Download Geo Unit 6

Geometry
Unit 6
Quadrilaterals
Unit 6
Properties of Polygons
• Formed by three or more consecutive
segments.
• The segments form the sides of the polygon.
• Each side intersects two other sides at its
endpoints.
• The intersections are the vertices of the
polygon.
• No three consecutive vertices of a polygon are
collinear.
Unit 6
Common Polygons
•
•
•
•
•
•
Triangle: three sides
Quadrilateral: four sides
Pentagon: five sides
Hexagon: six sides
Heptagon: seven sides
Octagon: eight sides
• Nonagon: nine sides
• Decagon: ten sides
• Dodecagon: twelve
sides
• n-gon: n sides
Unit 6
Convex vs. Concave
• Convex
• Concave
– All points of any
segment joining two
points in the interior of
the polygon must also be
in the interior of the
polygon.
– At least one segment
joining two points in the
interior of the polygon
also contain points in the
exterior of the polygon.
A
A
B
B
Unit 6
Polygon Interior/Exterior Angles
• Polygon Interior Angles
Theorem
– The sum of the interior angles of
a convex polygon with n sides is:
n  2180 
• Regular Polygon
• Polygon Exterior
Angles Theorem
– The sum of the measures of the
exterior angles, one at each
vertex, of a convex polygon is
360º.
1
– The measure of each interior
angle is:
n  2 180
n

2
8

– The measure of the Central Angle
is:
360
n
6
1
5
3
7
2
4
3
4
6
5
Unit 6
Quadrilaterals
• Parallelogram: a quadrilateral with both pairs of
opposite sides parallel.
• Rectangle: a parallelogram with four right angles.
• Rhombus: a parallelogram with four congruent sides.
• Square: a parallelogram with four right angles and
four congruent sides.
• Trapezoid: a quadrilateral with exactly one pair of
parallel sides.
• Kite: a quadrilateral with two distinct pairs of
congruent, adjacent sides.
Unit 6
Quadrilateral
Interior/Exterior Angles
• Quadrilateral Sum Theorem
– The sum of the interior angles of a quadrilateral is 360º.
• Quadrilateral Exterior Angles
– The sum of the measures of the exterior angles, one at each
vertex, of a convex quadrilateral is 360º.
• Regular Quadrilateral (Square)
– The measure of each interior angle is 90º.
Unit 6
Properties of Parallelograms
•
•
•
•
•
Opposite sides are parallel.
Opposite sides are congruent.
Opposite angles are congruent.
Consecutive angles are supplementary.
The diagonals bisect each other.
Unit 6
Proving a
Quadrilateral is a Parallelogram
• Prove that both pairs of opposite sides are parallel.
• Prove that both pairs of opposite sides are congruent.
• Prove that one pair of opposite sides is both congruent and
parallel.
• Prove that both pairs of opposite angles are congruent.
• Prove that consecutive angles are supplementary.
• Prove that the diagonals bisect each other.
• The quadrilateral formed by joining the midpoints of the
consecutive sides of another quadrilateral is a parallelogram.
Unit 6
Common Parallelograms
• Parallelogram: a quadrilateral with both pairs
of opposite sides parallel.
• Rectangle: a parallelogram with four right
angles.
• Rhombus: a parallelogram with four
congruent sides.
• Square: a parallelogram with four right angles
and four congruent sides.
Unit 6
Summary of Properties
• Rectangles
– Has all the properties of a parallelogram.
– All angles are right angles.
– The diagonals are congruent.
• Rhombuses
–
–
–
–
Has all the properties of a parallelogram.
All sides are congruent.
The diagonals are perpendicular.
The diagonals bisect opposite angles.
• Squares
– Has all the properties of a parallelogram, rectangle, and
rhombus.
Unit 6
Properties of a Trapezoid
• One pair of opposite sides are parallel and are called the bases.
• Each pair of base angles of an isosceles trapezoid is congruent
and the legs are congruent.
• The diagonals of an isosceles trapezoid are congruent.
base
legs
legs
base
Unit 6
Midsegment Theorem of
Trapezoids and Triangles
• Trapezoids
• Triangles
– The median (or midsegment)
of a trapezoid is parallel to the
bases.
– The length of the midsegment
is equal to half the sum of the
lengths of the bases (average).
base
legs
Midsegment
b1  b2 
– The midsegment of a triangle
is parallel to the third side.
– The length of the midsegment
of a triangle is one-half the
length of the third side.
– A line contains the midpoint
of one side of a triangle and is
parallel to another side bisects
the third side.
legs
2
base
Unit 6
Properties of a Kite
• a quadrilateral with exactly two pairs of consecutive congruent
sides
• diagonals are perpendicular
Unit 6