Name Class Date. Congruence and Transformations Going Deeper

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1 Name lass ate 4-1 ongruence and Transformations Going eeper ssential question: How can ou use transformations to determine whether figures are congruent? Two figures are congruent if the have the same size and shape. more formal mathematical definition of congruence depends on the notion of rigid motions. Two plane figures are congruent if and onl if one can be obtained from the other b rigid motions (that is, b a sequence of reflections, translations, and/or rotations.) 1 G-O.2.6 XMPL etermining If igures are ongruent Use the definition of congruence in terms of rigid motions to determine whether the two figures are congruent and eplain our answer. and have different sizes. Since rigid motions preserve distance, there is no sequence of rigid motions that will map to. Therefore, Houghton Mifflin Harcourt Publishing ompan You can map JLM to PQS b the translation that has the following coordinate notation: translation is a rigid motion. Therefore, LT 1a. Wh does the fact that and have different sizes lead to the conclusion that there is no sequence of rigid motions that maps to? M J L S P Q hapter Lesson 1

2 The definition of congruence tells ou that when two figures are known to be congruent, there must be some sequence of rigid motions that maps one to the other. You will investigate this idea in the net eample. 2 G-O.1. XMPL inding a Sequence of igid Motions or each pair of congruent figures, find a sequence of rigid motions that maps one figure to the other. You can map to ST b a reflection followed b a translation. Provide the coordinate notation for each. eflection: ollowed b... S T Translation: You can map G to HJ b a rotation followed b a translation. Provide the coordinate notation for each. J G otation: ollowed b... H Translation: LT 2a. plain how ou could use tracing paper to help ou find a sequence of rigid motions that maps one figure to another congruent figure. 2b. Given two congruent figures, is there a unique sequence of rigid motions that maps one figure to the other? Use one or more of the above eamples to eplain our answer. Houghton Mifflin Harcourt Publishing ompan hapter Lesson 1

3 3 G-O.1. XPLO Investigating ongruent Segments and ngles Use a straightedge to trace on a piece of tracing paper. Then slide, flip, and/or turn the tracing paper to determine if there is a sequence of rigid motions that maps to one of the other line segments. P S J epeat the process with the other line segments and the angles in order to determine which pairs of line segments and which pairs of angles, if an, are congruent. ongruent line segments: ongruent angles: Use a ruler to measure the congruent line segments. Use a protractor to measure the congruent angles. LT 3a. Make a conjecture about congruent line segments. Houghton Mifflin Harcourt Publishing ompan 3b. Make a conjecture about congruent angles. The smbol of congruence is. You read the statement UV XY as Line segment UV is congruent to line segment XY. ongruent line segments have the same length, so UV XY implies UV = XY and vice versa. ongruent angles have the same measure, so implies m = m and vice versa. ecause of this, there are properties of congruence that resemble the properties of equalit, and ou can use these properties as reasons in proofs. Properties of ongruence efleive Propert of ongruence Smmetric Propert of ongruence Transitive Propert of ongruence If, then. If and, then. hapter Lesson 1

4 PTI Use the definition of congruence in terms of rigid motions to determine whether the two figures are congruent and eplain our answer J M P L S Q Y Z T V U X or each pair of congruent figures, find a sequence of rigid motions that maps one figure to the other. Give coordinate notation for the transformations ou use. 4. N P. U 6. X M W V L M J 7. and GHJ. an ou conclude GHJ? plain. Houghton Mifflin Harcourt Publishing ompan hapter Lesson 1

5 Name lass ate 4-1 dditional Practice ppl the transformation to the polgon with the given vertices. Identif and describe the transformation. 1. M: (, ) ( 2, + 3) 2. M: (, ) (, ) ( 1, 3), (2, 1), (2, 4) P( 1, 2), Q( 2, 3), (1, 2) 3. M: (, ) (, ) G( 4, 3), H( 2, 3), J( 2, 1), ( 4, 1) 4. M: (, ) (2, 2) ( 2, 2), (1, 1), G(2, 2) Houghton Mifflin Harcourt Publishing ompan etermine whether the polgons with the given vertices are congruent.. ( 4, 4), ( 2, 4), ( 2, 2), ( 3, 1), 6. P(4, 4), Q( 4, 2), ( 2, 6); ( 4, 2); P(2, 6), Q(4, 6), (4, 4), J(2, 2), ( 2, 1), L( 1, 3) S(3, 3), T(2, 4) hapter 4 13 Lesson 1

6 Problem Solving 1. Irena is designing a quilt. She made this diagram to follow when making her quilt. What transformation(s) are used on the triangles to create the pattern in the quilt design? 2. n architect used this design for a stained glass window. What frieze transformation(s) is used to create the pattern in the window? 3. graphic artist incorporated the universal smbol for radiation for one of his designs. escribe the transformation(s) he used. 4. ichard developed a tessellating shape for floor tile. escribe the series of transformations he used to create the design. hoose the best answer.. team flag is made using a fabric with the design shown. What transformation is used? translation reflection rotation dilation 6. n art student used transformations in all her art. What transformation did she use for her design shown? translation G reflection H rotation J dilation Houghton Mifflin Harcourt Publishing ompan hapter Lesson 1