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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. B. A. C. D.

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4.5: Proving Quadrilateral Properties

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  1. 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

  2. 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

  3. Kites • Defn: A quadrilateral is a kite iff it has 2 distinct pairs of adjacent and congruent sides. 4.5: Proving Quadrilateral Props

  4. Anatomy of a Kite Ends: Vertices where the congruent sides intersect. 4.5: Proving Quadrilateral Props

  5. 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

  6. 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

  7. 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

  8. 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

  9. 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

  10. 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

  11. 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

  12. 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

  13. Assignment • Pages 248-249, # 42-66 (all) 4.5: Proving Quadrilateral Props

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