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Last Updated on April 27, 2023
There are many ways that triangles can be tested on the GRE. For example, you may have to find the area or perimeter of a triangle or use the Pythagorean theorem to find the length of a side. In this article, we’ll discuss similar triangles, which are presented in some tricky GRE questions. Our discussion will include the properties of similar triangles and some of the most common ways these triangles appear on the GRE.
Let’s start by learning what similar triangles are.
- What Are Similar Triangles?
- Properties of Similar Triangles
- Similar Triangles Examples
- Similar Triangles Problems
What Are Similar Triangles?
Similar triangles are triangles that have the same shape but not necessarily the same size. This occurs when their corresponding angles are of equal measure. Similar triangles that have both the same shape and size are referred to as congruent triangles.
KEY FACT:
Similar triangles have the same shape but not necessarily the same size.
Now let’s discuss the properties of similar triangles.
Properties of Similar Triangles
While we may feel that we can “eyeball” triangles to tell whether they are similar, this technique is not recommended, since many geometric figures on the GRE are not drawn to scale. Let’s instead refer to the following criteria that will help us to decide whether triangles are in fact similar. We will know two triangles are similar if any of the following are true:
- Two pairs of corresponding angles are congruent. If two angles are equal, the third angle must also be equal.
- Three pairs of corresponding sides are proportional. If all three sides are proportional in pairs, we know that all three angles must be equal in pairs as well.
- Two pairs of corresponding side lengths are proportional AND the angle between those sides is the same.
KEY FACT:
Triangles are similar if: (1) Two pairs of corresponding angles are equal; or (2) Three pairs of corresponding sides are equal; or (3) Two corresponding side lengths are equal and the angle between those sides is the same.
Let’s now review some common types of similar triangles that you can expect to see on the GRE.
Similar Triangles Examples
Similar Triangles – Example 1:
Looking at the figure above, we can determine that the two vertical angles, ACB and ECD, must be equal. As long as we know that any of the other pairs of corresponding angles are equal, we can determine that we have similar triangles. For instance, if we are given that angle BAC is equal in measure to angle CDE, then we know that angles ABC and DEC must be equal as well, and so the two triangles are similar.
Similar Triangles – Example 2:
Triangles AED and ABC are similar if side ED is parallel to side BC. With that information, we know that corresponding angles AED and ABC are equal and that corresponding angles ADE and ACB are equal. Since we have two pairs of congruent angles, we have similar triangles. (Note that angles BAC and EAD are in fact the same angle, and so of course they are also equal.)
Similar Triangles – Example 3:
In the above example, we have three similar triangles: ABC, ABD, and ACD. These three triangles are similar because each of them has a right angle and each of the two smaller triangles shares 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. Once again, because we see that each triangle shares two equal angles, we know that the three triangles are similar. If this is difficult to see, we can illustrate this by letting x = angle C, and then we can fill in the rest of the angles of the triangles.
Thus, the simplified angles are as follows:
We can see that each triangle has the angles x, 90, and 90 – x.
TTP PRO TIP:
Memorize the diagrams in which similar triangles are presented on the GRE.
Let’s now practice with two sample questions.
Similar Triangles Problems
Question 1
The measures of some interior angles and sides of the triangles ABC and DEF are indicated in the figure above. What is the length of side DE?
- 5
- 6
- 7
- 8
- 9
Solution:
Notice that angle ABC is 180 – (60 + 56) = 64 degrees. Similarly, angle DEF is 180 – (60 + 64) = 56 degrees. We see that both triangles have interior angles of 56°, 60°, and 64°. Recall that when all three interior angles of a triangle are the same as the interior angles of another triangle, those two triangles are similar. Recall also that sides of similar triangles are proportional.
In triangle ABC, the side opposite the 56° angle has a length of 3. In triangle DEF, the side opposite the 56° angle has a length of 6. Since one side of triangle DEF is twice the length of the corresponding side of triangle ABC, and since the two triangles are similar, it follows that all sides of triangle DEF are twice the length of the corresponding sides of triangle ABC. Notice that sides BC and DE are both opposite 60° angles, and so these sides are corresponding. It follows that the length of DE must be twice the length of BC. Thus, the length of side DE is 2 ⨉ 4 = 8.
Answer: D
Question 2
In the figure above, line segment DE is parallel to line segment BC, the length of AD is 5, the length of DB is 10, and the length of BC is 18.
Quantity A
The length of line segment DE
Quantity B
6
Solution:
Since DE is parallel to BC, it follows that angle ADE is congruent to angle ABC, and angle AED is congruent to angle ACB. Thus, triangle ADE and triangle ABC are similar. Notice that side AD of triangle ADE corresponds to side AB of triangle ABC, and side DE of triangle ADE corresponds to side BC of triangle ABC. Recall that the ratios of corresponding sides of two similar triangles are all equal. Therefore, the following must be true:
Length of side AD/Length of side AB = Length of side DE/Length of side BC
Note that side AB is the combination of segments AD and DB, and so the length of AB is 5 + 10 = 15. We can now let the length of side DE = n, and we have the following:
5/15 = n/18
Cross-multiplying, we have the following:
15n = 18 × 5
15n = 90
n = 6
Length of side DE = 6
We see that quantity A is equal to quantity B.
Answer: C
In this article, we have reviewed only a small portion of the topic of Geometry by convering similar triangles. If you would like to learn more about GRE Geometry questions, you can check out the top-rated Target Test Prep GRE Prep Course.
