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Classifying Quadrilaterals

Classifying Quadrilaterals. Lesson 6-1. Lesson Quiz. Judging by appearance, classify the quadrilaterals in Exercises 1 and 2 in as many ways as possible. 1. 2. quadrilateral, parallelogram, rectangle, rhombus, square. quadrilateral, kite. 4. What is the most precise name

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Classifying Quadrilaterals

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  1. Classifying Quadrilaterals Lesson 6-1 Lesson Quiz Judging by appearance, classify the quadrilaterals in Exercises 1 and 2 in as many ways as possible. 1. 2. quadrilateral, parallelogram, rectangle, rhombus, square quadrilateral, kite 4. What is the most precise name for the figure in Exercise 2? 3. What is the most precise name for the figure in Exercise 1? square kite 5. Find the values of the variables in the rhombus to the right. a = 60, x = 6, y = 2 6-2

  2. Properties of Parallelograms Lesson 6-2 Notes 6-2

  3. Properties of Parallelograms Lesson 6-2 Notes Given Defn of AIAs Defn of AIAs Defn of  Defn of  Reflexive POC AIA Thm AIA Thm ASA CPCTC 6-2

  4. Properties of Parallelograms Lesson 6-2 Notes 6-2

  5. Properties of Parallelograms Lesson 6-2 Notes Consecutive anglesare angles of a polygon that share a side. A proof of Theorem 6-2 uses the consecutive angles of a parallelogram, and the fact that supplements of the same angle are congruent. 6-2

  6. Properties of Parallelograms Lesson 6-2 Notes 6-2

  7. Properties of Parallelograms Lesson 6-2 Notes ABCD is a parallelogram. Given Definition of parallelogram Definition of AIAs 1&4 are AIAs; 2&3 are AIAs AIA Thm 14; 23 Opposite sides of a  are . ASA CPCTC 8. Definition of bisector 8. 6-2

  8. Properties of Parallelograms Lesson 6-2 Notes 6-2

  9. Use KMOQ to find m O. Q and O are consecutive angles of KMOQ, so they are supplementary. m O + m Q = 180 Definition of supplementary angles m O + 35 = 180 Substitute 35 for m Q. m O = 145 Subtract 35 from each side. Properties of Parallelograms Lesson 6-2 Additional Examples Using Consecutive Angles Quick Check 6-2

  10. x + 15 = 135 – x Opposite angles of a are congruent. 2x + 15 = 135 Add x to each side. 2x = 120 Subtract 15 from each side. x = 60 Divide each side by 2. Substitute 60 for x. m B = 60 + 15 = 75 m A + m B = 180 Consecutive angles of a parallelogram are supplementary. m A + 75 = 180 Substitute 75 for m B. m A = 105 Subtract 75 from each side. Properties of Parallelograms Lesson 6-2 Additional Examples Using Algebra Quick Check Find the value of x in ABCD. Then find m A. 6-2

  11. Find the values of x and y in KLMN. x = 7y – 16 The diagonals of a parallelogram bisect each other. 2x + 5 = 5y 2(7y – 16) + 5 = 5y Substitute 7y – 16 for x in the second equation to solve for y. 14y – 32 + 5 = 5y Distribute. 14y – 27 = 5y Simplify. –27 = –9y Subtract 14y from each side. 3 = y Divide each side by –9. x = 7(3) – 16 Substitute 3 for y in the first equation to solve for x. x = 5 Simplify. Properties of Parallelograms Lesson 6-2 Additional Examples Using Algebra Quick Check So x = 5 and y = 3. 6-2

  12. Properties of Parallelograms Lesson 6-2 Additional Examples Real-World Connection Theorem 6-4 states If three (or more) parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal. Explain how to divide a blank card into five equal rows using Theorem 6-4 and a sheet of lined paper. Place a corner of the top of the card on the first line of the lined paper. Place any other corner on the sixth line. Mark the points where the lines intersect one side of the card. Mark the points where the lines intersect the opposite side of the card. Connect the marks on opposite sides using a straightedge. If you use the same-side bottom corner, the lines are parallel to the top of the card. If you use the opposite corner, the lines are parallel to the diagonal of the card. Quick Check 6-2

  13. Properties of Parallelograms Lesson 6-2 Lesson Quiz Use parallelogram ABCD for Exercises 1–5 1. If AB = 3x + 11, BC = 2x + 19, and CD = 7x – 17, find x. 2. If mBAD = y and mADC = 4y – 70, find y. 3. If mABC = 2x + 100 and mADC = 6x + 84, find mBCD. 4. If mBCD = 80 and mCAD = 34, find mACD. 5. If AP = 3x, BP = y, CP = x + y, and DP = 6x – 40, find x and y. 7 50 72 46 x = 10, y = 20 6-2

  14. EFG GHE. Properties of Parallelograms Lesson 6-2 Check Skills You’ll Need (For help, go to Lessons 4-1 and 4-3.) Use the figure at the right. 1. Name the postulate or theorem that justifies the congruence 2. Complete each statement. a.FEG ? b.EFG ? c.FGE ? d. EF ? e.FG ? f. GE ? 3. What other relationship exists between FG and EH ? Check Skills You’ll Need 6-2

  15. 1. The triangles share the side EG, which is the included side between a pair of congruent corresponding angles. So, the triangles are congruent by the ASA Postulate. EFG Properties of Parallelograms Lesson 6-2 Check Skills You’ll Need Solutions 2.a. In the triangle congruence statement EFGGHE, follow the order of the letters for the angles: b. In the triangle congruence statement follow the order of the letters for the angles: c. In the triangle congruence statement follow the order of the letters for the angles: FEG HGE GHE, EFG GHE EFG GHE, FGE HEG 6-2

  16. EFG EFG Properties of Parallelograms Lesson 6-2 Check Skills You’ll Need d. In the triangle congruence statement use the e. In the triangle congruence statement use the f. In the triangle congruence statement use the Solutions (continued) GHE, position of the letters for the congruent segments: EF GH GHE, position of the letters for the congruent segments: FG HE EFG GHE, position of the letters for the congruent segments: GE EG 3. Since the alternate interior angles are congruent, FG || EH . 6-2

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