(8) Lesson 7.2 - Congruence
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This document contains a lesson on determining congruence between figures using transformations. It provides examples of identifying congruent triangles and determining the transformations needed to map one onto the other. It also gives examples of using congruence to find missing angle measures. The lesson explains that two figures are congruent if their corresponding sides are congruent and corresponding angles are congruent.
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(8) Lesson 7.2 - Congruence
- 1. Course 3, Lesson 7-2 Determine if the two figures are congruent by using transformations. Explain your reasoning. 1. 2. 3. Triangle ABC is congruent to triangle XYZ. Give a series of transformations that maps triangle XYZ onto triangle ABC.
- 2. Course 3, Lesson 7-2 ANSWERS 1. congruent; rotation 90° clockwise and then a translation 2. congruent; rotation 90° counterclockwise and then a translation 3. reflection then translation
- 3. HOW can you determine congruence and similarity? Geometry Course 3, Lesson 7-2
- 4. Course 3, Lesson 7-2 Common Core State Standards © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved. • 8.G.2 Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them. Mathematical Practices 1 Make sense of problems and persevere in solving them. 2 Reason abstractly and quantitatively. 3 Construct viable arguments and critique the reasoning of others. 4 Model with mathematics. Geometry
- 5. To • compare corresponding parts of congruent figures, • find missing measures of angles and sides of congruent figures Course 3, Lesson 7-2 Geometry
- 6. • corresponding parts Symbols • is congruent to Course 3, Lesson 7-2 Geometry
- 7. Course 3, Lesson 7-2 Geometry Words If two figures are congruent, their corresponding sides are congruent and their corresponding angles are congruent. Model Symbols Congruent Angles: Congruent Sides: ABC DEF ; ;A D B E C F ; ;AB DE BC EF CA FD
- 8. 1 Need Another Example? 2 3 Step-by-Step Example 1. Write congruence statements comparing the corresponding parts in the congruent triangles shown. Use the matching arcs and tick marks to identify the corresponding parts. Corresponding angles: ∠J ∠G, ∠L ∠I, ∠K ∠H Corresponding sides: JK GH, KL HI, LJ IG
- 9. Answer Need Another Example? Write congruence statements comparing the corresponding parts in the congruent triangles shown. ∠P ∠C, ∠Q ∠B, ∠R ∠D, RQ DB; RP DC; PQ CB
- 10. Determine any changes in the orientation of the triangles. The orientation is reversed so at least one of the transformations is a reflection. If you reflect ABC over the y-axis and then translate it down 2 units it coincides with XYZ. 1 Need Another Example? 2 3 Step-by-Step Example 2. Triangle ABC is congruent to XYZ. Write congruence statements comparing the corresponding parts. Then determine which transformations map ABC onto XYZ. The transformations that map ABC onto XYZ consist of a reflection over the y-axis followed by a translation of 2 units down. Analyze the figures to determine which angles and sides of the figures correspond. Corresponding angles: ∠A ∠X, ∠B ∠Y, ∠C ∠Z Corresponding sides: AB XY, BC YZ, CA ZX
- 11. Answer Need Another Example? Triangle RST is congruent to NML. Write congruence statements comparing the corresponding parts. Then determine which transformation(s) maps RST onto NML. ∠R ∠N, ∠S ∠M, ∠T ∠L, RS NM; RT NL; ST ML; Sample answer: If you reflect RST over the x-axis and then translate it to the left 6 units, it coincides with NML.
- 12. 1 Need Another Example? Step-by-Step Example 3. Miley is using a brace to support a tabletop. In the figure, BCE DFG. If m∠CEB = 50°, what is the measure of ∠FGD? Since ∠CEB and ∠FGD are corresponding parts in congruent figures, they are congruent. So, ∠FGD measures 50°.
- 13. Answer Need Another Example? In the figure below, quadrilateral ABCD is congruent to quadrilateral WXYZ. What is the measure of ∠X? 145°
- 14. How did what you learned today help you answer the HOW can you determine congruence and similarity? Course 3, Lesson 7-2 Geometry
- 15. How did what you learned today help you answer the HOW can you determine congruence and similarity? Course 3, Lesson 7-2 Geometry Sample answers: • Two figures are congruent if their corresponding sides are congruent and their corresponding angles are congruent. • On the coordinate plane, you can use the Distance Formula to find the lengths of sides of figures to see if corresponding sides are congruent.
- 16. What must be true about the corresponding angles and corresponding sides of two polygons for them to be congruent? Course 3, Lesson 7-2 Ratios and Proportional RelationshipsFunctionsGeometry