3.1: Symmetry in Polygons

• Definition of polygon: a plane figure with three or more segments which intersect two other segments at the endpoints. No two segments with a common endpoint are collinear.

• The segments are called sides, and the endpoints are called vertices.

A common endpoint; a vertex Segment;

side of polygon

Two similar octagons

Types of Symmetry

•Reflectional symmetry: A figure has reflectional symmetry if and only if its reflected image across a line (line of symmetry) coincides exactly with the preimage.

Types of Symmetry (Continued)

• Rotational symmetry: A figure has rotational symmetry if and only if it has at least one rotation image, not counting rotation images of 0° or multiples of 360°.

These two figures have rotational symmetry.

3.2: Properties of Quadrilaterals

•Quadrilateral: Any four sided polygon. Certain quadrilaterals have specific properties and these are called special quadrilaterals. They include trapezoids, parallelograms, rhombuses, rectangles and squares.

Trapezoids

• A trapezoid is a quadrilateral with one and only one pair of parallel sides

These two sides are parallel. They are the only set.

Parallelograms

• A parallelogram is a quadrilateral with two pairs of parallel sides.

• Conjectures: Opposite sides are congruent, opposite angles are congruent, consecutive angles are supplementary and the diagonals bisect each other.

Rhombuses

• Rhombus: A quadrilateral with four congruent sides

• Conjectures: All rhombuses are parallelograms, therefore all conjectures for the parallelogram holds true for the rhombus, diagonals are perpendicular bisectors.

Rectangles

•Rectangle: A quadrilateral with four right angles

•Conjecture: A rectangle is a parallelogram, therefore all conjectures for parallelogram hold true for rectangles, diagonals are congruent.

Squares

• Square: A quadrilateral with four congruent sides and four right angles

• Conjectures: A square is a rectangle, parallelogram and rhombus. All conjectures for those figures hold true for the square, Diagonals bisect each other, are congruent and are perpendicular bisectors.

3.3: Parallel lines and Transversals

• Parallel lines: two coplanar lines that do not meet no matter how far they might be extended

• Transversal: A line, ray or segment that intersects tow or more coplanar lines, rays or segments each at a different point.

The Beatles- Mathematicians in Disguise?

These lines on the road are parallel!

The Beatle are clearly forming a transversal on Abbey Road!

Postulates/ Theorems

• Corresponding Angles Postulate: If two lines cut by a transversal are parallel, then corresponding angles are congruent.

• Alternate Interior Angles Theorem: If two lines cut by a transversal are parallel, then alternate interior angles are congruent

• Alternate Exterior Angles theorem: If two lines cut by a transversal are parallel, then alternate exterior angles are congruent

Postulates/ Theorems (Continued)

• Same Side Interior Angles Theorem: If two lines cut by a transversal are parallel, then same side interior angles are supplementary.

87

65

43

21

Corresponding angles: 1+5, 2+6, 3+7, 4+8

Alternate interior angles: 3+6, 4+5

Alternate exterior angles: 1+8, 2+7

Same side interior angles: 3+5, 4+6

Same side exterior angles: 1+7, 2+8

3.4: Proving That Lines Are Parallel

• You know from 3.3 how to determine angle measures if you know the lines are parallel, but is the converse is also true?

• It is.

Theorems

• Converse of the Same-Side Interior Angles Theorem: If two lines are cut by a transversal in such a way that same side interior angles are supplementary, then the two lines are parallel.

• Converse of the Alternate Interior Angles theorem: if two lines are cut by a transversal in such a way that alternate interior angles are congruent, then the two lines are parallel.

Theorems

• Converse of the Alternate Exterior Angles Theorem: If two lines are cut by a transversal in such a way that alternate exterior angles are congruent, then the two lines are parallel.

• Theorem: If two coplanar lines are perpendicular to the same line then the two lines are parallel to each other.

• Theorem: If two coplanar lines are parallel to the same line, then the two lines are parallel to each other.

Review for Exam

• The website below offers important vocabulary terms and postulates which will help you review for the exam.

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Bibliography

"Parallels and Polygons." 1. Web. 8 Dec 2009. <http://www.mansfieldct.org/Schools/MMS/staff/labrec/ParallelsandPolygons.htm>.

Schultz, Hollowell, Ellis, Kennedy, Engelbrecht, Rutowski, . Geometry. Austin: Holt, Rinehart and Winston, 2001. 138-69. Print.