1. Over Lesson 6–1 5-Minute Check 1 A.180 B.162 C.144 D.126 Find the measure of an interior angle of a regular polygon that has 10 sides.

2. Over Lesson 6–1 5-Minute Check 2 A.135 B.150 C.165 D.180 Find the measure of an interior angle of a regular polygon that has 12 sides.

3. Over Lesson 6–1 5-Minute Check 3 A.3600 B.3420 C.3240 D.3060 What is the sum of the measures of the interior angles of a 20-gon?

4. Over Lesson 6–1 5-Minute Check 4 A.3060 B.2880 C.2700 D.2520 What is the sum of the measures of the interior angles of a 16-gon?

5. Over Lesson 6–1 5-Minute Check 5 A.21 B.15.25 C.12 D.10 Find x if QRSTU is a regular pentagon.

6. Over Lesson 6–1 5-Minute Check 6 A.pentagon B.hexagon C.octagon D.decagon What type of regular polygon has interior angles with a measure of 135°?

7. Then/Now 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.

8. Vocabulary parallelogram

9. Concept 1

11. Concept 2

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

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

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

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

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

17. Example 1 Use Properties of Parallelograms Answer: m  D = 32 m  D=m  BOpp.  s of a are . =32 Substitution

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

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

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

21. Concept 3

22. Example 2A 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

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

24. Example 2C Use Properties of Parallelograms and Algebra C. If WXYZ is a parallelogram, find the value of t. ΔWXY  ΔYZWDiagonal separates a parallelogram into 2  triangles.  YWX  WYZCPCTC m  YWX=m  WYZDefinition of congruence

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

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

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

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

29. Example 3 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)? Since the diagonals of a parallelogram bisect each other, the intersection point is the midpoint of Find the midpoint of Midpoint Formula

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

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

32. Example 4 Proofs Using the Properties of Parallelograms Write a two column proof. 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  DP and AP  CP. Therefore, AC and BD bisect each other. Given: are diagonals, and point P is the intersection of

33. Example 4 To complete the proof below, which of the following is relevant information? Prove:  LNO   NLM 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.