Download Review of Chapter 6 Section 1: Properties and Attributes of

Review of Chapter 6
Section 1: Properties and Attributes of Paralleograms
Sum of Interior Angles of a Polgyon: (n – 2) 180
15-gon: (15-2)180 = 2340°
Sum of Each Interior Angle of a Polgyon: (n -2) 180 ÷ n
15-gon: (15-2)180 ÷ 15 = 156°
Sum of Exterior angles of a Polygon: 360°
15-gon: 360°
Sum of Each Exetior Angle of a Polygon: 360°÷n
15-gon: 360 ÷ 15 = 24°
Convex: diagonals are on the interior of the polygon
Concave: diagonals are on the exterior of the polygon
1) Find sum of interior angles of a 25-gon:
2) Find the sum of each interior angle of a 25-gon:
3) Find the sum of the exterior angles of a 25-gon:
4) Find the sum of each exterior angle of a 25-gon:
5) Find the measure of each angle:
6) Find the measure of each exterior angle:
Hint # 5 & 6:
What do the
angles sum too?
7) Polygon (Yes or No?) If yes, name the polygon.
8) Is polygon concave or covex?
Section 2: Properties of Parallelograms
Opposite Sides are
Opposite Angles
Congruent
Congruent
ABCD is a paralleogram. Find
each measure.
Consecutive Angles
sum to 180°
Diagonals Bisect Each
Other
AD
m <B
BC = AD
m<A + m<B = 180
5x + 19 = 7x
6y + 5 + 10y – 1 = 180
19 = 2x
16y + 4 = 180
x = 9.5
16y = 176
AD = 7(9.5) = 66.5
y = 11 m< B = 6(11) + 5 = 71°
9) Find the following measures.
a. JK
b. LM
c. m<L
d. m<M
Find the following measures.
JG
EJ = JG
FJ = JH
3w = w + 8
4z – 9 = 2z
2w = 8
-9 = -2z
w=4
4.5 = z
JG = 4 + 8 = 12
FH = 2*JH = 2*2(4.5)=18
10) Find the following measures.
a. WV
b. YW
c. XZ
d. ZV
FH
Section 3: Conditons for Parallelograms
For this section, you take the properties you learned in the previous section and apply them to
whether a quadralilatral is a polyon and give an explanation of why or why not.
11) Show and explain why ABCD is a parallelogram when x = 7 and y = 4.
x + 14 = 7 + 14 = 21
5y – 4 = 5(4)-4 = 16
3x = 3(7) = 21
2y + 8 = 2(4) + 8 = 16
Explanation: ABCD is a paralleogram since the opposite sides are congruent.
12) Show that EFGH is a paralleogram when z = 11 and w = 4.5.
Explaination: _____________________________________________________________
13) Show that PQRS is a paralleogram when a = 2.4 and b = 9.
Explanation: _____________________________________________________________
14) Is this a parallelogram?
Why? ____________________________________________________
15) Is this a parallelogram?
Why? ______________________________________________________________________
16) Is this a paralleogram?
Why? _______________________________________________________
17) Is this a paralleogram?
Why? _______________________________________________________
18) Is this a paralleogram?
Why? ______________________________________________________
Section 4: Properties of Squares, Rectangles and Rhombi
Rectangle: A quadrilateral
Rhombus: A quadrilateral
with 4 right angles.
with 4 congruent sides.
Square: A quadrilateral with
4 right angles and 4
congruent sides.
Rectangles
If a quadrilateral is a rectangle then it’s a
paralleloram.
If a parallelogram is a rectangle then its
diagonals are congruent.
ABCD is a parallelogram
Rhombi
If a quadrilateral is a rhombus
then it is a parallelogram.
ABCD is a
parallelogram.
If the parallelogram is a
rhombus then its diagonals
are perpendicular.
If a parallelogram is a
rhombus then each diagonal
bisects a pair of opposite
angles.
19) RSTV is a rhombus. Find VT.
(Since all sides are congruent, set the expression equal to one another)
20) RSTV is a rhombus. Find m<WSR.
(m<SWT = 90° because the diagonals are perpendicular, so 2y + 10 = 90)
21) CDFG is a rhombus. Find:
a. CD.
b. Find m<GCH if m<GCD = (b + 3)° and m<CDF = (6b-40)°
Section 5: Conditions for Special Paralleograms
Conditions for Rectangles
If one angle of a parallelogram is right, then
the parallelogram is a rectangle.
If the diagonals of a parallelogram are
congruent then the parallelogram is a
rectangle.
Conditions for Rhombi
If one pair of consecutive
sides of a parallelogram is
congruent then the
parallelogram is a rhombus.
If the diagonals of a
parallelogram are
perpendicular then the
parallelogram is a rhombus.
If one diagonal bisects a pair
opposite angles of a
parallelogram then the
parallelogram is a rhombus.
22) Determine if the conclusion is valid. If not, explain why.
1st determine if ABCD is a parallelogram
Since
and the opposite sides are congruent ABCD is a paralleogram
nd
2 determine if ABCD is a rectangle.
Since
so <D is a right angle all the angles are right angles so it’s a rectangle.
rd
3 determine if ABCD is a rhombus
Since
and when the diagonals intersect they form a 90 degree angle it is a
rhombus.
Since ABCD is a rhombus and a rectangle then it is therefore a square because it has all
for sides congruent and all angles being right.
23)
24)
25)
26)
27)
Section 6: Properties of Kites and Trapezoids
Properties of Kites
Kite: a quadrilateral with exactly two pairs of congruent consecutive sides.
If a quadrilateral is a kite then the diagonals
are perependicular.
If a quadrilateral is a kite then exactly one pair
of opposite angles are congruent.
28) In kite EFGH, m<FEJ = 25°, and m<FGJ = 57°. Find each measure.
a. m<GFJ
m<FJG = 90°
m<GFJ + m<FGJ = 90°
m<GFJ + 57° = 90°
m<GFJ = 33°
b. m<JFE
Triangle FJE is a right triangle, so m<JFE + m<FEJ = 90°.
m<JFE + 25° = 90°
m<JFE = 65°
c. m<GHE
<GHE = <GFE
m<GFE = m<GFJ + m<JFE
m<GHE = 33° + 65° = 98°
29) In kite PQRS, m<PQR = 78° and m<TRS = 59°, find the following.
a. m<QRT
b. m<QPS
c. m<PSR
Isosceles Trapezoids
Isosceles Trapezoid: The base angles are congruent and the non-parallel sides are congruent.
If the quadrilateral is an
isoscelses trapezoid then the
base angles are congruent.
If a trapezoid has one pair of
base angles congruent, then
the trapezoid is isoscelses.
30) Find m<Y.
m<W + m<X = 180°
117° + X = 180
X = 63°
m<X = m<Y= 63°
31) RT = 24.1; QP = 9.6, find PS.
QS = RT
QS = 24.1
QP + PS = QS
9.6 + PS = 24.1
PS = 14.5
A trapezoid is isosceles if and
only if its diagonals are
congruent.
32) Find m<F.
33)
34) Find the value of y that makes the trapezoid isosceles.
(Hint: Set <E=<H)
35) JL = 5z + 3 and KM = 9z – 12. Find the value of z so that JKLM is isoceles.
36) Find the value of x so that PQST is isosceles.
Trapezoid Midsegment Theorem
The midsegment of a trapezoid is parallel to each base, and length is one half the sum of the
lenghts of the bases.
37) Find ST.
MN = ½ (ST + RU)
31 = ½ (ST + RU)
2 x 31 = ½ (ST + 38) x 2
62 = ST + 38
ST = 24
38) Find EH.