Ch. 6 – Polygons and Quadrilaterals

Jose Pablo Reyes

Ch. 6 – Polygons and Quadrilaterals

Polygon: Any plane figure with 3 o more sides

Parts of a polygon:

side – one of the segments that is part of the polygon

Diagonal – a line that connects two vertices that are not a side

Vertex – the point where two segments meet

Interior angle – the angle that is formed inside the polygon by two adjacent sides

Exterior angle – the angle that is formed outside the polygon by two adjacent sides

Describe what a polygon is. parts of a polygon.

Polygon Examples

Convex polygon – is a polygon in which all vertices point out

Concave polygon – is a polygon in which at least one angle is pointing into the center of the figure

Equiangular – polygon in which all angles are congruent

Equilateral – polygon in which all sides are congruent

Describe convenx and concave polygons , equilateral and equiangular.

Convex and Concave, Equilangular and Equilateral polygons – Examples

Interior angles theorem of quadrilaterals:

Interior angles theorem for quadrilaterals

Interior angle theorem of a quadrilateral Examples

Theorem 6-2-4 : If a quadrilateral is a parallelogram its diagonals bisect each other converse: if the diagonals of a quadrilateral bisect each other then it is a parallelogram

Theorem 6-3-2: If both pairs of opposite sides of a quadrilateral are congruent, then it is a parallelogram converse: if in a quadrilateral both pairs of opposite sides are congruent then it is a parallelogram

4 theorems of parallelograms and their converse

Theorem 6-6-2: If a quadrilateral is a kite, then its diagonals are perpendicularconverse: if in a quadrilateral the diagonals are perpendicular, then it is a kite

Theorem 6-5-2: If the diagonals of a parallelogram are congruent, then it is a rectangleconverse: if a rectangle has congruent diagonals, then it is a parallelogram

A quadrilateral is a parallelogram if; - opposite sides are parallel - opposite angles are congruent - opposite sides are congruent - adjacent angles are supplementary - diagonals bisect each other

Theorem 6.10:

How to prove a quadrilateral is a parallelogram – Theorem 6.10

Proving a quadrilateral is a parallelogram Examples

Squares :4 sided regular

polygon, that has all sides and angles

congruent int. angles = 90

2 pairs of parallel lines

Rhombuses:quadrilateral with 4 congruent sides its diagonals are

perpendicular

Rectangles:4 sided polygon with congruent

angles int. angles = 90diagonals are

congruent Two pairs of

parallel lines

Comparing – Rhombuses, Squares and rectangles

Comparing: Rhombuses, Squares and rectangles – Examples

Theorem 6-4-3: If a quadrilateral is a rhombus, then it is a parallelogram

Theorem 6-4-4: If a parallelogram is a rhombus, then its diagonals are perpendicular

Theorem 6-4-5: If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles

Rhombus Theorems

Rhombus Theorems – Examples

Theorem 6-4-1: If a quadrilateral is a rectangle, then it is a parallelogram

Theorem 6-4-2: If a parallelogram is a rectangle, then its diagonals are congruent

Rectangle Theorems

Rectangle Theorems – Examples

Trapezoid: A quadrilateral with one pair of parallel lines, each parallel side is called a base, and the parts that are not parallel are called legs.

Isosceles trapezoid: when the legs are congruent

Theorem 6-6-3: if a quadrilateral is an isosceles trapezoid, then each pair of base angles are congruent

Theorem 6-6-4: if a trapezoid has one pair of congruent base angles, then the trapezoid is isosceles

Theorem 6-6-5: a trapezoid is isosceles if and only if its diagonals are congruent

Trapezoid and its theorems

Trapezoid Theorems – Examples

Theorem 6-6-6: The midsegment of a trapezoid is parallel to each base, and its length is one half the sum of the lengths of the bases

Trapezoid mid segment theorem

Kite: A quadrilateral with two pairs of congruent consecutive sides

Theorem 6-6-1: if a quadrilateral is a kite, then its diagonals are perpendicular

Theorem 6-6-2: if a quadrilateral is a kite, then a pair of opposite angles are congruent

Kite and its Theorems

Kite – Examples

To find the areas of a square, rectangle, triangle, parallelogram, trapezoid, kite and rhombus

3 area postulates and how they are used