Post on 19-Jan-2016
Sum of Interior Angles of a Polygon
•Th. 6.1 – Polygon Interior Angles Theorem
•The sum of the measures of the interior angles of a convex n-gon is 180o(n – 2)
•Let = sum = 180o(n – 2)
Sum of Interior Angles of a Polygon
•Corollary to Th. 6.1
•The measure of each interior angle of a regular n – gon is 1/n(180o)(n – 2)
•m = 1/n(180o)(n – 2)
•Th. 6.2 Polygon Exterior Angles Theorem –
•The sum of the measures of the exterior angles, one from each vertex, of a convex polygon is 360o.
= 360
•Corollary to Th. 6.2 –
•The measure of each exterior angle of a regular n – gon is 1/n(360o).
•m = 1/n(360o).
Ex. 1. The measure of each angle of a regular n – gon is 160o. How many sides does the polygon have? (what is n?)
Ex. 2. The measure of each exterior angle of a regular polygon is 72o. How many sides does the polygon have?
Ex. 3 Find the measure of each interior angle of the quadrilateral shown below.
A B
CD
x + 30x + 90
x + 60 x
276/ 11 – 33 odd, 34
You are shown part of a convex n-gon. The pattern of congruent angles continues around the polygon. Find n. (hint, consider exterior angles)
142o
158o
Parallelograms
Proving Quadrilaterals are Parallelograms
W12
oN
– 5
62.5
8’
A B
CD
Definition – If both pairs of opposite sides of a quadrilateral are parallel, then the quadrilateral is a parallelogram.
AB || DC, AD || BC
A B
CD
Th. 6.7 – If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
AB DC, AD BC
A B
CD
Th. 6.8 – If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
A C, B D
A B
CD
Th. 6.9 – If an angle of a quadrilateral is supplementary to both consecutive angles, then the quadrilateral is a parallelogram.
A is supplementary to B and D
A B
CD
Th. 6.10 – If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
AC bisects DB, DB bisects AC,
AE EC, BE ED
E
A B
CD
Th. 6.11 – If one pair of opposite sides of a quadrilateral is parallel and congruent, then the quadrilateral is a parallelogram.
AB DC, AB || DC
P Q
U
RS
G:, PU UR P: PQRS is a parallelogram
1., PT UR 1. Given
If diagonals bisect each other the quadrilateral is a parallelogram
290/1 – 18, 20, 22 – 26
Sid Gilman
Bill Walsh
Sam Wyche Fassel
Mike Holmgren
Dennis Green
George Seifert
Wayne Coslet
MikeShanahan
BradMusgrave
Steve Mariucci
RayRhodes
MikeSherman
Quadrilaterals
Parallelograms TrapezoidsPerpendicular
DiagonalsOthers
No Properties
Rectangles Rhombus
Squares
Kites Others
TT
Special Parallelograms
Rhombus – a parallelogram that has all 4 sides congruent.
Rectangle – a parallelogram that has 4 right angles.
Square – a parallelogram that is both a rhombus and a rectangle.
Proving Special Parallelograms
6.12 – A parallelogram is a rhombus iff its diagonal are perpendicular.
6.13 – A parallelogram is a rhombus iff, each diagonal bisects a pair of opposite angles.
6.15 – A quadrilateral is a rhombus iff it has 4 congruent sides.
6.14 – A parallelogram is a rectangle iff, its diagonals are congruent.
CD
BA
AC BD
6.16 – A quadrilateral is a rectangle iff it has 4 right angles.
Proving Special Parallelograms
296/1 – 12, 21 – 24, 34
Trapezoids
A quadrilateral with exactly one pair of parallel sides.
Base
Base
LegLeg
Isosceles Trapezoid – Legs are congruent
Trapezoid has two pairs of base angles
Trapezoids
6.17 – If a trapezoid is isosceles, then each pair of base angles is congruent.
6.19 – If a trapezoid has one pair of congruent base angles, then it is isosceles.
6.18 - If a trapezoid is isosceles, then its diagonals are congruent.
6.20 – If a trapezoid has congruent diagonals, then it is isosceles.
Trapezoids
YX
D C
BA
Midsegment of a Trapezoid – connects the midpoints of the legs of the trapezoid
6.21 – The midsegment of a trapezoid is parallel to each base, and its length is half the sum of the lengths of the bases.
b1
b2
2)(
2
1 2121
bbbbXY
18
12
?
?
20
27
2x + 2
3x – 1
3x + 3
302/1 – 10, 11 – 31 odd, 37 – 41odd
A B
CD
AB DC, A C