that have the same shape but are different sizes. They have congruent corresponding angles and proportional corresponding sides. Similar triangles are a type of similar figure, and determining their similarity is much easier thanks to the triangle similarity theorems. These theorems, which are Angle - Angle (AA), Side - Angle - Side (SAS) and Side - Side - Side (SSS), make it possible to determine triangle similarity with minimal calculations. Before we go any further, let’s review key terms that will help these theorems make sense.
When parts of a figure are corresponding , this means that they are in the same location in each figure. Sides are proportional when the ratios between the corresponding sides are congruent. So, if you create ratios or fractions comparing all of the corresponding sides, each will have the same value and reduce to the same number.
When discussing congruent or similar figures, the included angle is the angle formed by the congruent or proportional sides. In the triangle below, since side AB is congruent to side DE and side BC is congruent to side EF, then angles B and E are the included angles.
B and E are included angles.
Triangle Similarity Theorems
Okay. Now that we’re refreshed on vocabulary, let’s further examine each of the similarity theorems.
We’ll begin with Angle - Angle (AA). For two triangles to be similar by Angle - Angle (AA), two angles of one triangle are congruent to two angles of another triangle. Look at triangle JKL below. Angle J is 52 degrees and angle K is 60 degrees. If we subtract 52 and 60 from 180 (the total number of degrees that all angles in a triangle must add up to), we will see that angle L is 68 degrees.
You can find one angle by subtracting the sum of the other two angles from 180.
Now, look at triangle MNO below. If angle M is congruent to angle J, and angle N is congruent to angle K, what can we say about angles L and O? We can say that they are congruent as well. With three pairs of congruent angles, the triangles are the same shape but different sizes, meaning that their sides are proportional and the triangles are similar.
These two triangles are similar.
For Side - Angle - Side (SAS), an angle of one triangle must be congruent to the corresponding angle of another triangle, and the lengths of the sides including these angles are in proportion. With congruent included angles, the proportional sides can’t fluctuate, and the third side in both triangles must be a specific length. So, with two sides already proportional, the lengths of the third sides must also be proportional, proving triangle similarity.
The last theorem is Side - Side - Side (SSS), which means that the three sets of corresponding sides of two triangles are in proportion. If the ratios of all corresponding sides are equal, then the sides are similar and so are the triangles.
When determining which theorem proves similarity, don’t overthink it; just look at the letters in each theorem. For triangles to be similar by Angle - Angle (AA), the measures of two angles in each triangle will be provided. If similar by Side - Angle - Side (SAS), then you will have the measures of two sides and the included angles of both triangles. For Side - Side - Side (SSS), you will have all three side lengths for both triangles. Let’s practice.
Are They Similar?
Is triangle ABC similar to triangle DEF?
Triangles for example 1
Let’s examine the given information. We have the lengths for two sides in both triangles and the measures of the included angles. This sounds like Side - Angle - Side (SAS). But, before concluding similarity by this theorem, we must check for congruent angles and proportional sides.
Angle B and angle E both measure 63 degrees, so the included angles are congruent. To set up the side proportions, always compare the two smallest sides together and the two largest sides together, going in the same order between triangles. For this example, our ratios are 3/6 and 5/8. If we convert to decimals, 3/6 = .5 and 5/8 = .625. Since these ratios are not equal, triangle ABC is not similar to triangle DEF.
For our next example, determine if triangle RST is similar to triangle WXY.
Triangles for example 2
Since we were given the lengths of all three sides in both triangles, Side - Side - Side (SSS) is the only theorem that can prove similarity. Let’s set up our proportions. The two smaller sides have a ratio of 6/3, the largest sides have a ratio of 10/5 and the remaining sides have a ratio of 8/4. From simplifying, all of the ratios equal two. Therefore, triangle RST is similar to triangle WXY by the Side - Side - Side (SSS) similarity theorem.
Let’s do one more. Is triangle CRE similar to triangle PHB?
Triangles for example 3
Based on the given information, these triangles can only be congruent by Angle - Angle (AA). But, it appears that only one pair of angles are congruent. So, you may think the triangles aren’t similar. Before coming to a conclusion, let’s calculate the measure of angle E. Subtracting 180 - 40 - 95, we find that angle E measures 45 degrees. With this information, we now see that angle R and angle H are congruent, as well as angle E and angle B. Therefore, triangle CRE is similar to triangle PHB by the Angle - Angle (AA) similarity theorem.
Similar triangles possess the same characteristics as other similar figures: congruent corresponding angles and proportional corresponding sides. The triangle similarity theorems, which are Angle - Angle (AA), Side - Angle - Side (SAS) and Side - Side - Side (SSS), serve as shortcuts for identifying similar triangles. When assessing similarity, always begin by examining the provided information, which will help you figure out which theorem to use when determining if triangles are similar or not.
After completing this lesson, you’ll be able to:
Define similar figures
Describe the characteristics required for triangles to be similar
Explain the three triangle similarity theorems
Determine whether two triangles are similar using Angle - Angle, Side - Angle - Side or Side - Side - Side theorems
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