§7.1 Quadrilaterals

§7.1 Quadrilaterals

The student will learn:

the definition of various quadrilaterals and

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theorems associated with quadrilaterals.

Quadrilateral

Definition

Let A, B, C, and D be four coplanar points. If no three of these points are collinear, and the segments AB, BC, CD. And DA intersect only at their end points, then the union of the four segments is called a quadrilateral. The segments are the sides and the points are the vertices. Segments AC and BD are the diagonals.

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Rectangle & Square

Definitions

If all four angles of a quadrilateral are right angles, then the quadrilateral is a rectangle.

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Definitions

If all four angles of a quadrilateral are right angles, then the quadrilateral is a rectangle.

If all four angles of a quadrilateral are right angles and all sides are congruent, then the quadrilateral is a square.

Parallelogram & Trapezoid

Definitions

A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel.

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A trapezoid is a quadrilateral in which exactly one pair of opposite sides are parallel. The parallel sides are called the bases.

Note

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Some of the previous definitions have classified quadrilaterals by sides and/or angles and/or diagonals. You will be asked in homework to organize quadrilaterals using those specifications.

Before we continue our study of quadrilaterals we will need the following information on parallelism.

Historical BackgroundEuclid’s Fifth. If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side which are the angles less than the two right angles.

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A

ln

m

1

2

1 + 2 < 180 then lines l and m meet on the A side of the transversal n.

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Playfair’s Postulate

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Given a line l and a point P not on l, there exist one and only one line m through P parallel to l.

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Equivalent Forms of the Fifth Area of a right triangle can be infinitely large.

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Angle sum of a triangle is 180.

Rectangles exist.

A circle can pass through three points.

Parallel lines are equidistant.

Given an interior point of an angle, a line can be drawn through the point intersecting both sides of the angle.

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Euclidean Parallelism

Definition. Two distinct lines l and m are said to be parallel, l || m, iff they lie in the same plane and do not meet.

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Theorem 1: Parallelism in Absolute Geometry

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If two lines in the same plane are cut by a transversal so that a pair of alternate interior angles are congruent, the lines are parallel.

Notice that this is a theorem and not an axiom or postulate.

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Parallelism in Absolute GeometryGiven: l, m and transversal t. and

1 ≅ 2 Prove: l ׀׀ m

BR

l

m

1

tA

2

Proof by contradiction.

(1) l not parallel to m, meet at R. Assumption

(2) 1 is exterior angle Def

(3) m 1 > m 2 Exterior angle inequality

(4) → ← Given 1 ≅ 2

TheoremIf parallel lines are cut by a transversal then the following relationships are true. (Without proof.)

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1 2

3 4

5 6

7 8

•Alternate interior angles are equal (3 & 6, 4 & 5.)

•Alternate exterior angles are equal (1 & 8, 2 & 7.)

•Corresponding angles are equal. (1 & 5, 2 & 6, 3 & 7, 4 & 8.)

Theorem

Each diagonal separates a parallelogram into two congruent triangles. Do you see the proof.

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Theorem

In a parallelogram, any two opposite sides are congruent.

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In a parallelogram, any two opposite angles are congruent.

TheoremIn a parallelogram, the diagonals bisect each other.

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In a parallelogram, any two consecutive angles are supplementary.

Prove!

TheoremA quadrilateral in which both pairs of opposite sides are congruent is a parallelogram.

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If two sides of a quadrilateral are parallel and congruent, then it is a parallelogram.

If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.

Prove!

TheoremIf a parallelogram has one right angle, then it has four right angles, and is a rectangle.

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In a rhombus the diagonals are perpendicular to one another.

If the diagonals of a quadrilateral bisect each other and are perpendicular, then it is a rhombus.

Prove!

TheoremQuadrilaterals may be proven congruent by one of the following means:

SASAS

ASASA

SASAA

SASSS

The proof is left to the student.

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Theorem – The Midline TheoremThe segment between the midpoints of two sides of a triangle is parallel to the third side and half as long.

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Hint:

CA

B

y

w

x

v

FD E

Cyclic Quadrilaterals

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Definitions

A quadrilateral is cyclic if its four vertices lie on a circle.

TheoremsThe perpendicular bisectors of the four sides of a cyclic quadrilateral are concurrent.

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The perpendicular bisectors of the two diagonals of a cyclic quadrilateral are concurrent at the center of the circle.

The opposite angles of a cyclic quadrilateral are supplementary.

Sketchpad on Campus

Assignment: §7.1

Add HW on parallelism