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The Quadrilateral Family Tree

The Quadrilateral Family Tree. Wednesday 1/5/11 – Thursday 1/6/11. SPECIAL PARALLELOGRAMS. Special Parallelograms. Rectangles, rhombuses, and squares. Rectangles. Rectangle. A quadrilateral with four right angles. Rectangle. Has all the properties of a parallelogram. Quadrilateral.

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The Quadrilateral Family Tree

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  1. The Quadrilateral Family Tree Wednesday 1/5/11 – Thursday 1/6/11 SPECIAL PARALLELOGRAMS

  2. Special Parallelograms Rectangles, rhombuses, and squares

  3. Rectangles

  4. Rectangle • A quadrilateral with four right angles

  5. Rectangle • Has all the properties of a parallelogram

  6. Quadrilateral 1. Four-sided polygon Parallelogram 1. 2. Opposite angles are congruent 3. Diagonals bisect each other 4. Consecutive angles are supp. RECTANGLE 1. 1. 1. Four right angles 2. 2. 3. 2. All properties above 3. 4. 3. Square 1. 2. 1. 2. 1. Opposite sides are congruent 3.

  7. Rectangle • What should be true about the overlapping triangles formed by the diagonals?

  8. Rectangle • What should be true about the overlapping triangles formed by the diagonals?

  9. Rectangle • The triangles are congruent by SAS since opposite sides of a parallelogram are congruent

  10. Rectangle • So the diagonals in a rectangle must be congruent by CPCTC

  11. Quadrilateral 1. Four-sided polygon Parallelogram 1. 2. Opposite angles are congruent 3. Diagonals bisect each other 4. Consecutive angles are supp. RECTANGLE 1. 1. 1. Four right angles 2. 2. 3. 2. All properties above 3. 4. 3. Diagonals are congruent Square 1. 2. 1. 2. 1. Opposite sides are congruent 3.

  12.  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

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

  14. Check It Out! Example 1b Carpentry The rectangular gate has diagonal braces. Find HK. Rect.  diags.  Rect.  diagonals bisect each other JL = LG JG = 2JL = 2(30.8) = 61.6

  15. Rhombuses

  16. Rhombus • A quadrilateral with four congruent sides

  17. Rhombus • Also has all the properties of a parallelogram

  18. Quadrilateral 1. Four-sided polygon Parallelogram 1. 2. Opposite angles are congruent 3. Diagonals bisect each other 4. Consecutive angles are supp. RHOMBUS RECTANGLE 1. 1. Four congruent sides 1. Four right angles 2. All properties above 2. 3. 2. All properties above 3. 4. 3. Diagonals are congruent Square 1. 2. 1. 2. 1. Opposite sides are congruent 3.

  19. Rhombus • What kind of triangles are formed by a diagonal?

  20. Rhombus • What kind of triangles are formed by a diagonal?

  21. Rhombus • What kind of triangles are formed by a diagonal?

  22. Rhombus • Each diagonal bisects a pair of opposite angles

  23. Rhombus • Each diagonal bisects a pair of opposite angles

  24. Quadrilateral 1. Four-sided polygon Parallelogram 1. 2. Opposite angles are congruent 3. Diagonals bisect each other 4. Consecutive angles are supp. RHOMBUS 1. Four congruent sides RECTANGLE 1. 2. All properties above 1. Four right angles 2. 3. Diagonals bisect opposite angles 2. All properties above 3. 3. Diagonals are congruent 4. Square 1. 2. 1. 2. 1. Opposite sides are congruent 3.

  25. Rhombus

  26. Rhombus • What should be true about angles 1 and 2? 2 1

  27. Rhombus • What would the angle measures have to be? 2 1

  28. Rhombus • What would the angle measures have to be?

  29. Quadrilateral 1. Four-sided polygon Parallelogram 1. 2. Opposite angles are congruent 3. Diagonals bisect each other 4. Consecutive angles are supp. RHOMBUS 1. Four congruent sides RECTANGLE 1. 2. All properties above 1. Four right angles 2. 3. Diagonals bisect opposite angles 2. All properties above 3. 3. Diagonals are congruent 4. Diagonals are Square 1. 2. 1. 2. 1. Opposite sides are congruent 3.

  30. Example 2A: Using Properties of Rhombuses to Find Measures TVWX is a rhombus. Find TV. WV = XT 13b – 9=3b + 4 10b =13 b =1.3 TV = XT TV =3b + 4 TV =3(1.3)+ 4 = 7.9

  31. Example 2B: Using Properties of Rhombuses to Find Measures TVWX is a rhombus. Find mVTZ. mVZT =90° Rhombus  diag.  14a + 20=90° a=5 mVTZ =mZTX mVTZ =(5a – 5)° mVTZ =[5(5) – 5)]° = 20°

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

  33. Check It Out! Example 2b CDFG is a rhombus. Find mGCH. mGCD = (b + 3)° and mCDF = (6b – 40)° Def. of rhombus mGCD + mCDF = 180° b + 3 + 6b –40 = 180° 7b = 217° b = 31°

  34. Check It Out! Example 2b Continued CDFG is a rhombus. Find mGCH. mGCD = (b + 3)° and mCDF = (6b – 40)° 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°

  35. Squares

  36. Square • A quadrilateral with four right angles (like a rectangle) and four congruent sides (like a rhombus)

  37. Square • So a square has all the properties of both a rectangle and a rhombus

  38. Quadrilateral 1. Four-sided polygon Parallelogram 1. 2. Opposite angles are congruent 3. Diagonals bisect each other 4. Consecutive angles are supp. Rhombus 1. Four congruent sides Rectangle 1. 2. All properties above 1. Four right angles 2. 3. Diagonals bisect opposite angles 2. All properties above 3. SQUARE 3. Diagonals are congruent 4. Diagonals are 1. All properties of a rectangle 1. 2. 1. Opposite sides are congruent 3. 2. All properties of a rhombus

  39. Parallelograms Rectangles Rhombuses Squares

  40. Quadrilateral 1. Four-sided polygon Kite TRAPEZOID 1. PARALLELOGRAM 1. Opposite sides are congruent 1. 2. 2. Opposite angles are congruent 3. 3. Diagonals bisect each other 4. Consecutive angles are supplementary Isosceles Trapezoid Rhombus 1. Rectangle 1. Properties of a parallelogram 1. All properties of a parallelogram 2. All sides are congruent 2. 3. Diagonals are perpendicular 2. Four right angles 3. 4. Diagonals bisect opposite angles 3. Diagonals are congruent Square 1. Properties of a rectangle 2. Properties of a rhombus

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