Geometry: From Triangles to Quadrilaterals and Polygons

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Geometry From Triangles to Quadrilaterals and
Polygons
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MA.912.G.3.2 Compare and contrast special
quadrilaterals on the basis of their properties.

• Block 24

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Convex quadrilaterals are classified as follows

• Trapezoid (Amer.) exactly one pair of opposite
  sides is parallel.
• Parallelogram both pairs of opposite sides are
  parallel.

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Convex quadrilaterals are classified as follows

• Rhomb (Rhombus) all four sides are of equal
  length
• Kite two adjacent sides are of equal length and
  the other two sides also of equal length

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Convex quadrilaterals are further classified as
follows

• Rectangle all four angles are right angles
• Square all four sides are of equal length and
  all four angles are equal (equiangular), with
  each angle a right angle.

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Quadrilaterals taxonomy
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Properties of quadrilaterals
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Parallelograms

• Parallelogram both pairs of opposite sides are
  parallel.

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Test for Parallelograms

• Parallelogram both pairs of opposite sides are
  parallel.
• In addition to that definition we have tests to
  determine if a quadrilateral is parallelogram
  like the following

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Test for Parallelograms

• The quadrilateral is a parallelogram
• If the opposite sides of a quadrilateral are
  congruent

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Test for Parallelograms

• The quadrilateral is a parallelogram
• If both pairs of opposite angles of a
  quadrilateral are congruent

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Test for Parallelograms

• The quadrilateral is a parallelogram
• If the diagonals of a quadrilateral bisect each
  other

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Test for Parallelograms

• The quadrilateral is a parallelogram
• If one pair of opposite sides is parallel and
  congruent

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Kite

• Kite two adjacent sides are of equal length and
  the other two sides also of equal length.
• This implies that one set of opposite angles is
  equal, and that one diagonal perpendicularly
  bisects the other.

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Rhomb

• Rhomb all four sides are of equal length.
• This implies that opposite sides are parallel,
  opposite angles are equal, and the diagonals
  perpendicularly bisect each other.

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Rectangle

• Rectangle (or Oblong) all four angles are right
  angles.
• This implies that opposite sides are parallel and
  of equal length, and the diagonals bisect each
  other and are equal in length.

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Square

• Square (regular quadrilateral) all four sides
  are of equal length (equilateral), and all four
  angles are equal (equiangular), with each angle a
  right angle.
• This implies that opposite sides are parallel (a
  square is a parallelogram), and that the
  diagonals perpendicularly bisect each other and
  are of equal length. A quadrilateral is a square
  if and only if it is both a rhombus and a
  rectangle.

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Trapezoid

• Trapezoid (Amer.) exactly one pair of opposite
  sides is parallel.
• The parallel sides are called bases, and the
  nonparallel sides are called legs
• If the legs are congruent then the trapezoid is
  called isosceles trapezoid.

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Test for Parallelograms and Coordinates

• If the quadrilateral is graphed on the coordinate
  plane you can use Distance Formula, Slope Formula
  and Midpoint Formula.
• The Slope Formula is used to determine if the
  opposite sides are parallel the Distance Formula
  is used to test opposite sides for congruency,
  the Midpoint Formula can be used to determine if
  the diagonals are bisecting each other

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Example

• We will check if a given quadrilateral on the
  coordinate plane is a parallelogram

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Question

• Prove that quadrilateral ABCD where
• A (-1,-1)
• B(3,0)
• C(4,2)
• D(0,1)
• Is a parallelogram

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Proof

• We will prove that ABCD and ADBC (opposite
  sides of quadrilateral are parallel)
• We will use the slope formula

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Proof

• We will prove that ABCD and ADBC
• We will use the slope formula

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Proof

• We will prove that ABCD and ADBC
• We will use the slope formula

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Proof

• We proved that both pairs of slopes are the same
  so ABCD and ADBC hence the quadrilateral
  ABCD is a parallelogram

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Now answer questions from the handout

• Find the diagonals of the parallelogram
• Find the intersection point of the diagonals
• Verify that the diagonals bisect each other

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Review and discussion

• Create the Venn diagram and ,,family tree of
  quadrilaterals
• Discus properties of quadrilaterals