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4.5: Proving Quadrilateral PropertiesPowerPoint Presentation

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.

4.5: Proving Quadrilateral Props

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

- 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

Ends: Vertices where the congruent sides intersect.

4.5: Proving Quadrilateral Props

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

- 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

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

- 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 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

- 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

- 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

- 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

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