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. B. A. C. D.

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4.5: Proving Quadrilateral Properties

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4 5 proving quadrilateral properties

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.

4.5: Proving Quadrilateral Props


4 5 proving quadrilateral properties

B

A

C

D

  • 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

4.5: Proving Quadrilateral Props


Kites

Kites

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

4.5: Proving Quadrilateral Props


Anatomy of a kite

Anatomy of a Kite

Ends: Vertices where the congruent sides intersect.

4.5: Proving Quadrilateral Props


Anatomy of a kite1

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.

4.5: Proving Quadrilateral Props


Properties of a kite theorem

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.

4.5: Proving Quadrilateral Props


Prove part a of the properties of a kite theorem

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

4.5: Proving Quadrilateral Props


Properties of a parallelogram theorem

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.

4.5: Proving Quadrilateral Props


Properties of a parallelogram theorem1

Properties of a Parallelogram Theorem

  • Prove part a.

  • Given: ABCD is a parallelogram.

  • Prove: ABC CDA

C

B

A

D

4.5: Proving Quadrilateral Props


Properties of a rhombus theorem

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.

4.5: Proving Quadrilateral Props


Properties of a rectangle theorem

Properties of a Rectangle Theorem

  • If a quadrilateral is a rectangle, then:

  • It is a parallelogram.

  • Its diagonals are congruent.

4.5: Proving Quadrilateral Props


Properties of a square theorem

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.

4.5: Proving Quadrilateral Props


Assignment

Assignment

  • Pages 248-249, # 42-66 (all)

4.5: Proving Quadrilateral Props


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