05.01 Triangle Congruence and Similarity

The Triangle Proportionality Theorem

In some of the practice questions, you have seen similar triangles contained within one another, instead of sitting side by side. What postulate or theorem makes these triangles similar? Take a look at the images below to see if you can find the answer! Afterward, try to find another characteristic these triangles share that does not involve one of the three similarity postulates or theorems. Three pairs of triangles are shown below. Take a look at the corresponding side lengths of each. Select the "Show Proportion" button to reveal the proportion between corresponding sides. Notice how both triangles share ∠A, which lies between the proportional sides. Therefore, these triangles are similar by the Side-Angle-Side Similarity Postulate. But, there is something else that's special about these triangles. Did you find it? What is significant about the two segments BC and DE? Select the Hint button below for a hint. What you may have noticed. Segments BC and DE are parallel to one another! You should also notice that the segments created by DE on the sides of the triangle are also proportional. Line AD/Line DB = Line AE/Line EC If a segment is parallel to one side of a triangle and intersects the other two sides of the triangle, the segment divides the sides of the triangle proportionally. This is the Triangle Proportionality Theorem!

In this lesson, you will be able to answer the following question: How do you prove each of the following theorems using either a two-column, paragraph, or flow chart proof? Triangle Proportionality Theorem The Converse of the Triangle Proportionality Theorem Pythagorean Theorem, using Similar Triangles The Converse of the Pythagorean Theorem

In this lesson, you will be able to answer the following question: How do you prove each of the following theorems using either a two-column, paragraph, or flow chart proof? Triangle Proportionality Theorem The Converse of the Triangle Proportionality Theorem Pythagorean Theorem, using Similar Triangles The Converse of the Pythagorean Theorem

Proving the Pythagorean Theorem with Similar Triangles

Pieces of Right Triangle Similarity Theorem

Pieces of Right Triangle Similarity Theorem

Pieces of Right Triangle Similarity Theorem As you have seen, all similar triangles have congruent angles and proportional sides. But right triangles with an altitude drawn from the vertex of the right angle have additional properties you should know. Take a look at the images below to see how showing the altitude BD creates several pairs of similar triangles.

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Remember, in order to determine congruence or similarity, you must first identify three congruent corresponding parts. There is one exception, the Angle-Angle (AA) Similarity Postulate, where you only need two angles to prove triangle similarity. The parts identified can be applied to the theorems below. See Attached Image

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Remember, order is important when writing proportions! Keep the measurements for one triangle all in the numerator or all in the denominator. If you have one measurement from one triangle in the numerator and another measurement from that same triangle in the denominator, you won't have a proportion.

Applications of the SAS Similarity Postulate

Side-Angle-Side Similarity Postulate What it says: If two or more triangles have corresponding, congruent angles, and the sides that make up these angles are proportional, then the triangles are similar. What it means: If corresponding angles are equal in two or more triangles, and the sides around that angle have the same simplified fraction, then the triangles are similar. What it looks like: See Attached Image Why it's important: If you only know one angle measure, but know the length of either side, then the Angle-Angle Similarity Postulate won't help you. But this one will!

Side-Side-Side Similarity Postulate

So far, you know of two postulates you can use to determine similarity in triangles. There is another postulate that allows you to determine if two or more triangles are similar, as well. See if you can figure out what it is! Look at triangles ABC and XYZ below. Are they similar to one another? Try setting up some proportions with corresponding sides. Remember that order matters! AB corresponds to XY, BC corresponds to YZ, and AC corresponds to XZ. Simplify those fractions to discover the relationship. Select the "Proportion" button for each image to verify your theory. What you may have noticed. Did you see that all three fractions made up by corresponding sides simplified to the same value? For these two triangles, they are similar according to the scale factor of 2. The second triangle will always be twice the size of the first triangle, which is why the ratio is equal to one half—because the first triangle (which was placed in the numerator of the proportion) is one half the size of the second triangle. The Side-Side-Side Similarity Postulate states that if all corresponding sides of two or more triangles form the same proportion, then the triangles are similar. Don't forget that order is important when writing proportions!

Proving the Triangle Proportionality Theorem

Triangle Proportionality Theorem: If a line is parallel to one side of a triangle and also intersects the other two sides, the line divides the sides proportionally.

Pieces of Right Triangles Similarity Theorem

What it says: If an altitude is drawn from the right angle of a right triangle, the two smaller triangles created are similar to one another and to the larger triangle. What it means: An altitude in a right triangle creates two smaller right triangles. All three of the triangles, the big one and the two smaller ones inside the big one, are similar to one another. What it looks like: See Attached Image Why it's important: With this theorem, you can prove the Pythagorean Theorem using similar triangles.

You have already looked at the Angle-Angle (AA) Similarity Theorem, and you know that dilations can be used to demonstrate similarity. This information can be used to prove additional theorems involving similarity. The game below is an excellent opportunity to explore when the use of each theorem or postulate is appropriate. Select the property, theorem, postulate, or formula that best fits the description given.

You have already looked at the Angle-Angle (AA) Similarity Theorem, and you know that dilations can be used to demonstrate similarity. This information can be used to prove additional theorems involving similarity. The game below is an excellent opportunity to explore when the use of each theorem or postulate is appropriate. Select the property, theorem, postulate, or formula that best fits the description given.

Side-Angle-Side Similarity Postulate

You learned how to identify triangles as congruent with one of four postulates or theorems: Side-Side-Side (SSS), Side-Angle-Side (SAS), Angle-Side-Angle (ASA), or Angle-Angle-Side (AAS). Just like there are multiple postulates and theorems to identify congruent triangles, there are multiple postulates and theorems to identify similar triangles. You have learned the Angle-Angle Similarity Postulate already; now find another using the similar triangles below. Pay special attention to the sides! Three pairs of similar triangles are shown below. Take a look at the corresponding side lengths of each. Select the "Show Scale Factor" button to reveal the scale factor between the triangles and the proportion between corresponding sides. How does the scale factor affect the relationship between these two triangles?

Proving the Converse of the Pythagorean Theorem

You learned to prove that the square of the hypotenuse really equals the sum of the squares of the two shorter legs on a right triangle using similar triangles. The converse of this can also be demonstrated with a proof. If a2 + b2 = c2, then the triangle is a right triangle. Converse of the Pythagorean Theorem: If the sum of the squares of the shorter sides is equal to the square of the longest side, then the triangle has a right angle.

Here are two examples of how the proportion should look.

AC/XZ = AB/XY or XZ/AC = XY/AB You wouldn't have a correct proportion if the triangles were mixed up between the numerator and denominator.

Applications of the SSS Similarity Postulate

Another theorem that will help you identify triangle similarity is the SSS Similarity Postulate. Remember how SSS worked in congruency? It's very similar here too! With similarity, though, it's all about proportions! Side-Side-Side Similarity Postulate What it says: If two or more triangles have three corresponding, proportional sides, then the triangles are similar. What it means: If all three corresponding sides of two or more similar triangles have the same simplified fraction, then the triangles are similar! What it looks like: See Attached Image Why it's important: If you only know all the side lengths, but none of the angle measures, then the Angle-Angle Similarity Postulate won't help you. But this one will!

Applications of the SAS Similarity Postulate Example 2

Are the two triangles similar, not similar, or can their similarity not be determined with the given information? This time we have a triangle within a triangle. In other words, are triangles ABC and HBG similar? BH equals 3, BG equals 4, BA equals 6, and BC equals 8. Notice that one angle is shared. Since this angle is congruent in both triangles, let's see how the sides forming that angle compare. Sometimes, it can be difficult to visualize these triangles together. So let's split them up, and draw them side by side. Notice that angle B is labeled in each triangle and has the identical measure of 103 degrees. Now let's label each of the sides, and compare the ratios of each set of corresponding sides. BH measures 3, BG measures 4, BA measures 6, and BC measures 8. The ratios of the corresponding sides would be and . As you can see, both ratios are equal because they simplify to 2. Therefore, these sides are proportional. Triangle ABC is similar to triangle HBG based on SAS.

Applications of the SAS Similarity Postulate Example 1

Are the two triangles similar, not similar, or can their similarity not be determined with the given information? To answer this question, we need to look closely at what is given and how that may help us determine which postulate to apply. First, notice angles B and E are congruent. Next, let's consider using the Side-Angle-Side Postulate because the angle given is congruent, and the length of each side on either side of the angle is given. Perhaps the sides are proportional. Let's compare the smaller sides, and set them up as a ratio. This gives the ratio 4 to 2. Notice that points B and E are both listed first. That is because these points correspond in each triangle. Similarly, let's compare the longer sides. This gives the ratio 6 to 3. Notice that again, A and F are both listed first in the ratio since they are corresponding points in both triangles. Are both ratios equal? If so, we would say that they are proportional. This gives us a proportion 4/2 = 6/3. Both of these fractions equal two and provide proof that these sides are proportional. Therefore, triangles ABC and FED are similar based on SAS.

The Side-Angle-Side Similarity Postulate states

The Side-Angle-Side Similarity Postulate states that if the sides of corresponding, congruent angles within two or more triangles are proportional, the triangles are similar. Take another look at the interactive. When the "Show Scale Factor" button is selected, it shows that the length of the left side of triangle ABC over the length of the left side of triangle XYZ is equal to the length of the bottom side of triangle ABC over the length of the bottom side of triangle XYZ. Simplify both of these fractions. Are they equal? If you haven't heard the word proportional in a while, you may want to review.

Use ΔABC, where a2 + b2 = c2, to prove that ∠ACB is a right angle.

The length of BC is a, the length of AC is b, and the length of AB is c, according to the given information. Construct right triangle SIT where legs ST and TI are congruent to legs AC and CB of triangle ABC, respectively. In other words, TI ≅ CB and ST ≅ AC by construction. ST has length i, TI has length s, and IS has length t. Because ΔSIT was constructed as a right triangle, t2 = s2 + i2 by the Pythagorean theorem Explanation: Even though you don't know that ΔABC is a right triangle, you do know that ΔSIT is a right triangle since it was constructed that way. As a result, you can apply the Pythagorean Theorem to this triangle. Substitute a for s and b for i in the Pythagorean Theorem so t2 = a2 + b2. Explanation: Remember, a = s and b = i by construction. So you are allowed to replace s with a and i with b. According to the given information, c2 = a2 + b2. By the Substitution Property of Equality, t2 = c2. Explanation: Both t2 and c2 equal the expression a2 + b2, so they must equal each other. Take the square root of both sides of the equation, according to the Square Root Property of Equality, to reach the equation t = c. By the Side-Side-Side Postulate ΔABC ≅ ΔSIT. In conclusion, ∠ACB is a right angle by CPCTC.