Download Geometry Notes- Unit 5

Geometry Unit 5 - Notes
Polygons
Syllabus Objective: 5.1 - The student will differentiate among polygons by their
attributes.
Review terms:
1) segment
2) vertex
3) collinear
4) intersect
Polygon- a plane figure bounded by a closed path, composed of a finite sequence of straight line
segments. These segments are called its sides, and the points where two edges meet are the
polygon's vertices.
Side - one of the line segments that make up the
polygon.
Polygon Parts
Vertex - point where two sides meet. Two or more of
these points are called vertices.
Diagonal - a line connecting two vertices that isn't a
side.
Interior Angle - Angle formed by two adjacent sides
inside the polygon.
Exterior Angle - Angle formed by two adjacent sides
outside the polygon.
Unit 5
Polygon Type
Number of Sides
Triangle
Quadrilateral
Pentagon
Hexagon
Heptagon
Octagon
Nonagon
Decagon
Hendecagon
Dodecagon
n-gon
3
4
5
6
7
8
9
10
11
12
n
Polygons
Page 1 of 20
*Take the opportunity to relate parts of the polygon name to real life objects and concepts.
For instance, a pentagon has five sides. Show a picture of the Pentagon. The building is
called the Pentagon due to its shape. Show a picture of an octopus. It has eight legs just
like an octagon has eight sides. A decade is ten years and a decagon has ten sides. A tricycle
has three wheels just like the triangle has three sides! A quadrilateral has four sides just
like a four-wheeler is called a “quad” for short.
Convex Polygon - a polygon for which there is no line containing a side of the polygon that also
contains a point in the interior of the polygon.
Polygon ABCD is a convex quadrilateral.
A
Interior
D
B
C
Concave Polygon - a polygon for which there is a line containing a side of the polygon that also
contains a point in the interior of the polygon.
Polygon EFGHIJKL is a concave octagon.
F
E
Interior
H
L
G
I
K
I always draw a teddy bear hiding in the cave. I’m not a good artist, but
they remember. Probably because I’m so bad!!
J
Examples: Determine which of the following figures is a polygon. If it is a
polygon, state whether it is a convex or concave polygon.
A
B
C
D
E
Solutions:
“A” and “C” are polygons. “A” is concave and “C” is convex.
“B” is not a polygon because some of its sides intersect more than two other sides.
“D” is not a polygon because it has a side that is not a segment.
“E” is not a polygon because it is not closed, two of its sides intersect only one
other side.
Unit 5
Polygons
Page 2 of 20
Equilateral - a polygon with all sides congruent to each other.
Equiangular - a polygon with all of its interior angles congruent to each other.
Regular - a polygon that is both equilateral and equiangular.
Examples: Determine if the polygon is convex or concave; appear to be
equiangular, equilateral, or regular and state the name of the polygon.
A
B
C
Solutions:
“A” is an equilateral, convex pentagon. It has five segments that are congruent to
each other, but it is not equiangular. Therefore, it is not a regular polygon.
“B” is equiangular and equilateral. Thus, it is a regular, convex triangle.
“C” is an equilateral, concave decagon. It is not equiangular, so it is not a regular
decagon.
Syllabus Objectives: 5.2 - The student will classify polygons by their special properties.
Diagonal - a segment that joins two nonconsecutive vertices in a polygon.
The green (internal) segments in the three polygons above are examples of diagonals.
Interior Angles of a Quadrilateral Theorem: The sum of the four angles of any quadrilateral
is 360°.
Unit 5
Polygons
Page 3 of 20
Types of Quadrilaterals:
Quadrilateral
Parallelogram
Trapezoid
Kite
Rhombus
Rectangle
Isosceles Trapezoid
Square
Quadrilateral – four sided polygon
When referring to the flow chart above, the quadrilateral at the top center is a four-sided
polygon. The four polygons on the left side all have opposite sides parallel and congruent.
Therefore, the parallelogram, rectangle, rhombus, and square are all parallelograms.
All parallelograms have opposite sides congruent, opposite angles are congruent, the diagonals
bisect each other, and consecutive angles are supplementary.
The quadrilaterals on the right side are not parallelograms. The trapezoid is a quadrilateral
that has one pair of opposite sides parallel. If the legs, or base angles, are congruent, then it is
an isosceles trapezoid. {Link concept to isosceles triangles.} The diagonals of an isosceles
trapezoid are congruent just like those of the rectangle; however they do not bisect each
other.
The diagonals of a kite are perpendicular like those in the rhombus; however they do not bisect
each other. Only one pair of opposite angles is congruent to each other. The other pair of
opposite angles is bisected by the diagonal that passes through them.
************************************************************************************
Parallelogram – A quadrilateral with both pairs of opposite sides parallel. The parallelogram
symbol is
.
Properties of Being a Parallelogram
Theorem: If a quadrilateral is a parallelogram, then its opposite sides are congruent.
{Students often confuse this with the definition of parallelogram. The definition (above)
does not indicate congruence of sides.}
.
Theorem: If a quadrilateral is a parallelogram, then its opposite angles are congruent.
Theorem: If a quadrilateral is a parallelogram, then its consecutive angles are supplementary.
Theorem: If a quadrilateral is a parallelogram, then its diagonals bisect each other.
Unit 5
Polygons
Page 4 of 20
************************************************************************************
Rectangle – A parallelogram with four right angles.
Properties of Being a Rectangle
The rectangle is equiangular. Its diagonals not only bisect each other but they are also
congruent to each other.
************************************************************************************
Rhombus – A parallelogram with four congruent sides.
Properties of Being a Rhombus
The rhombus is equilateral. Its diagonals not only bisect each other but they intersect
perpendicularly. The diagonals of a rhombus bisect a pair of opposite angles.
************************************************************************************
Square – A parallelogram with four congruent sides and four right angles.
Properties of Being a Square
The square is a regular quadrilateral. It has all the properties of the quadrilateral,
parallelogram, rectangle, and rhombus.
************************************************************************************
Trapezoid – A quadrilateral with exactly one pair of parallel sides, called bases. The nonparallel
sides are legs.
Midsegment of a trapezoid – a segment that connects the midpoints of the legs of a trapezoid.
The midsegment of a trapezoid is half the sum of the bases. {Link concept to the midsegment
of a triangle, imagine a trapezoid with one base length equal to 0. What would be the best name
for that figure? A triangle.}
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 – a quadrilateral that has two pairs of consecutive congruent sides, but in which opposite
sides are not congruent.
In a kite, the diagonals are perpendicular, one pair of opposite angles are congruent and the
other pair are bisected by the diagonal.
************************************************************************************
Unit 5
Polygons
Page 5 of 20
*Below is a table that shows the different types of quadrilaterals and the properties that
they have.
Property
Quad
║ogram
Rectangle
Rhombus
Square
Trapezoid
Isos. Trap.
Kite
4 sides
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
Interior angle
sum 360°.
Both pairs of
opp. sides & .
Exactly 1 pair of
opp. sides & .
Both pairs of
opp. sides ≅ .
Exactly 1 pair of
opp. sides ≅ .
X
X
X
X
≅.
All angles are ≅ .
All sides are
X
Exactly 1 pair of
opp. angles is.
Diagonals are ≅ .
X
X
X
X
X
X
X
⊥.
Diagonals bisect
each other.
Given:
X
X
Both pairs of opp.
angles are ≅ .
Diagonals are
X
X
X
X
X
X
X
X
X
T
W
.RSTW
1
Prove: ΔRST ≅ ΔTWR
4
Statements
3
2
R
Unit 5
X
S
Reasons
1) RSTW is a & ogram
1) Given
2) RS &WT
2) Def. of & ogram
3) ∠1 ≅ ∠2
3) 2
4) RT ≅ RT
4) Reflexive
5) RW & ST
5) Def. of & ogram
6) ∠3 ≅ ∠4
7) ΔRST ≅ ΔTWR
6) 2
& lines cut by trans., alt. int. ∠'s
& lines cut by trans., alt. int. ∠'s
≅
≅
7) ASA
Polygons
Page 6 of 20
Knowing the diagonal separates a parallelogram into 2 congruent triangles suggests some more
relationships.
o
The opposite sides of a parallelogram are congruent.
o
The opposite angles of a parallelogram are congruent.
Both of those can be proven by adding CPCTC to the last proof.
Many students mistakenly think the opposite sides of a parallelogram are equal by definition.
That’s not true, the definition states the opposites sides are parallel. This proof allows us to
show they are also equal or congruent.
**Inner Rectangle**
The Problem:
Let ABCD be a convex quadrilateral and let the angle bisectors of the angles at A,
B, C, D form the sides of a second quadrilateral PQRS.
What can you say about the original quadrilateral ABCD , if PQRS is to be a
rectangle?
Solution:
ABCD must be a parallelogram.
For suppose PA , PB are the bisectors of the angles at A and B. Then
m ∠APB = 90° if and only if m ∠PAB + m ∠PBA = 90° ,that is, if and only if
2m ∠PAB + 2m ∠PBA = 2 ( 90° ) → m ∠DAB + m ∠ABC = 180° , which in turn happens if
HJJG
HJJG
and only if the lines DA and BC are parallel. Similarly at the other vertices.
Syllabus Objective: 5.3 - The student will find the sum of the measures of the interior
angles of a polygon.
Review the Triangle Sum Theorem. {Show students how to derive the formula from
constructing diagonals in a polygon from a single vertex and applying the Triangle Sum
Theorem.}
Polygon Interior Angle Theorem: The sum of the measures of the interior angles of a convex
n-gon is (n - 2)(180°).
Unit 5
Polygons
Page 7 of 20
Corollary to Polygon Interior Angle Theorem: The measure of each interior angle of a regular
( n − 2 ) 180° .
n-gon is
n
Table of Polygon values:
Number of ∆s
Inside the
Polygon
Sum of the
Measures of the
Interior Angles
Polygon Type
Number of
Sides (n)
Triangle
3
1
1(180°) = 180°
Quadrilateral
4
2
2(180°) = 360°
Pentagon
5
Hexagon
6
Heptagon
7
5
5(180°) = 900°
Octagon
8
6
6(180°) = 1080°
Nonagon
9
7
7(180°) = 1260°
Decagon
10
8
8(180°) = 1440°
Hendecagon
11
9
9(180°) = 1620°
Dodecagon
12
10
10(180°) = 1800°
n-gon
n
n-2
(n - 2)(180°)
Measure of each
Interior Angle
(regular polygon)
1(180°)
3
2(180°)
4
3(180°)
5
4(180°)
6
5(180°)
7
6(180°)
8
7(180°)
9
8(180°)
10
9(180°)
11
10(180°)
12
(n - 2)(180°)
3(180°) = 540°
3
4(180°) = 720°
4
n
Example: Find the measure of each interior angle.
First, the sum of the interior angles is found to be (3)180 = 540°.
U
x – 8 + 3x – 11 + x + 8 + x + 2x + 7 = 540
8x – 4 = 540
8x = 544
x = 68
Solutions:
m ∠U = 60°, m ∠V = 193°, m ∠W = 76°, m ∠Y = 68°, and m ∠Z = 143° .
Unit 5
Polygons
V
(x - 8)°
Z
(3x - 11)°
W
(x + 8)°
(2x + 7)°
x°
Y
Page 8 of 20
Syllabus Objective: 5.4 - The student will solve problems involving properties of special
quadrilaterals.
Review the special properties of all quadrilaterals and special angles created when parallel lines
are cut by a transversal.
Example: Find the value of each variable in a parallelogram.
8
95°
a
11
x
b
x= ?
a= ?
y
y= ?
b= ?
Solutions:
x = 11
a = 85°
y=8
b= 95°
Example: Find the angle measurements of trapezoid ABCD. AB & CD {Review
AIA, AEA, CIA, and CA.}
A
D
45°
B
60°
C
m ∠DAB =___?___
Solutions:
m ∠DAB = 135°
m ∠CBA = ____?___
m ∠CBA =
120°
Example: If KLMN is a parallelogram, find the value of x and y.
Opp. sides ≅ , so 9x = 36 → x = 4.
Diag. bisect each other, so 3y – 7 = y + 15 →
2y = 22 → y = 11
L
K
3y-7
J
36
9x
y+15
N
M
**The Sliced Cube**
The picture shows a cube of side length 2 units. M and N are the mid-points of
sides DE and BG.
Unit 5
Polygons
Page 9 of 20
The Problem:
What sort of shape is ANHM ?
Calculate the lengths of AH and MN. Any comment about your previous answer
now?
Find the area of the shape ANHM.
Solution:
This concerns the ‘equal-diagonal’ property of a square.
Clearly all four sides of the required shape are equal in length, so the choice is
between a square (the obvious choice !) and a rhombus.
We can use Pythagoras to find that AH is 2 3 units in length, while MN is
2 2 units in length. ANHM cannot be a square if it has unequal diagonals, so it
must be a rhombus.
Finally, the area of a rhombus is half the product of its diagonals = 2 6 units².
Syllabus Objective: 5.5 - The student will write proofs involving special quadrilaterals.
Example: Write a two-column proof.
G
F
.
Given:
FGHJ
Prove: ∠F ≅ ∠H , ∠J ≅ ∠G
Statements
1)
H
J
Reasons
. FGHJ
1) Given
2) Def. of Parallelogram
2) FG & JH , FJ & GH
∠F and ∠J are supplementary,
3) ∠J and ∠H are supplementary,
∠H and ∠G are supplementary
3) ║ lines cut by a transversal, CIA
are supp.
4) ∠F ≅ ∠H , ∠J ≅ ∠G
4) ≅ supp. Th.
Syllabus Objective: 5.6 - The student will solve problems involving properties of polygons.
Review the Types of Quadrilaterals chart from p. 5 of these notes.
Reasoning with Quadrilaterals
The flow chart on page 3 of these notes can help students answer these.
Movement from the bottom towards the top will always result in a true statement.
Movement from the top towards the bottom will only sometimes be true.
Finally, movement sideways across the chart is never true.
Unit 5
Polygons
Page 10 of 20
Examples: Decide if the statement is Always, Sometimes, or Never true.
a) A rhombus is a kite.
b) A parallelogram is a rectangle.
c) A square is a quadrilateral.
a) Never
b) Sometimes
c) Always
Syllabus Objective: 5.7 - The student will perform constructions involving polygons.
(supplemental material required)
B
Example: Construct an equilateral triangle.
Mark a center, O, and draw a circle with a given radius.
Draw the diameter, AB .
Using A as the center and AO as the radius, draw an
arc that intersects the circle at C and D.
Connect B, C, and D to form a triangle.
O
C
A
B
Example: Construct a square.
Mark a center, O, and draw a circle with a given radius.
Draw the diameter, AB .
Draw another diameter, CD , which is the perpendicular
D
C
O
D
bisector of AB .
Connect A, D, B, and C to form a square.
A
Syllabus Objective: 5.8 - The student will find the measure of interior, exterior, and
central angles of a given regular polygon.
Discuss how to find the measure of each central angle. Guide students
into examples that show how the central angles create a full circle, so you
can divide 360° by how many central angles the polygon has.
Show an example with a triangle and have students create and
measure exterior angles. Total the three angles up and then do
another example with a rectangle. Students will be able to come to
the conclusion that the sum of the exterior angles of a polygon is
360°.
Compare and contrast interior, exterior, and central angle measurements of polygons in general.
It is very important to stress that students read carefully to understand
exactly what they are being asked to find; a total or a single angle
measurement. They must also be aware if the polygon is regular or not.
Unit 5
Polygons
Page 11 of 20
Polygon Exterior Angles Theorem: The sum of the measures of the exterior angles of a convex
polygon, one angle at each vertex, is 360° .
I always point out what a hard formula this one is!
Corollary to Polygon Exterior Angles Theorem: The measure of each exterior angle of a regular
360
.
n-gon is
n
Or… 180° − ( interior angle ) °
It depends what you already know! There’s no
reason to make it harder than necessary.
Central angle of a regular polygon – is an angle whose vertex is the center and whose sides
contain two consecutive vertices of the polygon. The central angle of a regular polygon can be
360
.
found by
n
What if students are told the measure of an interior angle and asked to find how many sides
the polygon has?
Example: The measure of an interior angle of a regular polygon is 156°. Find
the number of sides in the polygon.
156 =
( n − 2 ) 180
n = 15 .
n
(
)
→ 156n = n − 2 180 → 156n = 180n − 360 → −24n = − 360 →
OR
If an interior angle of a regular polygon is 156°, then the exterior angle will
measure 24°. Since the sum of the exterior angles will measure 360° and all
exterior angles are equal, one can divide 360 by 24 to get an answer of 15 sides.
Syllabus Objective: 5.9 - The student will utilize the distance, slope, and midpoint
formulas to classify a given quadrilateral.
Review:
1) distance formula
2) slope formula
3) midpoint formula
Students must be told that although a graph may look like a particular
quadrilateral, it should only give them a hint of the direction their
work should follow. There MUST be mathematical proof to justify
their answer. Review the distance, slope, and midpoint formulas. Also,
discuss how you know if two lines are parallel.
Unit 5
Polygons
Page 12 of 20
**What conditions are required to prove that a given quadrilateral is a
parallelogram?**
Theorem: If both pairs of opposite sides of a quadrilateral are congruent, then the
quadrilateral is a parallelogram.
Theorem: If both pairs of opposite angles of a quadrilateral are congruent, then the
quadrilateral is a parallelogram.
Theorem: If an angle of a quadrilateral is supplementary to both of its consecutive angles, then
the quadrilateral is a parallelogram.
Theorem: If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a
parallelogram.
Theorem: If one pair of opposite sides of a quadrilateral are congruent and parallel, then the
quadrilateral is a parallelogram.
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.
**Ok, I know I have a parallelogram. How do I get more specific?**
Theorem: A parallelogram is a rhombus if and only if its diagonals are perpendicular.
Theorem: A parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite
angles.
Theorem: A parallelogram is a rectangle if and only if its diagonals are congruent.
Unit 5
Polygons
Page 13 of 20
**Now I don’t have a parallelogram. Can I classify these quadrilaterals?**
Theorem: If a trapezoid is isosceles, then each pair of base angles is congruent.
Theorem: If a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid.
Theorem: A trapezoid is isosceles if and only if its diagonals are congruent.
Theorem: If a quadrilateral is a kite, then its diagonals are perpendicular.
Theorem: If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent.
Example 12: Use the figure below to prove that quadrilateral ABCD is an
isosceles trapezoid, given that BC & AD .
C
B
4
2
D
-
5
1
A
-
-
Proof:
It must be shown that two sides are parallel and the two legs are congruent. Given
BC & AD, it must only be proven that AB ≅ CD but not parallel. Begin by identifying
the ordered pair for each point on quadrilateral ABCD. A(-1,-1), B(-3,2), C(-1,4),
and D(2,2). Next, use the distance formula to determine the length of AB
and CD . Use the slope formula to show that the two segments have different
slopes.
Unit 5
Polygons
Page 14 of 20
AB =
(x 2 − x 1 )2 + ( y 2 − y 1 )2 =
( −3 − ( −1))2 + (2 − ( −1))2 =
CD =
(x 2 − x 1 )2 + ( y 2 − y 1 )2 =
(2 − ( −1))2 + (2 − 4)2 =
4 + 9 = 13.
9 + 4 = 13.
Therefore, AB = CD, or AB ≅ CD !
Slope of AB =
y 2 − y1
3
2 − ( −1)
.
=
=
x 2 − x1 −3 − ( −1) −2
Slope of CD =
y 2 − y1
−2
2−4
.
=
=
3
x 2 − x1 2 − ( −1)
Since they have different slopes, the two segments will intersect. It has now been
shown that there is exactly one pair of opposite sides parallel and that the legs,
AB and CD , are congruent.
It can now be determined that quadrilateral ABCD is an isosceles trapezoid.
Unit 5
Polygons
Page 15 of 20
This unit is designed to follow the Nevada State Standards for Geometry, CCSD
syllabus and benchmark calendar. It loosely correlates to Chapter 6 of McDougal
Littell Geometry © 2004, sections 6.1 – 6.6 & 11.1. The following questions were
taken from the 1st semester common assessment practice and operational exams
for 2008-2009 and would apply to this unit.
Multiple Choice
#
19.
Practice Exam (08-09)
Use the triangle below.
Operational Exam (08-09)
Use the triangle below.
x°
x°
( 3 x + 10 ) °
( 3 x + 3) °
70°
45°
36.
37.
What is the value of x?
A. 29
B. 33
C. 44
D. 49
How many sides does a nonagon have?
A. 7
B. 9
C. 11
D. 19
Which figure is a polygon?
A.
B.
C.
D.
A.
B.
C.
D.
What is the value of x?
15
20
25
70
How many sides does a heptagon have?
6
7
16
17
Which figure is a polygon?
A.
A.
B.
B.
C.
C.
D.
D.
Unit 5
Polygons
Page 16 of 20
38.
A hexagon is shown below.
A pentagon is shown below.
a°
a°
110°
100°
100°
150°
A.
B.
C.
D.
39.
A.
B.
C.
D.
What is the value of a?
90
100
130
150
Use the figure below.
The figure below is a hexagon with an
external angle.
60°
130°
What is the value of a?
100
110
120
150
x°
x°
100°
40°
What is the value of x?
A. 70
B. 60
C. 50
D. 40
40.
130°
150°
What is the value of x?
A.
20
B.
60
C.
80
D.
160
Rectangle ABCD is given below.
Parallelogram ABCD is given below.
A
11x + 9
A
B
10x – 4
26
31
D
6(x + 4)
A.
B.
C.
D.
41.
Unit 5
B
D
C
What is the value of x?
2
3
6
16
4x + 8
C
What is the value of x?
2
3
1
4
2
C.
2
9
5
D.
What is the measure of each exterior angle
of a regular octagon?
A. 144°
B. 135°
C. 45°
D. 36°
A.
B.
What is the measure of each exterior angle
of a regular hexagon?
A. 60°
B. 90°
C. 120°
D. 135°
Polygons
Page 17 of 20
42.
Which statement is true about a kite?
A. A kite has 4 congruent sides.
B. A kite has 2 pairs of parallel sides.
C. A kite has perpendicular diagonals.
D. A kite has congruent diagonals.
43.
Which statement below is true about an
isosceles trapezoid?
A. Both pairs of opposite sides are parallel.
B. Both pairs of opposite sides are congruent.
C. One pair of opposite sides is congruent and
the other is parallel.
D. One pair of opposite sides is both parallel
and congruent.
In the figure below, ΔKLM ≅ ΔABC .
44.
Which statement is true about a trapezoid?
A. A trapezoid has 1 acute angle and 3 obtuse
angles.
B. A trapezoid has 2 pairs of parallel sides and
4 sides.
C. A trapezoid has 4 right angles and 2 pairs of
parallel sides.
D. A trapezoid has 4 sides and exactly 1 pair of
parallel sides.
Which statement below is true?
A. All rhombi are squares.
B. All squares are rectangles.
C. All quadrilaterals are parallelograms.
D. All rectangles are rhombi.
In the figure below, ΔABC ≅ ΔXYZ .
A
L
8 cm
4 cm
47°
K
X
B
80°
45°
M
10 cm
A
C
5 cm
53°
C
Y
Z
B
45.
Which statement must be true?
A. AC = 8cm
B. BC = 6cm
C. m∠A = 53°
D. m∠C = 80°
Use the rhombus below.
B
Which statement must be true?
A. m∠X = 45°
B. m∠Z = 45°
C. YZ = 3cm
D. XY = 3cm
A
In trapezoid GHIJ below, GH & JI .
G
65°
( x + 10 ) °
E
C
A.
B.
C.
D.
Unit 5
H
J
A.
B.
C.
D.
D
What is m∠CDE ?
25°
65°
90°
115°
Polygons
x°
I
What is the value of x?
75°
85°
95°
105°
Page 18 of 20
46.
47.
A regular polygon has interior angles that
measure 144°. How many sides does this
polygon have?
A. 6
B. 8
C. 10
D. 12
Use the figure below.
A regular polygon has interior angles that
measure 150° . How many sides does this
polygon have?
A. 30
B. 15
C. 12
D. 10
Use the figure below.
x°
75°
41°
123°
x°
What is the value of x?
A. 23
B. 37
C. 43
D. 57
What is the value of x?
A. 64
B. 74
C. 116
D. 126
Unit 5
86°
Polygons
Page 19 of 20
Free Response
3.
Practice Exam (08-09)
Show that the quadrilateral QUAD, having vertices Q(–1,–1), U(3,2), A(6,–2), and D(2,–5); is a square.
Operational Exam (08-09)
3.
Prove that the quadrilateral QUAD, having vertices Q(–1,0), U(–3,6), A(0,7), and D(2,1) is a rectangle.
(A blank coordinate grid is provided.)
y
x
Unit 5
Polygons
Page 20 of 20