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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.

### 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.