You can demonstrate the congruence of two figures by using a rigid motion or a sequence of rigid motions to make the figures coincide.

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1 18 LESSON roperties of Rotations, Reflections, and Translations UNERSTN rigid motion changes the position of a figure without changing its shape or size. sequence of rigid motions can transform a figure into a congruent figure. Each point on the original figure matches a corresponding point on the transformed figure, called the image. If the vertices of the original figure are labeled with letters, such as, then the vertices of the image will be labeled with a prime symbol, (9), as in rotation is a turn of a figure about a point. Triangle J9K9L9 is a 90 -clockwise rotation of triangle JKL about point J. J K L K L reflection is a flip of a figure over a point or line. Quadrilateral M9N9O99 is a reflection of quadrilateral MNO over the vertical line. N M M N O O translation is a slide of a figure to a new location. entagon 9999E9 is a translation of pentagon E 4 units down and 5 units to the left. E E You can demonstrate the congruence of two figures by using a rigid motion or a sequence of rigid motions to make the figures coincide. uplicating any part of this book is prohibited by law. 104 omain 4: Geometry

2 onnect escribe the rigid motion that will make the two intersecting lines coincide. 1 Trace the vertical line on tracing paper. 2 Rotate it around point so that it completely covers the slanted line. 45 clockwise rotation of 45 is needed. You can use a protractor to verify the degree of rotation. 45 -clockwise rotation around point can be used to make the two lines coincide. escribe a rigid motion or sequence of rigid motions that could make the two line segments coincide. uplicating any part of this book is prohibited by law. Trace the line segments. Fold the tracing paper so that one segment folds onto the other. The line formed when the paper is folded is a line of reflection. One way to make the line segments coincide is to reflect either one onto the other over a line of reflection. TrY escribe another way to use one or more rigid motions that could make the two line segments coincide. Trace the figure and use the tracing to see if your way works. Lesson 18: roperties of Rotations, Reflections, and Translations 105

3 EXMLE escribe a sequence of rigid motions that could be used to make the 60 angles coincide. 1 Which rigid motions could be used? re the angles turned differently? Yes re the angles mirror images of one another? No re the angles in different locations? Yes Try translating the top angle down so the vertices coincide. Then rotate the image until the angles coincide. 2 Use tracing paper. Trace the top angle and move the tracing. Translate the angle tracing down 5 units so that the vertices of the angles coincide. 3 Rotate the tracing clockwise about the common vertex of the angles. 90 translation of the top angle 5 units down, followed by a 90 -clockwise rotation around the vertex, makes the angles coincide. ISUSS If you reverse the order of the two rigid motions (first rotate, then translate), would the angles coincide? Explain why or why not. uplicating any part of this book is prohibited by law. 106 omain 4: Geometry

4 1 EXMLE escribe how a reflection and a translation could be used to make parallel lines and coincide with parallel lines 00 and 0 0. Reflect and over a line to create a mirror image of the two lines. Lines and can be reflected over a horizontal line to form lines 99 and 99. oint corresponds to point 9, point corresponds to point 9, point corresponds to point 9, and point corresponds to point 9. 2 Look at the diagram below. uplicating any part of this book is prohibited by law. TrY escribe how a reflection and a translation could be used to make parallel lines and coincide with parallel lines 00 and 0 0. Use Math Tool: Grid aper to sketch the steps. Visualize sliding line 99 and line 99 so that each point moves 5 units to the right and 3 units up to make parallel lines 99 and 99 coincide with parallel lines 00 and 0 0. reflection across a horizontal line below line, followed by a translation to the right and up, makes parallel lines and coincide with parallel lines 00 and 0 0. Lesson 18: roperties of Rotations, Reflections, and Translations 107

5 ractice Identify the type of rigid motion needed to make the two figures coincide in one step Identify the line segment that could coincide with if the rigid motions described are performed. G F E K J H 4. is rotated 180 around point and then translated down. 5. is rotated 90 clockwise around point and then reflected across a vertical line. 6. is rotated 135 clockwise around point and then translated down. REMEMER You can use tracing paper and a protractor to help visualize the rigid motions described. Write true or false for each statement. If false, rewrite the statement so it is true. 7. Rigid motions do not change the shape of a figure. 8. Rigid motions may change the size of a figure. uplicating any part of this book is prohibited by law. 108 omain 4: Geometry

6 escribe in detail how one or more rigid motions could be used to make the leftmost figure coincide with the rightmost figure. Use the fewest rigid motions possible F F E E 11. K L L 12. Y Z X K X Y Z Solve. 13. ESRIE Use words and/or drawings to describe two different sequences of rigid motions that could be used to make the 50 angles shown coincide. 14. WRITE MTH Reflect the pair of parallel lines over the dashed line. Then explain how a rotation and a reflection can be used to move those reflected images so that they coincide with the original parallel lines. uplicating any part of this book is prohibited by law. Lesson 18: roperties of Rotations, Reflections, and Translations 109