Unit 8 - ParallelogramsPolygon & Quadrilaterals

Polygon

A closed figure bounded by three or more line segments

side of a polygon

segment

endpoints of segment

vertices of the polygon

diagonal of the polygon

segment joining two nonconsecutive vertices

Triangle

polygon with 3 sides

Quadrilateral

Polygon with 4 sides

Pentagon

Polygon with 5 sides

Hexagon

Polygon with 6 sides

Septagon

Polygon with 7 sides

Octagon

Polygon with 8 sides

Nonagon

Polygon with 9 sides

Decagon

Polygon with 10 sides

Dodecagon

polygon with 12 sides

N-gon

polygon with n sides

Regular Polygon

A polygon whose sides and angles are all congruent

Exterior Angle of a Polygon

An angle between any side of the polygon and a line extended from the next side

sum of the measures of the interior angles of a polygon with n sides

180(n-2)

Each interior angle of a regular polygon with n sides

(180(n-2))/n

sum of the measures of the exterior angles of any polygon

360

Each exterior angle of a regular polygon with n sides

360/n

Quadrilateral

a polygon is a quadrilateral if it contains 4 coplanar points, no 3 collinear, closed figure with 4 sides; convex or concave

Convex

diagonals always inside

Concave

diagonals sometimes outside

Kite

a quadrilateral with two pairs of adjacent sides congruent

diagonals of the kite

perpendicular

diagonal that divides the kite into two congruent triangle

perpendicular bisector of the other diagonal

diagonal of a kite formed between congruent sides

angle bisector

angles formed between two non congruent sides of a kite

are congruent

trapezoid

quadrilateral with one pair of opposite sides that are parallel (inclusive definition)

Isosceles Trapezoid

trapezoid with congruent base angles

trapezoid

quadrilateral in which exactly one pair of opposite sides are parallel (exclusive)

Isosceles Trapezoid

trapezoid with congruent legs

Legs

non parallel sides

In an isosceles trapezoid, base angles are

congruent

In an isosceles trapezoid, diagonals are

congruent

In an isosceles trapezoid, one pair of opposite sides are

congruent

How do we prove that a quadrilateral is an isosceles trapezoid?

If one pair of base angles of a trapezoid are congruent, then the trapezoid is isosceles.
If a trapezoid has congruent diagonals, then the trapezoid is isosceles.
If a trapezoid has congruent legs, then the trapezoid is isosceles

Parallelogram

quadrilateral in which both pairs of opposite sides are parallel

Each diagonal separates a parallelogram into

2 congruent triangles

In a parallelogram, both pairs of _________ are congruent

opposite sides

Any 2 ______ angles of a parallelogram are supplementary

consecutive

Any 2 ________ angles of a parallelogram are congruent

opposite

The diagonals of a parallelogram _______ each other

bisect

Ways to Prove a quadrilateral is a parallelogram:

Def of parallelogram (prove both pairs of opposite sides are parallel)
If both pairs of opposite sides of a quadrilateral are congruent, then it is a parallelogram.
If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.
If one pair of opposite sides of a quadrilateral are both parallel and congruent, then it is a parallelogram.

Rhombus

quadrilateral with 4 congruent sides; also parallelogram, kite, and isosceles trapezoid (inclusive)

Rectangle

A quadrilateral with 4 congruent angles. Therefore, each angle of a rectangles is a right angle.

Square

quadrilateral with 4 congruent sides and angles

In a rhombus, the diagonals _______ the angles

bisect

In a rhombus, the diagonals are

perpendicular

In a rectangle, the diagonals

are congruent

If a parallelogram has one _______, it is a rectangle.

right angle

If a parallelogram has ________, it is a rectangle.

congruent diagonals

If 2 ______ sides of a parallelogram are congruent, then it is a rhombus.

consecutive

If the diagonals of a parallelogram _______ the angles, then it is a rhombus.

bisects

If a _________ of a parallelogram are ___________, then it is a rhombus.

diagonals; perpendicular

If a quadrilateral is both a rectangle and a rhombus

, then it is a square.

If the diagonals of a _____ are perpendicular, then it is a square.

If the diagonals of a rectangle are perpendicular, then it is a square.

The midline theorem

The segment between the midpoints of two sides of a triangle is parallel to the third side and half its length.

The quadrilateral formed by joining the midpoints of consecutive sides of a quadrilateral is always

a parallelogram

The quadrilateral formed by joining the midpoints of consecutive sides of a rhombus is always a

rectangle

The quadrilateral formed by joining the midpoints of consecutive sides of a rectangle or any shape with congruent diagonals (isosceles trapezoid) is always a

rhombus

The segments joining the midpoints of opposite sides of any quadrilateral

bisect each other

If a transversal, t, Intersects 2 lines a and b in points A and B,

then a and b intercept the segment AB on the Transversal.

If 3 parallel lines intercept congruent segments on 1 transversal, T,

then they intercept congruent segments on every transversal T' which is parallel to T.

If 3 or more parallel lines intercept congruent segments on 1 transversal, t,

then they intercept congruent segments on any other transversal.

Converse of Midline Theorem:

If a line bisects one side of a triangle and is parallel to a second side, then it bisects the third side.

Median of a trapezoid

Segment joining the midpoints of the nonparallel sides.

The median of the trapezoid is _____ to the bases.

parallel

The median of a trapezoid ______ each diagonal of the trapezoid.

bisects