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Warm Up

Warm Up. Lesson Presentation. Lesson Quiz. Warm Up Find the value of each variable. 1. x 2. y 3. z. 2. 18. 4. Objectives. Solve problems using the properties of: Parallelograms Rectangles Rhombuses Squares Kites Trapezoids. Vocabulary. Parallelogram Rectangle Square kite

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Warm Up

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  1. Warm Up Lesson Presentation Lesson Quiz

  2. Warm Up Find the value of each variable. 1.x2.y 3.z 2 18 4

  3. Objectives • Solve problems using the properties of: • Parallelograms • Rectangles • Rhombuses • Squares • Kites • Trapezoids

  4. Vocabulary Parallelogram Rectangle Square kite trapezoid Square Rhombus

  5. California Standards 7.0 Students prove and use theorems involving the properties of parallel lines cut by a transversal, the properties of quadrilaterals, and the properties of circles.

  6. Any polygon with four sides is a quadrilateral. However, some quadrilaterals have special properties. These special quadrilaterals are given their own names.

  7. Helpful Hint Opposite sides of a quadrilateral do not share a vertex. Opposite angles do not share a side.

  8. A quadrilateral with two pairs of parallel sides is a parallelogram. To write the name of a parallelogram, you use the symbol .

  9. In CDEF, DE = 74 mm, DG = 31 mm, and mFCD = 42°. Find CF. opp. sides Example 1: Properties of Parallelograms CF = DE Def. of segs. CF = 74 mm Substitute 74 for DE.

  10. In CDEF, DE = 74 mm, DG = 31 mm, and mFCD = 42°. Find mEFC. cons. s supp. Example 2: Properties of Parallelograms mEFC + mFCD = 180° mEFC + 42= 180 Substitute 42 for mFCD. mEFC = 138° Subtract 42 from both sides.

  11. opp. s  Ex. 4: Using Properties of Parallelograms to Find Measures WXYZ is a parallelogram. Find YZ. YZ = XW Def. of  segs. 8a – 4 = 6a + 10 Substitute the given values. Subtract 6a from both sides and add 4 to both sides. 2a = 14 a = 7 Divide both sides by 2. YZ = 8a – 4 = 8(7) – 4 = 52

  12. cons. s supp. Ex. 5: Using Properties of Parallelograms to Find Measures WXYZ is a parallelogram. Find mZ. mZ + mW = 180° (9b + 2)+ (18b –11) = 180 Substitute the given values. Combine like terms. 27b – 9 = 180 27b = 189 b = 7 Divide by 27. mZ = (9b + 2)° = [9(7) + 2]° = 65°

  13. diags. bisect each other. TEACH! Example 4 EFGH is a parallelogram. Find JG. EJ = JG Def. of  segs. 3w = w + 8 Substitute. 2w = 8 Simplify. w = 4 Divide both sides by 2. JG = w + 8 = 4 + 8 = 12

  14. diags. bisect each other. TEACH! Example 5 EFGH is a parallelogram. Find FH. FJ = JH Def. of  segs. 4z – 9 = 2z Substitute. 2z = 9 Simplify. z = 4.5 Divide both sides by 2. FH = (4z – 9) + (2z) = 4(4.5) – 9 + 2(4.5) = 18

  15. A second type of special quadrilateral is a rectangle. A rectangleis a quadrilateral with four right angles.

  16. Since a rectangle is a parallelogram, a rectangle “inherits” all the properties of parallelograms.

  17.  diags. bisect each other Example 1: Craft Application A woodworker constructs a rectangular picture frame so that JK = 50 cm and JL = 86 cm. Find HM. Rect.  diags.  KM = JL = 86 Def. of  segs. Substitute and simplify.

  18. Check It Out! Example 1a Carpentry The rectangular gate has diagonal braces. Find HJ. Rect.  diags.  HJ = GK = 48 Def. of  segs.

  19. Like a rectangle, a rhombus is a parallelogram. So you can apply the properties of parallelograms to rhombuses.

  20. Example 2A: Using Properties of Rhombuses to Find Measures TVWX is a rhombus. Find TV. WV = XT Def. of rhombus 13b – 9 = 3b + 4 Substitute given values. 10b = 13 Subtract 3b from both sides and add 9 to both sides. b = 1.3 Divide both sides by 10.

  21. Example 2A Continued TV = XT Def. of rhombus Substitute 3b + 4 for XT. TV = 3b + 4 TV = 3(1.3) + 4 = 7.9 Substitute 1.3 for b and simplify.

  22. Check It Out! Example 2a CDFG is a rhombus. Find CD. CG = GF Def. of rhombus 5a = 3a + 17 Substitute a = 8.5 Simplify GF = 3a + 17 = 42.5 Substitute CD = GF Def. of rhombus CD = 42.5 Substitute

  23. Check It Out! Example 2b CDFG is a rhombus. Find the measure. mGCH if mGCD = (b + 3)° and mCDF = (6b – 40)° Def. of rhombus mGCD + mCDF = 180° b + 3 + 6b – 40 = 180° Substitute. 7b = 217° Simplify. b = 31° Divide both sides by 7.

  24. TEACH! Example 2b Continued mGCH + mHCD = mGCD Rhombus  each diag. bisects opp. s 2mGCH = mGCD Substitute. 2mGCH = (b + 3) Substitute. 2mGCH = (31 + 3) Simplify and divide both sides by 2. mGCH = 17°

  25. A square is a quadrilateral with four right angles and four congruent sides. In the exercises, you will show that a square is a parallelogram, a rectangle, and a rhombus. So a square has the properties of all three.

  26. Helpful Hint Rectangles, rhombuses, and squares are sometimes referred to as special parallelograms.

  27. When you are given a parallelogram with certain properties, you can use the theorems below to determine whether the parallelogram is a rectangle.

  28. Below are some conditions you can use to determine whether a parallelogram is a rhombus.

  29. Remember! You can also prove that a given quadrilateral is a rectangle, rhombus, or square by using the definitions of the special quadrilaterals.

  30. A kiteis a quadrilateral with exactly two pairs of congruent consecutive sides.

  31. TEACH! Example 2b In kite PQRS, mPQR = 78°, and mTRS = 59°. Find mQPS. QPS  QRS Kite  one pair opp. s   Add. Post. mQPS = mQRT + mTRS mQPS = mQRT + 59° Substitute. mQPS = 51° + 59° Substitute. mQPS = 110°

  32. TEACH! Example 2c In kite PQRS, mPQR = 78°, and mTRS = 59°. Find each mPSR. Polygon  Sum Thm. mSPT + mTRS + mRSP = 180° mSPT = mTRS Def. of s mTRS + mTRS + mRSP = 180° Substitute. 59° + 59°+ mRSP = 180° Substitute. Simplify. mRSP = 62°

  33. A trapezoidis a quadrilateral with exactly one pair of parallel sides. Each of the parallel sides is called a base. The nonparallel sides are called legs. Base anglesof a trapezoid are two consecutive angles whose common side is a base.

  34. If the legs of a trapezoid are congruent, the trapezoid is an isosceles trapezoid. The following theorems state the properties of an isosceles trapezoid.

  35. Example 3A: Using Properties of Isosceles Trapezoids Find mA. mC + mB = 180° Same-Side Int. s Thm. 100 + mB = 180 Substitute 100 for mC. mB = 80° Subtract 100 from both sides. A  B Isos. trap. s base  mA = mB Def. of  s mA = 80° Substitute 80 for mB

  36. Example 3B: Using Properties of Isosceles Trapezoids KB = 21.9m and MF = 32.7. Find FB. Isos.  trap. s base  KJ = FM Def. of segs. KJ = 32.7 Substitute 32.7 for FM. Seg. Add. Post. KB + BJ = KJ 21.9 + BJ = 32.7 Substitute 21.9 for KB and 32.7 for KJ. BJ = 10.8 Subtract 21.9 from both sides.

  37. Lesson Quiz: Part I In PNWL, NW = 12, PM = 9, and mWLP = 144°. Find each measure. 1.PW2. mPNW 18 144°

  38. Lesson Quiz: Part II QRST is a parallelogram. Find each measure. 2.TQ3. mT 71° 28

  39. Lesson Quiz: Part III 5. Three vertices of ABCD are A (2, –6), B (–1, 2), and C(5, 3). Find the coordinates of vertex D. (8, –5)

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