Download Ch 8 Notes

Geometry Definitions, Postulates, and Theorems
Chapter 8: Quadrilaterals
Section 8.1: Find Angle Measures in Polygons
Standards: 12.0 Students find and use measures of sides and of interior and exterior angles of triangles and polygons to
classify figures and solve problems.
Recall: A polygon is a plane figure that is formed by three or more segments called sides. Each side
intersects exactly two other sides – one at each endpoint called a vertex.
Ex. In each polygon below, draw all the diagonals from one vertex. Notice that this divides each
polygon into triangular regions.
***Theorem 8.2 – Polygon Interior Angles Theorem
.
Corollary to Theorem 8.1 – Interior Angles of a Quadrilateral
Corollary to Theorem 8.1 – The measure of each interior angle of a
regular n-gon is
# of Sides
3
4
5
6
7
8
9
10
12
n
Name of polygon
Sum of Interior Angles
Ex. Find the sum of the measures of the
interior angles of a 24 sided polygon.
Ex. The sum of the measures of the interior
angles of a convex polygon is 2340 .
Classify the polygon by the number of sides.
Ex. Find the value of x.
Ex. What is the measure of one interior angle
in a regular 20-sided polygon?
72°
136°
x°
127°
108°
Calculating Exterior Angles:
1200
600
1200
1200
600
600
***Theorem 8.2 – Polygon Exterior Angles Theorem
Sum
Regular
each 
int. 
Corollary to Theorem 8.2 – The measure of each exterior angle of a
regular n-gon is
ext. 
Ex. What is the value of x in the diagram shown?
89o
Ex. Each exterior angle of the regular n-gon
has a measure of 12 . Find the value of n.
67o
2xo
xo
Ex. A convex hexagon has exterior angles with
measures 34 , 49 , 58 , 67 , and 75 . What
is the measure an exterior angle at the
sixth vertex?
Ex. Find the measure of one exterior angle
of a regular 20-sided polygon.
Section 8.2: Use Properties of Parallelograms
Standards: 7.0 Students prove and use theorems involving the properties of parallel lines cut by a transversal, the
properties of quadrilaterals, and the properties of circles.
***Parallelogram –
***Theorem 8.3 – IF a quadrilateral is a parallelogram,
***Theorem 8.4 – IF a quadrilateral is a parallelogram,
Ex. In Parallelogram ABCD, find the values of x and y.
a.
D
A o
y
12
x4
B
125 o
C
b.
A
18
2 xo
D
60 o B
y5
C
***Theorem 8.5 – IF a quadrilateral is a parallelogram,
***Theorem 8.6 – IF a quadrilateral is a parallelogram,
Section 8.3: Show that a Quadrilateral is a Parallelogram
Standards: 7.0 Students prove and use theorems involving the properties of parallel lines cut by a transversal, the
properties of quadrilaterals, and the properties of circles.
***Theorem 8.7 – IF both pairs of opposite sides of a quadrilateral are congruent,
***Theorem 8.8 – IF both pairs of opposite angles of a quadrilateral are congruent,
***Theorem 8.9 – IF one pair of opposite sides of a quadrilateral are congruent
***Theorem 8.10 – IF the diagonals of a quadrilateral bisect each other,
Section 8.4: Properties of Rhombuses, Rectangles, and Squares
Standards: 7.0 Students prove and use theorems involving the properties of parallel lines cut by a transversal, the
properties of quadrilaterals, and the properties of circles. 17.0 Students prove theorems by using coordinate geometry,
including the midpoint of a line segment, the distance formula, and various forms of equations of lines and circles.
Rhombus –
Rectangle –
Square –
Rhombus Corollary – A quadrilateral is a rhombus
Rectangle Corollary – A quadrilateral is a rectangle
Square Corollary – A quadrilateral is a square
***Theorem 8.11 – A parallelogram is a rhombus
***Theorem 8.12 – A parallelogram is a rhombus
***Theorem 8.13 – A parallelogram is a rectangle
Section 8.5: Use Properties of Trapezoids and Kites
Standards: 7.0 Students prove and use theorems involving the properties of parallel lines cut by a transversal, the
properties of quadrilaterals, and the properties of circles.
Trapezoid –
Bases –
Base Angles –
Legs –
Isosceles Trapezoid –
***Theorem 8.14 – IF a trapezoid is isosceles,
***Theorem 8.15 – IF a trapezoid has a pair of congruent base angles,
***Theorem 8.16 – A trapezoid is isosceles
Midsegment of a Trapezoid –
***Theorem 8.17 – Midsegment Theorem for Trapezoids
***Theorem 8.18 – IF a quadrilateral is a kite,
P
Y
T
L
***Theorem 8.19 – IF a quadrilateral is a kite,
A
Section 8.6: Identify Special Quadrilaterals
Standards: 7.0 Students prove and use theorems involving the properties of parallel lines cut by a transversal, the
properties of quadrilaterals, and the properties of circles. 12.0 Students find and use measures of sides and of interior and
exterior angles of triangles and polygons to classify figures and solve problems.
Quadrilateral
Parallelogram
Rectangle
Rhombus
Trapezoid
Kite
Isosceles Trapezoid
Square
Each quadrilateral in the diagram has the properties of the quadrilateral linked above it. For example,
a rhombus has the properties of a parallelogram and a quadrilateral.
Ex. What quadrilateral has perpendicular diagonals?
Ex. What quadrilateral has congruent diagonals?
Ex. What quadrilateral has both pairs of opposite sides congruent?