Last Updated on April 20, 2023
Triangles are a favorite topic of SAT test question writers. It should be no surprise, then, to encounter several triangle questions on the test, as triangles can be used to test such concepts as area, perimeter, Pythagorean theorem, and many Trigonometry concepts. In this article, we’ll discuss what it means for two triangles to be similar, including the properties of such triangles and some of the ways these triangles most commonly appear on the SAT.
Here are the topics we’ll cover:
- What Are Similar Triangles?
- Recognizing Similar Triangles
- Similar Triangles Examples
- SAT Similar Triangles Questions
- In Conclusion
- What’s Next?
Let’s start by learning what similar triangles are.
What Are Similar Triangles?
Two triangles are called similar if they have the same shape but not necessarily the same size. In particular, triangles are similar when their corresponding angles are of equal measure. As an example, take a look at the following pair of similar triangles.
Note that, while one triangle is larger than the other, the two triangles have the same shape overall. Also note that all three angles of the first triangle have the same measures as the corresponding angles of the other triangle.
If two similar triangles also have corresponding sides with equal lengths, then they are called congruent triangles.
KEY FACT:
Similar triangles have three corresponding angles of equal measure.
Now let’s discuss how to determine whether two triangles are similar.
Recognizing Similar Triangles
Two triangles are similar if any of the following statements are true:
- Two pairs of corresponding angles are congruent. If the measures of two corresponding angles are equal, then the corresponding third angles must also be equal.
- All three corresponding sides, when paired, are proportional. If this is the case, then the three angles will also be equal in pairs.
- Two pairs of corresponding side lengths are proportional AND the corresponding angles between those sides have the same measure.
KEY FACT:
If any one of the following is true for two triangles, then the triangles are similar: (1) Of the three corresponding angles, two pairs are equal; (2) All three corresponding sides, when paired, are proportional; (3) two corresponding side lengths are proportional and the corresponding angles between those sides have the same measure.
Let’s now review some common configurations of similar triangles that you can expect to see on the SAT.
Similar Triangles Examples
Example 1
In the figure above, the two vertical angles, ACB and ECD, must be equal. If we can determine that any of the other pairs of corresponding angles are equal, we can state that triangles DCE and BAC are similar triangles.
For example, if angle BAC is equal in measure to angle CDE, we know that angles ABC and DEC must be equal as well, so the two triangles are similar.
Example 2
Triangles AED and ABC are similar if side ED is parallel to side BC. If that is the case, corresponding angles AED and ABC are equal, and corresponding angles ADE and ACB are equal.
Since we would then have two pairs of congruent angles, triangles BAC and EAD would be similar.(Note that triangles BAC and EAD both contain the apex angle A, which makes the third pair of equal angles.)
Example 3
In the above example, we have three similar triangles: ABC, ABD, and ACD. These triangles are similar because each has a right angle, and the two smaller triangles each share a common angle with the larger triangle ABC.
Angle B is common to both triangle ABC and triangle ADB, and Angle C is common to both triangle ABC and triangle ACD. Because each triangle shares two equal angles, the three triangles are similar.
Since this may be difficult to visualize, we can label the triangles by letting x = angle C. Then, we can fill in the rest of the angle measures of all of the triangles.
Thus, the simplified angles are as follows:
We can see that each triangle has the angles x, 90, and 90 – x. Thus, the three triangles are similar.
TTP PRO TIP:
Know the configurations of similar triangles that are most often tested on the SAT.
Let’s now practice with three sample questions.
SAT Similar Triangles Questions
Question 1
Triangles BAC and EDF are shown above. What is the length of side DE?
- 5
- 6
- 7
- 8
Solution:
We are given two triangles in the diagram, each with two given angle measures.
In triangle BAC, two of the angles are 60 and 56. Thus, since a triangle has a total of 180 degrees, the third angle must be 180 – 56 – 60 = 64 degrees.
In triangle EDF, two of the angles are 64 and 60. Thus, the third angle must be 180 – 60 – 64 = 56 degrees.
We can now see that both triangles contain angles measuring 56, 60, and 64 degrees. Since the two triangles have the same angle measures, they are similar. Additionally, the corresponding sides of each triangle must be proportional.
So, let’s look at two corresponding sides to determine the ratio between their lengths. Side AB, with a length of 3, is opposite the 56-degree angle, and side DF, with a length of 6, is opposite the 56-degree angle in the other triangle. Thus, the ratio of a side of triangle BAC to a corresponding side of triangle EDF is 3/6, which simplifies to 1/2. In other words, the corresponding sides of triangle EDF are double the sides of triangle BAC. Since side BC, with a length of 4, is opposite the 60-degree angle, side DE, which is also opposite a 60-degree angle, must be twice the length of BC, or 8.
Answer: D
Question 2
In the figure above, line segment DE is parallel to line segment BC. What is the length of side DE?
- 3
- 6
- 9
- 12
Solution:
Since side DE is parallel to side BC, we can determine that triangles DAE and BAC have angles with the same three measures. Specifically, angle ADE = angle ABC, and angle AED = angle ACB. Also, both triangles share angle DAE. Thus, triangles BAC and DAE are similar.
Since we know that the two triangles are similar, we next need to determine the ratio of the corresponding pairs of sides. Sides AD and AB are opposite congruent angles, so let’s make a ratio of their lengths. Note that, while we’re not given the length of side AB, we can find it by adding the lengths of sides AD and DB. So, the length of side AB is 5 + 10 = 15.
Therefore, we know that the ratio of corresponding sides of triangles DAE and BAC is 5/15 = 1/3. Thus, each side of the smaller triangle is 1/3 the length of the corresponding side of the larger triangle. Since side BC has a length of 18 and is opposite angle BAEC, side DE, which is opposite angle DAE, has a length of 18 ⨉ 1/3 = 6.
Answer: B
Question 3
Triangles CAB and FDE are shown above. The lengths of their sides are also given. Which of the following is equal to the ratio of e to g?
- w/y
- x/w
- y/w
- x/y
Solution:
First, we should note that since triangles CAB and FDE have the same angle measures of 28, 64, and 90 degrees, the triangles are similar. Thus, the ratio of the lengths of two sides in triangle CAB must equal the ratio of the lengths of the corresponding sides in triangle FDE.
Now recognize that side AB in triangle CAB corresponds to side DF in triangle FDE, as they are both opposite 64-degree angles. Similarly, side AC in triangle CAB corresponds to side FE in triangle FDE, as they are both opposite 90-degree angles.
Thus, we can say that AB/AC = DF/FE.
Since AB = e, AC = g, DF = w, and FE = y, we can say that e/g = w/y.
Answer: A
In Conclusion
Similar triangles might not have the same size, but they definitely have the same shape. One of the most frequent questions that students ask about similar triangles is this: What are the 3 ways to prove triangles are similar?”
Three different relationships between triangles will tell us that they are similar:
- Two pairs of corresponding angles are congruent.
- Three pairs of corresponding sides are proportional.
- Two pairs of corresponding side lengths are proportional AND the corresponding angles between those sides have the same measure.
SAT questions about solving similar triangles will most often ask us to determine the length of a particular side or to state the ratio of the lengths of the sides of two similar triangles.
What’s Next?
Knowledge of similar triangles is only one of the many topics you need to learn in order to score well on the SAT. If you would like to learn more about getting a great SAT Math score, check out this blog. And if you want a great SAT preparation experience, try our TTP SAT Course.
