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Common Core Standards: Math

High School: Geometry

Similarity, Right Triangles, and Trigonometry HSG-SRT.A.2

2. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

Students should already be familiar with transformations, at least to the extent that they know that it takes more than an, "Abracadabra!" to change or move a shape. It helps if they know what translation, reflection, rotation, and dilation are, though. Magic tricks are fun and all, but not exactly useful when determining similarity.

Students should already know that if they can perform any rigid transformation to carry one shape exactly onto another, the two shapes are congruent. If we throw dilation into the mix (whether it's contraction or expansion), we can safely say that the two shapes are similar.

While those basic rules apply to pretty much any two-dimensional shape you can imagine (although it might be a pain to verify that two dodecagons are similar), triangles get special treatment. In fact, triangles are like the royal family of geometry...only without the corgis.

Because they're so small and easy, we can use triangles to really get to the heart of what similarity is. Let's say we have two triangles, ΔABC and ΔPQR, that are congruent to each other.

Then, we can contract ΔABC so that it's half the size of ΔPQR. (That's a scale factor of 0.5. Hint, hint.) Dilation is still a similarity transformation, so even though they aren't congruent anymore, the two triangles are still similar.

Students should realize that even though the side lengths of ΔABC have changed, its angles have not. They might say something like, "Hey, those two triangles have similar-looking angle measures. They're just not the same size." If you hear something like that, it's a good sign.

This means two triangles are similar when all their corresponding angles are equal. In fact, students can take this to the next level. Whenever any two polygons of differing sizes have the same set of angle measures within them, the polygons are similar.

So that's angles. What about side lengths? Our scale factor tells us how we relate side lengths of similar triangles to each other. Ratios of corresponding side lengths should always give the same scale factor. If two shapes are similar, these ratios of corresponding side lengths will all equal the same number: the scale factor.

A good way to make sure students have cemented all this knowledge is to have the students measure the sides of similar shapes with a ruler and the angles with a protractor. Then, give another set of shapes that are almost similar. Students can then realize that the measures of the angles in the figure affect the scale ratio they calculate using the side lengths.

Drills

  1. Which of the other triangles is similar to ΔABC and why?

    Correct Answer:

    Answer Explanation:

    The angles in any triangle must add up to 180°, so ∠B has to be 100°. That means we have to figure out which triangle has angle measures of 30°, 50°, and 100°. Subtract the given angles in each triangle from 180°, and we can see that ΔDEF is a 30°-55°-95° triangle, ΔGHJ is the other 30°-50°-100° triangle, and ΔMNP is a 30°-40°-110° triangle. The only triangle that has the same angles is ΔGHJ, so it's the only similar triangle.


  2. Two frame houses are built, a taller one next to a shorter one. If the frame houses are to be similar in their construction, what should the dimensions of the bigger house be?

    Correct Answer:

    x = 55 ft, y = 44 ft

    Answer Explanation:

    The two shapes have to be similar, which means their sides must all have the same scale factor. We can find the scale factor by dividing the side length of the larger house by the corresponding side of the smaller house. In other words, our scale factor is 66 ÷ . To find x, we can multiply the corresponding side by this same scale factor: 30 × . We could calculate the same for y = 24 × 116 = 44, but we already have our answer.


  3. Find the value of x given that the two triangles are similar and isosceles.

    Correct Answer:

    9.7 inches

    Answer Explanation:

    Since the two triangles are similar, we can set up a proportion involving the known sides. In other words, we can say that 512 is our scale factor, and multiply the scale factor by the corresponding side of the larger triangle to find x. If we do the math right, we should get 23.3 × 512 = 9.7 inches.


  4. Are the two quadrilaterals given below similar or not? (The sum of all angles in any quadrilateral is always 360°.)

    Correct Answer:

    Yes, the two quadrilaterals are similar

    Answer Explanation:

    The sum of the angles in any quadrilateral is 360°. It says so right there. If we add up the angle measures in ABCD and subtract from 360°, we'll find ∠C = 80°. Similarly (no pun intended), if we add up the angle measures in FGHJ and subtract from 360°, we'll find ∠J = 100°. Both quadrilaterals have the exact same angle measures, so yes. They're similar.


  5. Assuming that AB || DC and that the two triangles in the figure are similar, which triangle is similar to ΔABE?

    Correct Answer:

    ΔCDE

    Answer Explanation:

    Uh...is this some sort of trick question? No, not really. When we write that two triangles are similar, we want to be very specific about the angles we're talking about. We need to write the similar triangle so that its vertices match up with ∠A, then ∠B, and finally ∠E. We can see that ∠AEB and ∠CED are vertical angles, so that makes them congruent. Then, since AB || DC, we can assume AC is a transversal that cuts through both parallel lines. That makes ∠A ≅ ∠C. From there, since the angles in any triangle add up to 180°, that makes ∠B ≅ ∠D. Now match up the corresponding angles, and we'll see that (A) is correct.


  6. Mike has to build two similar tents for his buddy Jim. Both tents have a 70° angle at the top, but Mike forgets what the bottom angles are supposed to be. He asks Jim for the blueprints, and Jim gives him the diagrams below. What is the value of x?

    Correct Answer:

    9

    Answer Explanation:

    Since ΔABC ~ ΔDEF, we have to figure out which angles from the two triangles correspond. Follow the order of the vertices in the triangle names. It turns out ∠B ≅ ∠E, so we can set their angle measures equal and solve for x. Once we do, we'll have 7x – 2 = 4x + 25, so solving for x gives us x = 9. Thank goodness we haven't forgotten our algebra, huh?


  7. Mike has to build two similar tents for his buddy Jim. Both tents have a 70° angle at the top, but Mike forgets what the bottom angles are supposed to be. He asks Jim for the blueprints, and Jim gives him the diagrams below. What are the angles of the tents?

    Correct Answer:

    49°, 61°, and 70°

    Answer Explanation:

    We can eliminate (D) right from the start because we know both triangles have an angle of 70°. That leaves (A), (B), and (C). If we plug x = 9 back into the given relationships, we'll find ∠B and ∠E are 61° angles. From there, we can find the value of ∠F, which turns out to be a 49° angle. That means ∠C is also 49°. It makes sense that the angles of both tents are equal because (spoiler alert!) they're similar.


  8. Mike has to build two similar tents for his buddy Jim. Both tents have a 70° angle at the top, but Mike forgets what the bottom angles are supposed to be. He asks Jim for the blueprints, and Jim gives him the diagrams below. What are the side lengths of the larger tent?

    Correct Answer:

    9 ft, 10.44 ft, and 11.21 ft

    Answer Explanation:

    We can set up a proportion to find the scale factor between the two triangles. Dividing the two known sides gives us a scale ratio of  ≈ 1.5. That means we should expect the larger tent to have side lengths that are 1.5 times larger that the small tent. If we multiply each of the small tent's side lengths by this scale factor, we should get 6 × 1.5 = 9 ft and 6.96 × 1.5 = 10.44 ft.


  9. In the diagram below, what is the measure of ∠E if the two triangles are similar?

    Correct Answer:

    60°

    Answer Explanation:

    Since the triangles are similar, they have the same angle measures. We have two 30°-60°-90° right triangles on top of each other in this diagram. We hope you can see that ∠B has to be 60°. We can see that ∠C and ∠F are both 90°, but how can we be sure which of the other two angles is which? If you don't want to just go with your gut, we recommend looking at the side opposite the angle. Since it's clear that DF is longer than FE, we know ∠E is greater than ∠D. That means ∠D is 30° and ∠E is 60°.


  10. Which of the triangles below is similar to ΔXYZ?

    Correct Answer:

    Answer Explanation:

    Basically, this involves a lot of side comparison. If two triangles are similar, all the corresponding side lengths will have the same scale factor. That means, if ΔXYZ ~ ΔPMN, then:

    This is only true for ΔSRT, since . All the other ones? Not so much.