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Splash Screen. Five-Minute Check (over Lesson 6–1) CCSS Then/Now New Vocabulary Theorems: Properties of Parallelograms Proof: Theorem 6.4 Example 1: Real-World Example: Use Properties of Parallelograms Theorems: Diagonals of Parallelograms

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  1. Splash Screen

  2. Five-Minute Check (over Lesson 6–1) CCSS Then/Now New Vocabulary Theorems: Properties of Parallelograms Proof: Theorem 6.4 Example 1: Real-World Example: Use Properties of Parallelograms Theorems: Diagonals of Parallelograms Example 2: Use Properties of Parallelograms and Algebra Example 3: Parallelograms and Coordinate Geometry Example 4: Proofs Using the Properties of Parallelograms Lesson Menu

  3. Find the measure of an interior angle of a regular polygon that has 10 sides. A. 180 B. 162 C. 144 D. 126 5-Minute Check 1

  4. Find the measure of an interior angle of a regular polygon that has 12 sides. A. 135 B. 150 C. 165 D. 180 5-Minute Check 2

  5. What is the sum of the measures of the interior angles of a 20-gon? A. 3600 B. 3420 C. 3240 D. 3060 5-Minute Check 3

  6. What is the sum of the measures of the interior angles of a 16-gon? A. 3060 B. 2880 C. 2700 D. 2520 5-Minute Check 4

  7. Find x if QRSTU is a regular pentagon. A. 21 B. 15.25 C. 12 D. 10 5-Minute Check 5

  8. What type of regular polygon has interior angles with a measure of 135°? A. pentagon B. hexagon C. octagon D. decagon 5-Minute Check 6

  9. Content Standards G.CO.11 Prove theorems about parallelograms. G.GPE.4 Use coordinates to prove simple geometric theorems algebraically. Mathematical Practices 4 Model with mathematics. 3 Construct viable arguments and critique the reasoning of others. CCSS

  10. You classified polygons with four sides as quadrilaterals. • Recognize and apply properties of the sides and angles of parallelograms. • Recognize and apply properties of the diagonals of parallelograms. Then/Now

  11. parallelogram Vocabulary

  12. Concept 1

  13. Concept 2

  14. A. CONSTRUCTION In suppose mB = 32, CD = 80 inches, BC = 15 inches. Find AD. Use Properties of Parallelograms Example 1A

  15. AD = BC Opposite sides of a are . Use Properties of Parallelograms = 15 Substitution Answer:AD = 15 inches Example 1

  16. B. CONSTRUCTION In suppose mB = 32, CD = 80 inches, BC = 15 inches. Find mC. Use Properties of Parallelograms Example 1B

  17. mC + mB = 180 Cons. s in a are supplementary. Use Properties of Parallelograms mC + 32= 180 Substitution mC = 148 Subtract 32 from each side. Answer:mC = 148 Example 1

  18. C. CONSTRUCTION In suppose mB = 32, CD = 80 inches, BC = 15 inches. Find mD. Use Properties of Parallelograms Example 1C

  19. mD = mB Opp. s of a are . Use Properties of Parallelograms = 32 Substitution Answer:mD = 32 Example 1

  20. A. ABCD is a parallelogram. Find AB. A. 10 B. 20 C. 30 D. 50 Example 1A

  21. B. ABCD is a parallelogram. Find mC. A. 36 B. 54 C. 144 D. 154 Example 1B

  22. C. ABCD is a parallelogram. Find mD. A. 36 B. 54 C. 144 D. 154 Example 1C

  23. Concept 3

  24. Use Properties of Parallelograms and Algebra A. If WXYZ is a parallelogram, find the value of r. Opposite sides of a parallelogram are . Definition of congruence Substitution Divide each side by 4. Answer:r = 4.5 Example 2A

  25. 8s = 7s + 3 Diagonals of a bisect each other. Use Properties of Parallelograms and Algebra B. If WXYZ is a parallelogram, find the value of s. s = 3 Subtract 7s from each side. Answer:s = 3 Example 2B

  26. Use Properties of Parallelograms and Algebra C. If WXYZ is a parallelogram, find the value of t. ΔWXY  ΔYZW Diagonal separates a parallelogram into 2  triangles. YWX  WYZ CPCTC mYWX = mWYZ Definition of congruence Example 2C

  27. Use Properties of Parallelograms and Algebra 2t = 18 Substitution t = 9 Divide each side by 2. Answer:t = 9 Example 2C

  28. A. If ABCD is a parallelogram, find the value of x. A. 2 B. 3 C. 5 D. 7 Example 2A

  29. B. If ABCD is a parallelogram, find the value of p. A. 4 B. 8 C. 10 D. 11 Example 2B

  30. C. If ABCD is a parallelogram, find the value of k. A. 4 B. 5 C. 6 D. 7 Example 2C

  31. Midpoint Formula Find the midpoint of Since the diagonals of a parallelogram bisect each other, the intersection point is the midpoint of Parallelograms and Coordinate Geometry What are the coordinates of the intersection of the diagonals of parallelogram MNPR, with vertices M(–3, 0), N(–1, 3), P(5, 4), and R(3, 1)? Example 3

  32. Parallelograms and Coordinate Geometry Answer: The coordinates of the intersection of the diagonals of parallelogram MNPR are (1, 2). Example 3

  33. A. B. C. D. What are the coordinates of the intersection of the diagonals of parallelogram LMNO, with verticesL(0, –3), M(–2, 1), N(1, 5), O(3, 1)? Example 3

  34. Given: are diagonals, and point P is the intersection of Prove:AC and BD bisect each other. Proof: ABCD is a parallelogram and AC and BD are diagonals; therefore, AB║DC and AC is a transversal. BAC  DCA and ABD  CDB by Theorem 3.2. ΔAPBΔCPD by ASA. So, by the properties of congruent triangles BP DPand AP  CP. Therefore, AC and BD bisect each other. Proofs Using the Properties of Parallelograms Write a paragraph proof. Example 4

  35. Given:LMNO, LN and MO are diagonals and point Q is the intersection of LN and MO. A.LO  MN B.LM║NO C.OQ  QM D.Q is the midpoint of LN. To complete the proof below, which of the following is relevant information? Prove:LNO  NLM Example 4

  36. End of the Lesson

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