4.5: Proving Quadrilateral Properties

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4.5 Proving Quadrilateral Properties

• Expectations
• G1.4.2 Solve multi-step problems and construct
  proofs involving quadrilaterals.
• G2.3.1 Prove triangles are congruent.
• G2.3.2 Use congruent triangles to prove
  additional theorems.
• L3.3.1 Know the basic format of an proof.

2

• In the figure below, AC is the diameter of the
  circle, B is a point on the circle, AB is
  congruent to BC and D is the midpoint of AC. What
  is the degree measure of angle ABD?
• 30
• 45
• 60
• 90
• Cannot be determined from the given
  information

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Kites

• Defn A quadrilateral is a kite iff it has 2
  distinct pairs of adjacent and congruent sides.

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Anatomy of a Kite
Ends Vertices where the congruent sides
intersect.
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Anatomy of a Kite
The diagonal of a kite with its endpoints at the
ends of the kite is the symmetry diagonal for the
kite.
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Properties of a Kite Theorem

• If a quadrilateral is a kite, then
• The symmetry diagonal bisects the angles at the
  ends of the kite.
• Its diagonals are perpendicular.

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Prove part a of the Properties of a Kite Theorem
Given ABCD is a kite. Prove AC bisects ?DAB
and ?DCB
D
A
C
B
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Properties of a Parallelogram Theorem

• If a quadrilateral is a parallelogram, then
• Each diagonal forms 2 congruent triangles.
• Both pairs of opposite angles are congruent.
• Each pair of opposite sides are congruent.
• Diagonals bisect each other.
• Consecutive angles are supplementary.

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Properties of a Parallelogram Theorem

• Prove part a.
• Given ABCD is a parallelogram.
• Prove ?ABC ??CDA

C
B
A
D
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Properties of a Rhombus Theorem

• If a quadrilateral is a rhombus, then
• It is a parallelogram and a kite.
• Its diagonals are perpendicular.
• Its diagonals bisect opposite angles.

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Properties of a Rectangle Theorem

• If a quadrilateral is a rectangle, then
• It is a parallelogram.
• Its diagonals are congruent.

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Properties of a Square Theorem

• If a quadrilateral is a square, then
• It is a parallelogram, rectangle, rhombus and
  kite.
• Its diagonals are perpendicular, congruent, they
  bisect each other and they bisect the angles at
  opposite ends of the square.

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Assignment

• Pages 248-249, 42-66 (all)