Kite and Trapezoid Properties

1
Kite and Trapezoid Properties

• Recall that a kite is a quadrilateral with two
  distinct pairs of congruent consecutive sides
• One way to look at a kite is as two isosceles
  triangles sharing a common base
• The vertex angles of a kite are the angles
  between the congruent sides
• The nonvertex angles are the other two angles of
  the kite
• Recall that the vertex angle bisector of an
  isosceles triangle is a line of symmetry
• Is there a similar line of symmetry for a kite?
• What does the line of symmetry tell you about
  the nonvertex angles?

2
Kite and Trapezoid Properties

• C-35 Kite Angles Conjecture
• The nonvertex angles of a kite are congruent
• In the diagram at right, DBEN _at_ DBYN by SSS,
• so ÐY _at_ ÐE by CPCTC
• C-38 Kite Angle Bisector Conjecture
• The vertex angles of a kite are bisected by a
  diagonal
• As shown above, DBEN _at_ DBYN by SSS,
• so Ð1 _at_ Ð2 by CPCTC and Ð3 _at_ Ð4 by CPCTC

3
Kite and Trapezoid Properties

• C-36 Kite Diagonals Conjecture
• The diagonals of a kite are perpendicular to
  each other
• In the diagram, Ð1 _at_ Ð2 by the Kite Angle
  Bisector Theorem, so DBUY _at_ DBUE by SAS, and ÐBUY
  _at_ ÐBUE by CPCTC.
• Since ÐBUY and ÐBUE form a linear pair, they
  both
• measure 90, so the diagonals are perpendicular
• C-37 Kite Diagonal Bisector Conjecture
• The diagonal connecting the vertex angles of a
  kite is the perpendicular bisector of the other
  diagonal
• As shown above, DBUY _at_ DBUE by SAS, so segments
  UY and UE are congruent by CPCTC, making diagonal
  BN the perpendicular bisector of diagonal EY

U
4
Kite and Trapezoid Properties

• Recall that a trapezoid is a quadrilateral with
  exactly one pair of parallel sides
• The bases of a trapezoid are the parallel sides
• The base angles of a trapezoid are the pairs of
  angles that share a common base
• A trapezium is a quadrilateral with no parallel
  sides
• There are two bones in your wrist called the
  trapezoid and trapezium because of their shapes
• An isosceles trapezoid is a trapezoid in which
  the two non-parallel sides are congruent

5
Kite and Trapezoid Properties

• What is the sum of the two consecutive base
  angles
• on the same side of a trapezoid?
• C-39 Trapezoid Consecutive Angles Conjecture
• The consecutive angles between the bases of a
• trapezoid are supplementary
• In the diagram at right, TRAP is a trapezoid with
  parallel sides
• RA and TP. TR and AP are transversals between
  parallel lines.
• Using AP, Ð1 and Ð3 are alternate interior
  angles, so Ð1 _at_ Ð3.
• Since Ð2 and Ð3 are a linear pair, the sum of
  their measures is 180.
• Since Ð1 _at_ Ð3, the sum of Ð1 and Ð2 is also 180,
  so Ð1 and Ð2 are supplementary.

6
Kite and Trapezoid Properties

• In an isosceles trapezoid the two non-parallel
  sides are congruent
• Are there any other congruencies associated with
  isosceles trapezoids?
• C-40 Isosceles Trapezoid Conjecture
• The base angles of an isosceles trapezoid are
  congruent
• In the diagram at right, Ð1 _at_ Ð2 and Ð3 _at_ Ð4
• C-41 Isosceles Trapezoid Diagonals Conjecture
• The diagonals of an isosceles trapezoid are
  congruent
• Create DTAP and DPRT by drawing diagonals in
  trapezoid TRAP.
• Using the Isosceles Trapezoid Conjecture, DTAP _at_
  DPRT by SAS,
• so diagonals TA and PR are congruent by CPCTC