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# 4.5: Proving Quadrilateral Properties - PowerPoint PPT Presentation

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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Presentation Transcript

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

• 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

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

Anatomy of a Kite

Ends: Vertices where the congruent sides intersect.

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.

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.

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

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.

Properties of a Parallelogram Theorem
• Prove part a.
• Given: ABCD is a parallelogram.
• Prove: ABC CDA

C

B

A

D

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.

Properties of a Rectangle Theorem
• If a quadrilateral is a rectangle, then:
• It is a parallelogram.
• Its diagonals are congruent.