Download x - Mrs. Malinda Young, M.Ed

CHAPTER 8
QUADRILATERALS
By Margo Pennington and Ashlynn Kolarik
SECTION 1
 A diagonal of a polygon is a segment that joins two nonconsecutive
vertices.
 Diagonals of polygons form triangles which can be useful in
calculating the sum of interior angles.
 Theorem 8.1- Polygon Interior Angles Theorem: The sum of the
measures of the interior angles of a convex n-gon is (n-2)*180◦
 Corollary to Theorem 8.1- Interior Angles of a Quadrilateral: The sum
of the measures of the exterior angles of a quadrilateral is 360◦
 Example 1- Find the sum of the interior angle measures
A pentagon has 5 sides. Use the polygon interior angles theorem to
find the sum of interior angles.
(5-2)*180◦
=540◦
TRY THIS!
Find the sum of the measures of the interior
angles of a convex heptagon.
SOLUTION
(n-2)180
n=7
5*180
=900◦
CONTINUED
 Example 2: Find the number of sides given the sum of interior angles.
The sum of the measures of the interior angles of a convex polygon is
720◦.
(n-2)*180◦=720◦
(n-2)=4
N=6, the polygon is a hexagon.
 Example 3: Find the value of x shown in the diagram.
x + 68 + 126 + 106 = 360
x + 300 = 360
x = 60
 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◦
TRY THIS!
Find the value of x.
∠1=2x°
∠2=72°
∠3=19°
∠4=5x-2°
∠5=31°
SOLUTION
2x + 5x + 72 + 19 + 31- 2 = 360
7x + 120 = 360
7x = 240
x ≈ 34.3
SECTION 2
 A parallelogram is a quadrilateral with both pairs of opposite sides parallel
 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.
 Example 1: Find the values of x and y.
x-7
x-7=23
y◦
x=30
90°
y=90
23
 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.
TRY THIS!
Find the values of x and y.
AB=y+12
BC=-4x+11
CD=2y-1
AD=2x-3
SOLUTION
y + 12 = 2y - 1
y = 13
2x – 3 = 4x + 11
x = -7
SECTION 3
 Theorem 8.7: If both pairs of opposite sides of a quadrilateral are
congruent, then the quadrilateral is a parallelogram.
 Theorem 8.8: If both pairs of opposite angles of a quadrilateral are
congruent, then the quadrilateral is a parallelogram.
 Example 1: Solve a real-world problem.
A carpenter wants to build a table that is a parallelogram. Find the value
of x and y that make both pairs of opposite angles congruent.
x+1=101
x+1
80
x=100
y=80
101
y
TRY THIS!
Find the values of x and y that make the quad a p-gram.
SOLUTION
5y + 15 = 110
5y = 95
y = 19
2x + 40 = 70
2x = 30
x = 15
CONTINUED
TRY THIS!
 Find the value of x.
SOLUTION
x + 40 = 2x + 18
x = 22
CONTINUED
 Section Wrap- Up~ Proving that A Quadrilateral is a Parallelogram:
1.) Opposite sides are parallel.
2.) Opposite sides are congruent.
3.) Opposite angles are congruent.
4.) One pair of opposite sides are congruent and parallel.
5.) Diagonals bisect each other.
SECTION 4
 A rhombus is a parallelogram with four congruent sides.
 A rectangle is a parallelogram with four right angles.
 A square is a parallelogram with four congruent sides and four right
angles.
 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.
CONTINUED
 Comparing and Contrasting Parallelograms:
TRY THIS!
Classify the Quadrilaterals
a.)
b.)
SOLUTION
a.) Rhombus because all sides are congruent.
b.) Square because all sides are congruent
and all angles are 90°.
CONTINUED
 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.
 Example 2: Solve a real world problem
An inspector wants to confirm that a certain lap pool is a rectangle. Can he
assume based on the measurements that it is one?
88
92
92
88
e He cannot assume this is a rectangle because the angles should each be
ninety degrees.
PROPERTIES OF QUADRILATERALS
Rectangle
Property
All sides congruent
Rhombus
Square
X
X
Diagonals bisect each other
X
X
X
X
Both pairs opp. Sides congruent
X
X
X
X
Diagonals are Congruent
X
X
X
X
Both pairs of opp. Sides parallel
X
X
X
X
X
X
Diagonals are perpendicular
1 pair of opp. Sides are parallel.
X
X
X
X
1 pair opp. angles are congruent
X
X
X
X
X
X
X
All angles congruent
Kite
Trapezoid
X
X