How Do Dilations Map Angles?

New York State Common Core Math Geometry, Module 2, Lesson 9

Worksheets for Geometry

Student Outcomes

• Students prove that dilations map an angle to an angle with equal measure.
• Students understand how dilations map triangles, squares, rectangles, trapezoids, and regular polygons.

How Do Dilations Map Angles?

Classwork

Exploratory Challenge/Exercises 1–4

1. How do dilations map triangles?
   a. Make a conjecture.
   b. Verify your conjecture by experimenting with diagrams and directly measuring angles and lengths of segments.
   
2. How do dilations map rectangles?
   a. Make a conjecture.
   b. Verify your conjecture by experimenting with diagrams and directly measuring angles and lengths of segments.

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3. How do dilations map squares?
   a. Make a conjecture.
   b. Verify your conjecture by experimenting with diagrams and directly measuring angles and lengths of segments.
   
4. How do dilations map regular polygons?
   a. Make a conjecture.
   b. Verify your conjecture by experimenting with diagrams and directly measuring angles and lengths of segments

Exercises 5–6

5. Recall what you learned about parallel lines cut by a transversal, specifically about the angles that are formed.
6. A dilation from center 𝑂 by scale factor 𝑟 maps ∠𝐵𝐴𝐶 to ∠𝐵′𝐴′𝐶′. Show that 𝑚∠𝐵𝐴𝐶 = 𝑚∠𝐵′𝐴′𝐶′.

Discussion

The dilation theorem for angles is as follows:

DILATION THEOREM: A dilation from center 𝑂 and scale factor 𝑟 maps an angle to an angle of equal measure. We have shown this when the angle and its image intersect at a single point, and that point of intersection is not the vertex of the angle.

Lesson Summary

• Dilations map angles to angles of equal measure.
• Dilations map polygonal figures to polygonal figures whose angles are equal in measure to the corresponding angles of the original figure and whose side lengths are equal to the corresponding side lengths multiplied by the scale factor.

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