Unit 5. Butterflies, Pinwheels, & Wallpaper. I can identify types of angles when parallel lines are cut by a transversal.

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1 Unit 5 Butterflies, Pinwheels, & Wallpaper I can identify types of angles when parallel lines are cut by a transversal. Investigation 4 Practice Problems Lesson 1: Focus on Dilations #1 6 Lesson 2: Checking Similarity without Transformations #7 15 Lesson 3: Using Similar Triangles #16 22 The Problems of the first three Investigations focused on ideas and techniques for recognizing and making symmetric patterns. They also focused on comparing figures to see if they have identical shape and size. You know, from earlier work in Stretching and Shrinking, that interesting things happen when you enlarge or reduce a figure. For example, in one Investigation, you used a rubber band tool to enlarge the logo for a mystery club. In another Investigation, you compared various transformations of Mug Wump. Which characters are members of the Wump family, which are impostors? By measuring the figures that result from stretching or shrinking operations, you learned about. These are actions that change the size, but not the shape, or geometric figures. In this Investigation, you will review and extend your understanding of dilations and other similarity transformations.

2 Lesson 1: Focus on Dilations I can identify types of angles when parallel lines are cut by a transversal. The key part of any similarity transformation is a. The diagram shows how a, centered at point P with scale factor 3 2 or 1.5, transform triangle ABC to triangle XYZ. The two triangles are. In everyday language, a dilation is usually an enlargement with a scale factor greater than 1. How are dilations different in math? A scale factor greater than 1 causes stretching, while a scale factor less than 1 causes shrinking. In fact, the diagram above shows how a dilation with center P and scale factor 2, or about 0.67, transforms 3 triangle XYZ to triangle ABC. This Investigation will help you answer the following question: How do dilations affect the size and shape of the figures they transform? In this Problem, you will review the properties of stretching and shrinking transformations by working with figures on a coordinate grid.

3 Problem 4.1 A. Use the figure below. Draw the image of quadrilateral PQRS after a dilation with center (0, 0) and scale factor 3. Label corresponding points P, Q, R, and S. 1. How do the side lengths of quadrilateral P Q R S compare to those of quadrilateral PQRS? 2. How do the angle measures of quadrilateral P Q R S compare to those of quadrilateral PQRS? 3. How does the perimeter of quadrilateral P Q R S compare to that of quadrilateral PQRS? 4. How does the area of quadrilateral P Q R S compare to that of quadrilateral PQRS? 5. How do the slopes of the sides of quadrilateral P Q R S compare to the slopes of the sides of quadrilateral PQRS? 6. What rule of the form (x, y) (, ) shows how coordinates of corresponding points are related under a dilation with center (0, 0) and scale factor 3?

4 B. On the drawing from Question A, draw. 1. What similar triangles do you see? Explain. 2. Suppose point Z = (2z, 4z). How are points P and Z related to each other? Use coordinates to find the slopes of why your discovery makes sense.. What do you notice? Explain C. Use the figure below. Draw the image of quadrilateral PQRS after a dilation with center (0, 0) and scale factor ½. 1. How do the side lengths of quadrilateral P Q R S compare to those of quadrilateral PQRS? 2. How do the angle measures of quadrilateral P Q R S compare to those of quadrilateral PQRS?

5 3. How does the perimeter of quadrilateral P Q R S compare to that of quadrilateral PQRS? 4. How does the area of quadrilateral P Q R S compare to that of quadrilateral PQRS? 5. How do the slopes of the sides of quadrilateral P Q R S compare to the slopes of the sides of quadrilateral PQRS? 6. What rule of the form (x, y) (, ) shows how coordinates of corresponding points are related under a dilation with center (0, 0) and scale factor ½? D. Use your results from Questions A and B to write conjectures about the effects of a dilation with scale factor k on a polygon. If necessary, try dilating a few more figures. (Begin your conjectures with: When two polygons are related by a dilation with scale factor k, ) E. If you dilate a figure with reflectional or rational symmetry, will the resulting image have the same symmetry? Explain. Lesson 2: Checking Similarity without Transformations I can identify types of angles when parallel lines are cut by a transversal. In Investigation 2, you learned how to determine whether two triangles are congruent without actually moving one onto the other. That suggests there might be comparable strategies for testing the similarity of triangles. If two triangles are similar, corresponding angles are congruent and the corresponding sides are related by a scale factor k.

6 For example, in triangles ABC and XYZ, you can measure to check that: In this Problem, you will explore where you need to measure all the sides and angles of the triangles to determine similarity. This Investigation will help you answer the following question: Can you conclude that two triangles are similar if you know the measures of only one, two or three pairs of corresponding parts? Explain. Problem 4.2 The students at Palms Middle School have different ideas about whether they need to measure all the sides and angles of two triangles to determine similarity. A. Kevin and Ming think you only need to measure the angles of each triangle. They argue, The angles give a polygon its shape. If two triangles have congruent corresponding angles, they will have the same shape and be similar. Do you agree with their reasoning? Why or why not?

7 B. Owen and Natasha agree that you need to measure angles, but have a different argument. They claim, If two triangles have congruent corresponding angles, then you can flip, turn, and/or slide the smaller triangle onto the larger triangle as shown below. Then line segment AC is parallel to line segment XZ, so a dilation centered at point Y will stretch triangle ABC exactly onto triangle XYZ. This proves that the original triangles are similar. 1. Do you agree with their reasoning? Why or why not? 2. How can they conclude that line segments AC and XZ are parallel? 3. How can they conclude that a dilation centered at point Y would stretch triangle ABC onto triangle XYZ? C. Kelly and Rico think you need to measure only two pairs of corresponding angles to guarantee similarity. They argue, If you know the measures of two angles of a triangle, then you can subtract the sum of the two angles from 180 to find the measure of the third angle. If you know that the sum of two angles in one triangle is equal to the sum of two corresponding angle sin another, then the third angles of those triangles must be equal in measure. Do you agree with their reasoning? Why or why not?

8 D. Complete this statement in a way that combines the ideas of the Palms Middle School students: If angles in one triangle are equal in measure to corresponding angles in another triangle, then. E. The shortcut for proving that two triangles are similar help to verify other observations you might have made. 1. In the figure, the equation of the line is y = ax + b and points P, Z, and W are on the line. Complete the diagram and find similar triangles. Explain why they are similar. 2. How can you use the similar triangles to explain why the slope of line segment PZ equals the slope of line segment PW (or the slope of line segment ZW)? Explain. Lesson 3: Using Similar Triangles I can identify types of angles when parallel lines are cut by a transversal. Similar triangles have the same shape, but are usually a different size. You can use the relationships between corresponding parts of similar triangles to solve measurement problems. For example, the diagram on the next page shows a method for calculating the height of an object that is difficult to measure directly. Place a mirror on a leveled spot at a convenient distance from the object. Back up from the mirror until you can see the reflection on the top of the object in the center of the mirror. The two triangles in the diagram are similar. To find the object s height, you need to measure three distances and use similar triangles.

9 Problem 4.3 A. Jim and Sue use the mirror method to estimate the height of traffic light near their school. They make the following measurements: 1. Sketch the situations to show the similar triangles formed. Label any parts of the triangles with the known measurements. 2. Explain how you know the triangles are similar. 3. Use properties of similarity to estimate the height of the traffic light.

10 B. Jim and Sue also use the mirror method to estimate the height of the gymnasium in their school. They make the following measurements: 1. Sketch the situation to show the similar triangles formed. Label any parts of the triangles with the known measurements. 2. Use properties of similarity to estimate the height of the gymnasium. C. Find an object that is too tall for you to measure directly. Use the mirror method to estimate its height. Make a sketch and explain how you used properties of similar triangles to estimate the height.