Download 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.
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4.5: Proving Quadrilateral Props
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?
B
A. 30°
B. 45
C. 60
D. 90
A
C
D
E. Cannot be determined
from the given
information
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4.5: Proving Quadrilateral Props
Kites
Defn: A quadrilateral is a kite iff it has 2
distinct pairs of adjacent and congruent
sides.
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4.5: Proving Quadrilateral Props
Anatomy of a Kite
Ends: Vertices
where the
congruent sides
intersect.
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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.
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4.5: Proving Quadrilateral Props
Properties of a Kite Theorem
If a quadrilateral is a kite, then:
a. The symmetry diagonal bisects the angles
at the ends of the kite.
b. Its diagonals are perpendicular.
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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
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4.5: Proving Quadrilateral Props
Properties of a Parallelogram
Theorem
If a quadrilateral is a parallelogram, then:
a) Each diagonal forms 2 congruent
triangles.
b) Both pairs of opposite angles are
congruent.
c) Each pair of opposite sides are congruent.
d) Diagonals bisect each other.
e) Consecutive angles are supplementary.
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4.5: Proving Quadrilateral Props
Properties of a Parallelogram
Theorem
Prove part a.
Given: ABCD is a parallelogram.
Prove: ABC CDA
C
B
A
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D
4.5: Proving Quadrilateral Props
Properties of a Rhombus
Theorem
If a quadrilateral is a rhombus, then:
a. It is a parallelogram and a kite.
b. Its diagonals are perpendicular.
c. Its diagonals bisect opposite angles.
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4.5: Proving Quadrilateral Props
Properties of a Rectangle
Theorem
If a quadrilateral is a rectangle, then:
a. It is a parallelogram.
b. Its diagonals are congruent.
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4.5: Proving Quadrilateral Props
Properties of a Square Theorem
If a quadrilateral is a square, then:
a. It is a parallelogram, rectangle, rhombus
and kite.
b. Its diagonals are perpendicular, congruent,
they bisect each other and they bisect the
angles at opposite ends of the square.
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4.5: Proving Quadrilateral Props
Assignment
Pages 248-249, # 42-66 (all)
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4.5: Proving Quadrilateral Props