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Understanding Congruence: Properties of Parallel and Perpendicular Lines

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jean
16 diapositives · 319 Views · Sep 16, 2014

In this lesson, students will explore the properties of congruence in polygons and learn how to determine congruent angles and sides between two congruent shapes. Key concepts include the matching of corresponding sides and angles of polygons, using congruence statements, and applying these principles to solve problems. Students will engage with examples and practice writing congruence statements for various polygon pairs, reinforcing their understanding of this essential geometric concept.

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Understanding Congruence: Properties of Parallel and Perpendicular Lines
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  1. 7-6 Congruence Standard: M8Ga. Students will understand and apply the properties of parallel and perpendicular lines and understand the meaning of congruence. Element d. Understand the meaning of congruence: that all corresponding angles are congruent and all corresponding sides are congruent. Essential Question: Given 2 polygons that are congruent, how do you determine the congruent angles and sides? Course 3

  2. 7-6 Congruence Course 3 A correspondence is a way of matching up two sets of objects. If two polygons are congruent, all of their corresponding sides and angles are congruent. In a congruence statement, the vertices in the second polygon are written in order of correspondence with the first polygon.

  3. Helpful Hint 7-6 Congruence Marks on the sides of a figure can be used to show congruence. Course 3

  4. 7-6 Congruence 55 55 Course 3 Additional Example 1A: Writing Congruent Statements Write a congruence statement for each pair of polygons. The first triangle can be named triangle ABC. To complete the congruence statement, the vertices in the second triangle have to be written in order of the correspondence. A@Q, so A corresponds to Q. B@R, so B corresponds to R. C@P, so C corresponds to P. The congruence statement is triangle ABC@ triangle QRP.

  5. 7-6 Congruence Course 3 Additional Example 1B: Writing Congruent Statements Write a congruence statement for each pair of polygons. The vertices in the first pentagon are written in order around the pentagon starting at any vertex. D@M, so D corresponds to M. E@N, so E corresponds to N. F@O, so F corresponds to O. G@P, so G corresponds to P. H@Q, so H corresponds to Q. The congruence statement is pentagon DEFGH@ pentagon MNOPQ.

  6. 7-6 Congruence Course 3 Check It Out: Example 1A Write a congruence statement for each pair of polygons. The first trapezoid can be named trapezoid ABCD. To complete the congruence statement, the vertices in the second trapezoid have to be written in order of the correspondence. A B | 60° 60° || |||| 120° 120° ||| D C A@S, so A corresponds to S. Q R ||| 120° 120° B@T, so B corresponds to T. || |||| C@Q, so C corresponds to Q. 60° 60° | D@R, so D corresponds to R. T S The congruence statement is trapezoid ABCD@ trapezoid STQR.

  7. 7-6 Congruence WX @ KL a + 8 = 24 –8 –8 a = 16 Course 3 Additional Example 2A: Using Congruence Relationships to Find Unknown Values In the figure, quadrilateral VWXY@ quadrilateral JKLM. Find a. Subtract 8 from both sides.

  8. 7-6 Congruence ML @ YX 6b = 30 6b = 30 6 6 Course 3 Additional Example 2B: Using Congruence Relationships to Find Unknown Values In the figure, quadrilateral VWXY@ quadrilateral JKLM. Find b. Divide both sides by 6. b = 5

  9. 7-6 Congruence J @V 5c = 85 5c = 85 5 5 Course 3 Additional Example 2C: Using Congruence Relationships to Find Unknown Values In the figure, quadrilateral VWXY@ quadrilateral JKLM. Find c. Divide both sides by 5. c = 17

  10. 7-6 Congruence IH @ RS 3a = 6 3a = 6 3 3 Course 3 Check It Out: Example 2A In the figure, quadrilateral JIHK@ quadrilateral QRST. Find a. Divide both sides by 3. 3a I H a = 2 6 4b° S R 120° J 30° Q K c + 10° T

  11. 7-6 Congruence H @S 4b = 120 4b = 120 4 4 Course 3 Check It Out: Example 2B In the figure, quadrilateral JIHK@ quadrilateral QRST. Find b. Divide both sides by 4. 3a I H b = 30° 6 4b° S R 120° J 30° Q K c + 10° T

  12. 7-6 Congruence K @T c + 10 = 30 c + 10 = 30 –10 –10 Course 3 Check It Out: Example 2C In the figure, quadrilateral JIHK@ quadrilateral QRST. Find c. Subtract 10 from both sides. 3a I H c = 20° 6 90° 4b° S R 120° 90° J 30° c + 10° Q K T

  13. 7-6 Congruence 1. Find XY. 3. Find CD. Course 3 Lesson Quiz In the figure, WXYZ@ABCD 10 80° 2. Find mB. 8 90° 4. Find mZ.

  14. 7-6 Congruence Course 3 Now, let’s work on Practice A and then we’ll check together. Use your red pen to check the ones that are correct and correct the ones that are wrong. Raise your hand if you have a question. • B • D • A • C • d = 10 • t = 55° 5. Triangle ABC Triangle DEF • Parallelogram ABCD Parallelogram NMTR

  15. 7-6 Congruence Course 3 Now, let’s work on Practice B and then we’ll check together. Use your red pen to check the ones that are correct and correct the ones that are wrong. • Triangle JKL Triangle TQP • Quadrilateral BDLK Quadrilateral JHRP • Quadrilateral BJLT Quadrilateral KDYP • Hexagon ABCDEF Hexagon VTZYXW • a = 4 6. b = 9 7. c = 10 • 8. x = 19 9. y = 105° 10. z = 15°

  16. 7-6 Congruence Course 3 Closure Essential Question: Given 2 polygons that are congruent, how do you determine the congruent angles and sides? Tell me in your own words…