Journal 6By: Maria Jose Diaz-Duran

Describe what a polygon is. Include a discussion about the parts of a polygon. Also

compare and contrast a convex with a concave polygon. Compare and contrast equilateral and

equiangular. Give 3 examples of each.• A polygon is closed figure

formed by 3 or more segments.

Convex Concave

No points to the center of the figure, no diagonal to the exterior also a regular polygon is always convex.

If it goes to the center of the figure, contains points in the exterior of a polygon

Equilateral Equiangular

All sides are congruent A polygon in which all angels are congruent

Examples

Explain the Interior angles theorem for quadrilaterals. Give

at least 3 examples.

• The sum of the interior angle measures of a convex polygon with n sides is (n-2)180

ExamplesA. Find the sum of the interior angel measures of a convex octagon:(n-2)180(8-2)1801080B. Find the measures of each interior angle of a regular nonagonStep 1: (n-2)180(9-2)1801260Step 2: Find the measures of 1 interior angle1260/9= 140C. Find the measure of each interior angle of quadrilateral PQRS.(4-2)180=360M<p + M<Q + M<R +M<S=360 C+ 3C+ C+ 3C=3608C=360C=45

Describe the 4 theorems of parallelograms and their converse

and explain how they are used. Give at least 3 examples of each.Theorems Converse

If a quadrilateral is a parallelogram, then its opposite sides are congruent

If its opposite sides are congruent, then a quadrilateral is a parallelogram

If a quadrilateral is a parallelogram, then its opposite angles are congruent

If the opposite angles are congruent, then a quadrilateral is a parallelogram

If a quadrilateral is a parallelogram, then its consecutive angles are supplementary

If its consecutive angles are supplementary then a quadrilateral is a parallelogram

If a quadrilateral is a parallelogram, then its diagonal bisect each other

If its diagonals bisect each other then a quadrilateral is a parallelogram

Examples

If a quadrilateral is a parallelogram, then its opposite sides are congruent

If a quadrilateral is a parallelogram, then its opposite angles are congruent

If a quadrilateral is a parallelogram, then its consecutive angles are supplementary

If a quadrilateral is a parallelogram, then its diagonal bisect each other

Describe how to prove that a quadrilateral is a parallelogram. Include an explanation about theorem 6.10. Give

at least 3 examples of each.

Theorems

If one pair of opposite sides of a quadrilateral are parallel and congruent, then the quadrilateral is a parallelogram

If both pairs of opposite sides of a quadrilateral are congruent then the quadrilateral is a parallelogram

If both pair of opposite angels of a quadrilateral are congruent then then quadrilateral is a parallelogram.

Examples

If one pair of opposite sides of a quadrilateral are parallel and congruent, then the quadrilateral is a parallelogram

If both pairs of opposite sides of a quadrilateral are congruent then the quadrilateral is a parallelogram

If both pair of opposite angels of a quadrilateral are congruent then then quadrilateral is a parallelogram.

Compare and contrast a rhombus with a square with a rectangle. Describe the

rhombus, square and rectangle theorems. Give at least 3 examples of

each.Rectangle Square Rhombus

A quadrilateral with four right angles.

A quadrilateral with four right angles and four congruent sides

A quadrilateral with four congruent sides.

Theorems Theorems Theorems

If a quadrilateral is a rectangle, then it is a parallelogram

A quadrilateral is a square if and only if it is a rhombus and a rectangle.

If a quadrilateral is a rhombus then its is a parallelogram

If a parallelogram is a rectangle then its diagonals are congruent

If a parallelogram is a rhombus then its diagonals are perpendicular

If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles.

Examples

Describe a trapezoid. Explain the trapezoidal theorems.

Give at least 3 examples each

 Trapezoid

A quadrilateral with exactly one pair of parallel sides, each of the parallel sides is called a base and the nonparallel sides are called legs.

Theorems Isosceles Trapezoids:

if a quadrilateral is an isosceles trapezoid, then each pair of base angles are congruent.

if a trapezoid has one pair of congruent base angles, then the trapezoid is isosceles.

A trapezoid is isosceles if and only if its diagonals are congruent

Trapezoid Midsegment Theorems:

The midsegment of a trapezoid is parallel to each base and its length is one half the sum of the lengths of the bases.

Examples

Describe a kite. Explain the kite theorems. Give at least 3

examples of each.

 Kite

A quadrilateral with exactly two pairs of consecutive sides.

Theorems

If a quadrilateral is a kite, then its diagonals are perpendicular.

If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent.

Examples

If a quadrilateral is a kite, then its diagonals are perpendicular.

If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent.

Describe how to find the areas of a square, rectangle, triangle,

parallelogram, trapezoid, kite and rhombus. Give at least 3 examples of each.

 Areas:Square To find the area of a square,

multiply the lengths of two sides together (XxX=X2)

Rectangle To find the area of a rectangle, just multiply the length times the width: (LxW)

Triangle 1/2 xbxh or bxh/2

Parallelogram To find the area of a parallelogram, just multiply the base length (b) times the height (h) (BxH)

Trapezoid ½ H(B+b)

Kite To find the area of a kite, multiply the lengths of the two diagonals and divide by 2 (1/2AB)

Rhombus To find the area of a rhombus, multiply the lengths of the two diagonals and divide by 2 (1/2AB)

Examples

Describe the 3 area postulates and how they are

used. Give at least 3 examples of each.

• 1. Area of a Square Postulate: The area of a square is the square of the length of a side.

• 2. Area Congruence Postulate: If two closed figures are congruent then they have the same area.

• 3. Area Addition Postulate: The area of a region is the sum of the areas of its non-overlapping parts.

Examples

Area of a Square Postulate

Area Congruence Postulate

Area Addition Postulate