Download Geometry 1 – Unit One: Essentials of Geometry, Practice

Geometry 1 – Unit One: Essentials of Geometry, Practice
Use the diagram to the right to answer
Questions #1 – #6.
B
A
E
1.
Name three points that are collinear.
2.
Name three points that are coplanar but not collinear.
3.
HJJG
Give another name for AE.
5.
HJJG
HJJG
The intersection of AC and BD is __________. 6.
C
D
4.
Name a ray that passes through point A.
Name a pair of opposite rays.
Use the number line below to answer Questions #7 – #10.
N
O
P
Q
R
S
T
U
V
-3
-2
-1
0
1
2
3
4
5
7.
Find the length of OR.
9.
Name a segment that is congruent to NR.
8.
Find the length of PV .
10.
Use the Segment Addition Postulate to complete this statement: OQ + QT = __________ .
11.
The endpoints of EF are E (2, 3) and F (4, 9). Find the coordinates of the midpoint of EF . Then
find the length of EF .
12.
The endpoints of XY are X (−4, − 2) and Y (2, 6). Find the coordinates of the midpoint of XY . Then
find the length of XY .
13.
Point M is the midpoint of LC. LM has a length of ( x + 2) cm and MC has a length of (2 x − 3) cm.
Find the value of x and the length of LC.
Use the diagram to the right to answer
Questions #14 – #18. Find the measure of
each angle, and then classify the angle based
on its measurement.
14.
∠ LNK
15.
∠ JNL
16.
∠ JNM
17.
∠ MNK
18.
Name a pair of congruent angles.
Use the diagram to the right to answer
Questions #19 – #27.
19.
Give another name for ∠ 4.
A
B
C
20.
Name a pair of adjacent angles.
21.
Name a pair of vertical angles.
22.
Name two angles that form a linear pair.
4
5
8
F
6
7
E
D
23.
If ∠ 7 = 125°, then ∠ 8 = __________ °.
24.
If ∠ 6 = 54°, then ∠ 8 = __________ °.
25.
∠ AFC = 129°, ∠ 4 = (6x + 10)°, and ∠ 5 = (9x – 16)°. Find the value of x and the measure of ∠ 4.
26.
Use the Angle Addition Postulate to complete this statement: ∠ AFB + __________ = ∠ AFC.
27.
JJJG
∠ BFD = 110°. If FC is the bisector of ∠ BFD, find the measure of ∠ 6.
28.
∠ X and ∠ Y are complementary. ∠ X is twice as large as ∠ Y. Find the measure of both angles.
29.
∠ X and ∠ Y are supplementary. ∠ X has a measure of (5x + 1)° and ∠ Y has a measure
of (12x – 8)°. Find the measure of both angles.
*********************************ANSWERS*********************************
1.
Collinear means points that are all part of the same line. So, two possible answers would be points
A, E, & C and points D, B, & E.
2.
Coplanar means points that are all part of the same plane. You need to find points that are part of the
same plane, but not part of the same line. So one possible answer would be points A, E, and B.
Note – this is not the only correct answer.
3.
HJJG HJJG
AC or EC Note – these are not the only correct answers.
4.
JJJG JJJG
EA or CA Note – these are not the only correct answers.
5.
Intersection is another word for where the two lines meet. The answer is point E.
6.
Opposite rays are two rays that have the same initial point and are collinear. So, two possible answers
JJJG
JJJG
JJJG
JJJG
would be EA and EC and EB and ED.
7.
Find the length of OR using the Ruler Postulate. 1 – (–2) = 3
8.
Find the length of PV using the Ruler Postulate. 5 – (–1) = 6
9.
NR has a length of 4 units. So, you must find another line segment with a length of 4 units. So, two
possible answers would be QU and PT . Note – these are not the only correct answers.
10.
OT
11.
⎛ 2 + 4 3 + 9 ⎞ ⎛ 6 12 ⎞
Midpoint = ⎜
,
⎟=⎜ ,
⎟ = ( 3, 6 )
2 ⎠ ⎝2 2 ⎠
⎝ 2
EF =
12.
2
2
= 22 + 62 = 4 + 36 = 40 = 6.3 units
⎛ −4 + 2 −2 + 6 ⎞ ⎛ −2 4 ⎞
Midpoint = ⎜
,
⎟ = ⎜ , ⎟ = ( −1, 2 )
2 ⎠ ⎝ 2 2⎠
⎝ 2
XY =
13.
( 4 − 2 ) + ( 9 − 3)
( 2 − ( −4 ) ) + ( 6 − ( −2 ) )
2
2
= 62 + 82 = 36 + 64 = 100 = 10 units
The midpoint of a segment is the point that divides a segment into two congruent segments. Therefore,
LM = MC. x + 2 = 2 x − 3 → x = 5. Then substitute 5 in for x.
LM = 5 + 2 = 7, MC = 2 ( 5 ) − 3 = 10 − 3 = 7. LC = LM + MC = 7 + 7 = 14
14.
JJJG
JJJG
NL points at 90°. NK points at 145°. ∠ LNK = 145° − 90° = 55°. An angle with a measure of less than
90° is an acute angle.
15.
JJJG
JJJG
NL points at 90°. NJ points at 180°. ∠ JNL = 180° − 90° = 90°. An angle with a measure of 90° is a
right angle.
16.
JJJJG
JJJG
NM points at 0°. NJ points at 180°. ∠ JNM = 180° − 0° = 180°. An angle with a measure of 180° is a
straight angle.
17.
JJJJG
JJJG
NM points at 0°. NK points at 145°. ∠ JNM = 145° − 0° = 145°. An angle with a measure of more
than 90° is an obtuse angle.
18.
Congruent angles are two angles that have the same measure. ∠ JNL = 90° and ∠ LNM = 90°.
Therefore, ∠ JNL ≅ ∠ LNM.
19.
∠ AFB or ∠ BFA
20.
Adjacent angles are two angles that are next to each other without sharing any interior points.
Therefore, some possible answers would be ∠ 4 & ∠ 5, ∠ 5 & ∠ 6, and ∠ 6 & ∠ 7. Note – these are
not the only correct answers.
21.
Vertical angles are angles formed by the intersection of two lines that are not adjacent. Therefore, ∠ 8
and ∠ 6 are vertical angles. Another pair of vertical angles would be ∠ AFC and ∠ DFE. Note – these
are not the only correct answers.
22.
A linear pair of angles is two angles that are adjacent and are supplementary. Therefore, ∠ 8 and ∠ 7
form a linear pair. Another linear pair of angles would be ∠ AFB and ∠ BFD. Note – these are not the
only correct answers.
23.
∠ 7 and ∠ 8 form a linear pair, which means they are supplementary. ∠ 8 = 180° – 125° = 55°
24.
∠ 6 and ∠ 8 are vertical angles. Vertical angles always have the same measure. If ∠ 6 = 54°, then
∠ 8 = 54°.
25.
∠ AFC = ∠ 4 + ∠ 5 → 129 = ( 6 x + 10 ) + ( 9 x − 16 ) → 129 = 15 x − 6 → 135 = 15 x → x = 9
To find the measure of ∠ 4, replace x with 9. ∠ 4 = 6 ( 9 ) + 10 = 54 + 10 = 64°
26.
∠ BFC or ∠ CFB
27.
An angle bisector is a ray that divides an angle into two congruent angles.
1
1
∠ 6 = ∠ BFD → ∠ 6 = (110° ) = 55°
2
2
28.
Complementary angles are two angles that add up to 90°.
∠ X = ( 2 x ) °, ∠ Y = ( x ) ° → ∠ X + ∠ Y = 90° → 2 x + x = 90 → 3 x = 90 → x = 30
∠ X = 2 ( 30 ) ° = 60°, ∠ Y = 30°
29.
Supplementary angles are two angles that add up to 180°.
∠ X + ∠ Y = 180° → ( 5 x + 1) + (12 x − 8 ) = 180 → 17 x − 7 = 180 → 17 x = 187 → x = 11
∠ X = 5 (11) + 1 = 55 + 1 = 56°, ∠ Y = 12 (11) − 8 = 132 − 8 = 124°