chapter 8: quadrilaterals guided notes -...
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Geometry Fall Semester
Name: ______________
Chapter 8: Quadrilaterals
Guided Notes
CH. 8 Guided Notes, page 2
8.1 Find Angle Measures in Polygons
Term Definition Example
consecutive vertices
nonconsecutive
vertices
diagonal
Theorem 8.1 Polygon Interior Angles Theorem
The sum of the measures of the interior
angles of a convex n-gon is S = 180(n – 2)º.
Corollary to Theorem 8.1
Interior Angles of a
Quadrilateral
The sum of the measures of the interior
angles of a quadrilateral is 360°.
Theorem 8.2 Polygon Exterior Angles Theorem
The sum of the measures of the exterior
angles of a convex polygon, one angle at
each vertex, is 360°.
CH. 8 Guided Notes, page 3
Convex Polygon Number of Sides Number of Triangles
Sum of Angle Measures
Triangle 3 1 (1 • 180) = 180º Quadrilateral 4 2 (2 • 180) = 360º
Pentagon 5 3 (3 • 180) = 540º Hexagon 6 4 (4 • 180) = 720º Heptagon 7 5 (5 • 180) = 900º Octagon 8 6 (6 • 180) = 1080º n-gon n n – 2 180(n – 2)º
Examples:
1. The sum of the measures of the interior angles of a convex polygon is
1260° . Classify the polygon by the number of sides.
2. Find the value of x in each of the diagrams . a).
b).
CH. 8 Guided Notes, page 4 3. The base of a lamp is in the shape of a regular 15-gon. Find the measure of each interior angle and the measure of each exterior angle.
CH. 8 Guided Notes, page 5
8.2 Use Properties of Parallelograms
Term Definition Example
parallelogram
Theorem 8.3
If a quadrilateral is a parallelogram, then its
opposite sides are congruent.
Theorem 8.4
If a quadrilateral is a parallelogram, then its
opposite angles are congruent.
Theorem 8.5
If a quadrilateral is a parallelogram, then its
consecutive angles are supplementary.
Theorem 8.6
If a quadrilateral is a parallelogram, then its
diagonals bisect each other.
Example:
1. Find the values of x and y.
CH. 8 Guided Notes, page 6 2. As shown, a gate contains several parallelograms. Find
m!ADC when
m!DAB = 65°. 3. The diagonals of parallelogram
STUV intersect at point
W . Find the coordinates of
W.
CH. 8 Guided Notes, page 7
8.3 Show that a Quadrilateral is a Parallelogram
Term Definition Example
Theorem 8.7 (Converse of
Thm 8.3)
If both pairs of opposite sides of a
quadrilateral are congruent, then the
quadrilateral is a parallelogram.
Theorem 8.8 (Converse of
Thm 8.4)
If both pairs of opposite angles of a
quadrilateral are congruent, then the
quadrilateral is a parallelogram.
Theorem 8.9
If one pair of opposite sides of a
quadrilateral is congruent and parallel, then
the quadrilateral is a parallelogram.
Theorem 8.10
(Converse of
Thm 8.6)
If the diagonals of a quadrilateral bisect
each other, then the quadrilateral is a
parallelogram.
A quadrilateral is a parallelogram if any one of the following is true.
1. Both pairs of opposite sides are parallel. Definition 2. Both pairs of opposite sides are congruent. Theorem 8.7 3. Both pairs of opposite angles are congruent. Theorem 8.8 4. A pair of opposite sides is both congruent and parallel. Theorem 8.9 5. Diagonals bisect each other. Theorem 8.10
CH. 8 Guided Notes, page 8 Examples: 1. In the diagram at the right,
AB and
DC represent adjustable supports of a basketball hoop. Explain why
AD is always parallel to
BC . 2. The headlights of a car have the shape shown at the right. Explain how you know that
!B "!D .
3. For what value of x is quadrilateral PQRS a paralleogram?
CH. 8 Guided Notes, page 9 4. Show that quadrilateral KLMN is a parallelogram.
CH. 8 Guided Notes, page 10
8.4 Properties of Rhombuses, Rectangles, and Squares
Term Definition Example
rhombus
rectangle
square
Rhombus Corollary
A quadrilateral is a rhombus if and only if it
has four congruent sides.
Rectangle Corollary
A quadrilateral is a rectangle if and only if it
has four right angles.
Square Corollary
A quadrilateral is a square if and only if it is
a rhombus and a rectangle.
Theorem 8.11
A parallelogram is a rhombus if and only if
its diagonals are perpendicular.
Theorem 8.12
A parallelogram is a rhombus if and only if each diagonal
bisects a pair of opposite angles.
Theorem 8.13
A parallelogram is a rectangle if and only if its
diagonals are congruent.
CH. 8 Guided Notes, page 11
If a quadrilateral is a rectangle, then the following properties hold true.
1. Opposite sides are congruent and parallel. 2. Opposite angles are congruent. 3. Consecutive angles are supplementary. 4. Diagonals are congruent and bisect each other. 5. All four angles are right angles.
Examples: 1. For any rhombus
RSTV , decide whether the statement is always or sometimes true. Draw a sketch and explain your reasoning. a)
!S "!V b)
!T "!V 2. Classify this special quadrilateral. Explain your reasoning.
3. You are building a frame for a painting. The measurements of the frame are shown at the right. a) The frame must be a rectangle. Given the measurements in the diagram, can you assume that it is? b) You measure the diagonals of the frame. The diagonals are about 25.6 inches. What can you conclude about the shape of the frame?
CH. 8 Guided Notes, page 12
8.5 Use Properties of Trapezoids and Kites
Term Definition Example
trapezoid
parts of
trapezoids
1. Bases—
2. Legs—
3. Base Angles—
isosceles trapezoid
Theorem 8.14
If a trapezoid is isosceles, then each pair of
base angles is congruent.
Theorem 8.15
If a trapezoid has a pair of congruent base
angles, then it is an isosceles trapezoid.
Theorem 8.16
A trapezoid is isosceles if and only if its
diagonals are congruent.
CH. 8 Guided Notes, page 13 midsegment of a
trapezoid (median)
Theorem 8.17 Midsegment Theorem for Trapezoids
The midsegment of a trapezoid is parallel to
each base and its length is one half the sum
of the lengths of the bases.
kite
Theorem 8.18
If a quadrilateral is a kite, then its
diagonals are perpendicular.
Theorem 8.19
If a quadrilateral is a kite, then exactly one
pair of opposite angles is congruent.
Examples: 1. Show that
CDEF is a trapezoid.
CH. 8 Guided Notes, page 14 2. A shelf fitting into a cupboard in the corner of a kitchen is an isosceles trapezoid. Find
m!N,m!L, and
m!M.
3. In the diagram,
MN is the midsegment of trapezoid
PQRS. Find
MN.
4. Find
m!T in the kite shown at the right.
CH. 8 Guided Notes, page 15
8.6 Identify Special Quadrilaterals
Quadrilateral Hierarchy Diagram
Properties of Quadrilaterals Property Parallelogram Rectangle Rhombus Square Kite Trapezoid
All sides are ! . Exactly 1 pair of opposite sides are ! .
Both pairs of opposite sides are ! .
Both pairs of opposite sides are //.
Exactly one pair of opposite sides are //.
All angles are ! . Exactly one pair of opposite angles are ! .
Both pairs of opposite angles are ! .
Diagonals are ! . Diagonals are ! . Diagonals bisect each other.
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