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9.
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Tell whether each statement is true Always (A), Sometimes (S), or Never (N).
a. The acute !s of a right
are complementary.
b. The supplement of one of the !s of a
the other two !s of the .
c. A
A
is equal in measure to the sum of
N
contains two obtuse !s.
d. If one of the !s of an isosceles
A
is 60°, the
is equilateral.
e. If the sides of one are doubled to form another , each ! of the second
is twice as large as the corresponding ! of the first
A
N
11.
Find m!MRP, m!ORP, and m!MOR.
R
M
10°
O
P
m!MRP = 90 - 10 = 80°
m!ORP = 80/2 = 40°
m!MOR = 90 - 40 = 50°
15.
The measures of the two !s of a are in the ratio of 2:3. If the third ! is 4° larger than the
larger of the other two !s, find the measure of an exterior angle at the third vertex.
Find the indicated angle.
2x
2x + 3x + (3x + 4) = 180
x = 22
3x
3x + 4
?
3x + 4 = 3(22) + 4 = 70
m exterior ! = 180 - 70 = 110°
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16.
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CD bisects !ACB
BD is one of the trisectors of !ABC.
Find m!D
A
30°
m!ABC = m!ACB =
(180 - 30)
2
= 75°
m!DBC = either 25° or 50°
m!D = either 180 - 25 - 37.5 = 117.5°
or 180 - 50 - 37.5 = 92.5
D
B
C
17.
Given: EFGH is a rectangle
FH = 20
J, K, M, and O are midpoints.
a. Find the perimeter of JKMO.
E
K
b. What is the most descriptive
name for JKMO?
F
KJ & MO are a midlines of
J
EFH and
H
O
M
G
GHF, respectively. Therefore, KJ = MO = 10.
Diagonal EG (not shown) can be drawn to show that KM = JO = 10 by similar reasoning.
Perimeter of JKMO = 10 * 4 = 40.
The best description for JKMO is....rhombus!
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18.
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Given: m!PST = (x + 3y)°
m!P = 45°
m!R = (2y)°
m!PSR = (5x)°
P
45°
Find: m!PST
R
(2y)°
(5x)° (x + 3y)°
S
T
(5x) +(x + 3y) = 180
(5x) + (2y) + 45 = 180
x = 15, y = 30
m!PST = (15) + 3(30) = 105°
19.
Prove that the midpoint of the hypotenuse of a right
Right ABC
D is midpoint of
hypotenuse AC
Given:
D is equidistant
from A, B, & C
Prove:
is equidistant from all three vertices.
D
ABC with D midpoint of hypot. AC
Reasons
1. Given
2. Draw BD to E so that BD " DE
2. Auxiliary Lines
3. Draw AE & CE
3. Auxiliary Lines
4. AD " DC
5. ABCE is a parallelogram
4. Definition of Midpoint
5. A quad with diags that bisect each other is
a parallelogram
6. !ABC is right
7. ABCE is a rectangle
6. Definition of right
7. A parallelogram with a right ! is a rectangle
8. BE " AC
8. Diagonals of a rectangle are "
9. AD " DC " BD
10. D is equidistant to A, B, & C
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B
Statements
1. Right
E
A
9. Division Property of " Segments
10. Definition of Equidistant
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20.
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Prove that if the midpoints of a quadrilateral are joined in order, the figure formed is a !.
Q
Given:
Prove:
Z
Quadrilateral QRST
with midpoints W,
X, Y, & Z
W
WXYZ is a !
T
Y
R
X
S
Statements
Reasons
1. Quad QRST with midpoints W, X, Y, & Z
1. Given
2. Draw QS & RT
2. Auxiliary Lines
3. WZ
WX
RT; XY
QS; ZY
RT
QS
3. Midline Theorem
4. WZ
XY; WX
ZY
4. Transitive Property of
5. Definition of !
5. WXYZ is a !
Segments
21.
Given: AB " AC
AE " DE " DB " BC
m!BDE = 180 - 4x
m!BDC = 180 - (180 - 4x) - x = 3x
Find: m!A
x + x + 2x + 3x = 180
B
x
2x
3x
C
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2x
3x
x=
E
x
180
7
A
x
D
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