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SPECIAL QUADRILATERALS

SPECIAL QUADRILATERALS. NON-PARALLELOGRAMS. Quadrilaterals. The Tricky Trapezoid. Definition: A quadrilateral with exactly one pair of opposite sides parallel. Special Property (Corollary)

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SPECIAL QUADRILATERALS

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  1. SPECIAL QUADRILATERALS NON-PARALLELOGRAMS

  2. Quadrilaterals

  3. The Tricky Trapezoid

  4. Definition: • A quadrilateral with exactly one pair of opposite sides parallel.

  5. Special Property (Corollary) • If a quadrilateral is a trapezoid, then the pairs of base-to-base consecutive interior angles are supplementary. Trapezoids Exactly 1 pair of opposite sides parallel Base-to-base consecutive interior angles are supplementary 1 2 1 2

  6. Midsegments of Trapezoids • The midsegment is the segment connecting the midpoints of the legs of a trapezoid. Trapezoids Exactly 1 pair of opposite sides parallel Base-to-base consecutive interior angles are supplementary

  7. Theorem #49: • The midsegment of a trapezoid is: • 1) Parallel to the bases of the trapezoid • 2) Length = ½ (sum of the bases) • ½ (b1 + b2) Trapezoids Exactly 1 pair of opposite sides parallel Base-to-base consecutive interior angles are supplementary Midsegments Parallel to the bases ½ (sum of the bases) b1 b2

  8. Trapezoids “HOT FACTS” • 4 Sides – Quadrilateral • Odd Looking Quad! • Exactly 1 pair of opposite sides parallel • Base-Base Consecutive angles supplementary • Midsegments! • Parallel to the bases • ½ the sum of the bases

  9. Proving Trapezoids are Quadrilateral Dude, are you serious?

  10. Definition: • If a quadrilateral has exactly 1 pair opposite sides parallel, then the quadrilateral is a trapezoid. Trapezoids EXACTLY 1 pair of opposite sides paralleltrapezoid

  11. Midsegments: • If a quadrilateral has a midsegment that is parallel to both bases and is ½ the sum of the bases, then the quadrilateral is a trapezoid. Trapezoids EXACTLY 1 pair of opposite sides paralleltrapezoid Midsegments parallel AND ½*(sum of the bases) b1 b2

  12. Area of a Trapezoid • Theorem #55: • Area = ½*height*(sum of the bases) • A = ½*h*(b1 + b2) b1 h b2

  13. What about Special Trapezoids? You did know they exist, right?

  14. Definition of an Isosceles Trapezoid: • A trapezoid whose legs are congruent. Isosceles Trapezoids Legs congruent

  15. Theorem #46: • A trapezoid is isosceles if and only if each pair of base angles are congruent. Isosceles Trapezoids Legs congruent Each pair of base angles are congruent

  16. Theorem #48: • A trapezoid is isosceles if and only if its diagonals are congruent. Isosceles Trapezoids Legs congruent Each pair of base angles are congruent Diagonals congruent

  17. Isosceles Trapezoids “HOT FACTS” • 4 Sides – Quadrilateral • Bottom part of an isosceles triangle! • Exactly 1 pair of opposite sides parallel (Bases) • Base-Base Consecutive angles supplementary • Midsegments! • Parallel to the bases • ½ the sum of the bases • Legs are congruent • Each pair of base angles are congruent • Diagonals congruent

  18. Everything you ever wanted to know about Trapezoids and Isosceles Trapezoids…You now know If you did things right, you should have only used 1 sheet of paper, right?

  19. The “Kean” Kite

  20. Definition: • A quadrilateral that has 2 pairs of consecutive sides congruent. Kite 2 pairs of consecutive sides congruent

  21. Theorem #50: • A quadrilateral is a kite if and only if its diagonals are perpendicular. Kite 2 pairs of consecutive sides congruent Diagonals perpendicular

  22. Theorem #51: • A quadrilateral is a kite if and only if it has exactly 1 pair of opposite angles congruent. Kite 2 pairs of consecutive sides congruent Diagonals perpendicular Exactly one pair of opposite angles congruent

  23. Theorem #51 ½ (or #A): • A quadrilateral is a kite if and only if its long diagonal bisects the short diagonal. Kite 2 pairs of consecutive sides congruent Diagonals perpendicular Exactly one pair of opposite angles congruent Long diagonal bisects the Short diagonal

  24. Kites “HOT FACTS” • 4 Sides – Quadrilateral • 2 Isosceles triangles with same bases • 2 pairs of consecutive sides are congruent • Diagonals perpendicular • Exactly 1 pair of opposite angles congruent • Long diagonal bisects the Short diagonal

  25. Proving a Quadrilateral is a Kite Why not just fly one!

  26. Definition: • A quadrilateral that has 2 pairs of consecutive sides congruent. Kite 2 pairs of consecutive sides congruent  Kite

  27. Theorem #50: • A quadrilateral is a kite if and only if its diagonals are perpendicular. Kite 2 pairs of consecutive sides congruent  Kite Diagonals perpendicular  Kite

  28. Theorem #51: • A quadrilateral is a kite if and only if it has exactly 1 pair of opposite angles congruent. Kite 2 pairs of consecutive sides congruent  Kite Diagonals perpendicular  Kite Exactly one pair of opposite angles congruent  Kite

  29. Theorem #51 ½ (or #A): • A quadrilateral is a kite if and only if its long diagonal bisects the short diagonal. Kite 2 pairs of consecutive sides congruent  Kite Diagonals perpendicular  Kite Exactly one pair of opposite angles congruent  Kite Long diagonal bisects the Short diagonal  Kite

  30. Area of a Kite • Theorem #56: • Area = ½*product of the diagonals • A = ½*d1*d2 d1 d2

  31. Everything you ever wanted to know about Kites…You now know If you did things right, you should have only used 1 sheet of paper, right?

  32. Quadrilaterals

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