Download This copy is for me, Peter Lin, only.

Honors Geometry
Issue 1
Super Mathter
November 10, 2004
(: 301-520-6030
Fax: 301-251-8645
Name:
For class info, visit www.MathEnglish.com
Direct your questions and comments to
[email protected]
Peter Lin
Peter Lin
SUPPLEMENTARY AND COMPLEMENTARY ANGLES........................................................ 2
VERTICAL ANGLES ...................................................................................................................... 3
ANGLE BISECTORS....................................................................................................................... 6
PARALLEL LINES AND ANGLES...............................................................................................10
STANDARD TEST .........................................................................................................................12
–1–
This copy is for me, Peter Lin, only.
Honors Geometry
Issue 1
[Linear Pair Postulate]
Supplementary and Complementary
Two angles of a linear pair are
Angles
supplementary to each other.
When the two sides of an angle form a
1. What should be the value of x in the
straight line, the angle is called a straight
figure?
angle or flat angle. Specifically, the angle is
measured to be 180°.
180o
x
A
B
O
Straight angle is 180°. Right angle is 90°.
right angle notation, it
stands for 90 o
A right angle is special, it also has a special
notation. Instead of writing 90° at the
angle, we draw a small square near the
angle vertex. Both ∠AOB and ∠BOC are
right angles in the figure. Each angle of a
rectangle is a right angle.
B
90o
A
90o
O
C
90o
If the sum of two angles is 90°, we called
them complementary angles. Namely,
∠1 + ∠2 = 90° ó ∠1 and ∠2 are
complementary.
If the sum of two angles is 180°, we called
them supplementary angles. Namely,
∠1 + ∠2 = 180° ó ∠1 and ∠2 are
supplementary.
–2–
This copy is for me, Peter Lin, only.
50o
Honors Geometry
Issue 1
Question set [4 - 7]
Vertical Angles
Two lines L and M intersect. Given that ∠1
= 105°.
Definition: vertical angle
In the figure, ∠1 and ∠2 are called vertical
angles since their sides form two crossing
lines and they stay on the opposite sides of
the vertex.
L
1
2
4
3
M
O
2
1
vertical angles
4. Find the measure of ∠2 and state the
reason.
O
4
3
N O T vertical angles
Then ∠1 = ∠2, namely, vertical angles are
congruent.
2. For ∠1, name the vertical angle and
adjacent angle.
5. Find the measure of ∠3 and state the
reason.
3
1
4
2
6. ∠1 and ∠3 are called _ _ _ _ _ _ _ _ _
_ _ _ _, so are ∠2 and ∠4. Each pair of
angles are _ _ _ _ _ _ _ _ _.
3. List all pairs of adjacent and vertical
angles.
6
5
1
2
3
4
7. Find the measure of ∠4 and state the
reason.
–3–
This copy is for me, Peter Lin, only.
Honors Geometry
Issue 1
12. An angle is equal to its complementary
Question set [8 - 17]
angle, what is the measure of this angle?
Conceptual and computational problems.
8. If two angles are complementary to the
same angle, then they must be
congruent. Is this statement always
true?
13. What is the complementary angle for
35°? What is the supplementary for
35°?
9. If an angle has 60° as its complementary
angle, what is its supplementary angle?
14. An angle is two times the
complementary angle of 40°. What is
the measure of this angle?
10. An angle is twice of its complementary
angle, what is the measure of this angle?
15. Given: line EF bisects ∠AOB as shown
in the figure, namely, ∠3 = ∠4.
Prove: ∠1 = ∠2.
11. An angle is twice its supplementary
angle, what is the measure of this angle?
C
A
O
E
1
3
2
4
D
–4–
This copy is for me, Peter Lin, only.
F
B
Honors Geometry
16. Both ∠AOC and ∠BOD are right
angles. ∠BOC = 40°. Find the measure
of ∠AOD.
Issue 1
B
C
A
D
40o
O
17. As the figure shows, OA bisects
∠COB, which is a right angle. Find the
measures of ∠1.
C
A
1
D
B
O
E
18. What is the measure of ∠AOB?
A
25o
O
(60 - x)o
(3x + 15)o
B
–5–
This copy is for me, Peter Lin, only.
Honors Geometry
Issue 1
21. What should be the value for x in the
Angle Bisectors
figure?
Definition: angle bisector
Angle bisector divides an angle equally.
The ray OB divides the angle ∠AOB
60
x
evenly into two congruent angles: ∠1 and
∠2, so OB is called the angle bisector of
∠AOB.
o
A
B
1
OB bisects an
angle
2
O
C
22. Find the value of x.
3x+5
Question set [19 - 22]
Find the value of x in each of the
following.
19. What is the measure of x in the figure?
2x-5
x
40o
Question set [23 - 28]
Conceptual and computational problems.
20. ∠AOD is a straight angle (180°), OB
and OC divide the entire angle into
three congruent angles, what should be
the value for x?
C
23. ∠1 and ∠2 are called __ __ __ __ __ __
__ __ __ __.
B
1
x
D
x
x
O
A
–6–
This copy is for me, Peter Lin, only.
2
Honors Geometry
Issue 1
24. Two angles are __________ if they add
28. Is it true that a straight angle is twice a
right angle?
up to be 90°.
25. Two angles are __________ if they add
up to be 180°.
Question set [29 - 30]
∠AOB
and ∠BOC are linear pair. DO
bisects ∠AOB and EO bisects ∠BOC.
E
B
2
C
2
O
D
A
29. Given that ∠AOB = 40°, what is the
measure of ∠2?
26. What angles are supplementary to ∠1?
1
1
3
4
30. Prove that ∠1 + ∠2 = 90° regardless of
the measure of ∠AOB.
27. If BD bisects ∠ABC, what is the
measure of ∠ABD?
A
D
o
Question set [31 - 35]
Computational problems.
o
C
B
–7–
This copy is for me, Peter Lin, only.
Honors Geometry
Issue 1
31. BD bisects ∠ABC. If ∠ABD = 30°,
34. ∠COE is bisected by OD and ∠AOC
find the value for x.
is bisected by OB. Find the value for x.
A
C
D
D
B
xo
x
20o
30o
E
A
O
C
B
35. P is a point on AE. BP bisects ∠APC.
DP bisects ∠CPE. Find the measure of
∠DPB.
32. ∠ABC is bisected by BD in the figure.
If ∠ABE = 100°, find the value for x.
B
C
D
C
E
B
xo
100o
x
o
E
D
o
x
A
P
A
Name of an angle:
33. As in the following figure, is it true
that x = y?
A
xo
1
40o
O
40o
B
An angle is formed by one vertex and two
sides connected by the vertex. As in the
figure, the angle can be expressed as ∠AOB
or ∠1 in short.
yo
Definition: congruent angle
Congruent angles are equal in measure.
–8–
This copy is for me, Peter Lin, only.
Honors Geometry
Issue 1
is congruent to
2
1
is not congruent to
3
4
When two angles ∠1 and ∠2 are measured
to be the same, we called them congruent
angles, or we say ∠1 is congruent to ∠2.
–9–
This copy is for me, Peter Lin, only.
Honors Geometry
Issue 1
There are two such pairs: (∠3, ∠6) and
Parallel Lines and Angles
(∠4, ∠5).
Definition: corresponding angles
Alternate exterior angles:
Two lines L1 and L2 (not necessarily
parallel) are cut by a transversal. Get
1
2
L
familiar with the following terms.
1
1
3
5
7
2
L1
7
L2
8
4
There are two such pairs: (∠2, ∠7) and
(∠1, ∠8).
6
L2
8
Consecutive exterior angles:
There are four such pairs: (∠1, ∠5), (∠3,
∠7), (∠2, ∠6), (∠4, ∠8).
THEOREM A
[Corresponding Angles Postulate]
If L1 and L2 are parallel and cut by a
transversal then corresponding angles are
congruent.
1
2
L1
7
L2
8
There are two such pairs: (∠1, ∠7) and
(∠2, ∠8).
Consecutive and alternate angles
The term consecutive pair refers to both
angles falling on the same side of the
transversal.
Consecutive interior angles:
L1
6
L2
There are two such pairs: (∠3, ∠5), (∠4,
∠6).
L1
co
ns
ec
ut
iv
e
5
4
co
ns
ec
ut
iv
e
3
L2
The term alternate pair refers to either of
the angle falling at the opposite side of the
transversal.
Alternate interior angles:
L1
6
L2
L1
al
te
rn
at
e
5
4
al
te
rn
at
e
3
– 10 –
This copy is for me, Peter Lin, only.
L2
Honors Geometry
Interior and exterior angles
The term interior pair refers to both angles
falling in the interior strip formed by L1
and L2.
Issue 1
L1
Interior
pairs
L2
The term exterior pair refers to both angles
falling in the exterior strip formed by L1
and L2.
L1
exterior
pairs
L2
36. In each of the following problems use
the information to name the segments
that must be parallel. If there is no such
segment, write none.
A
4
B 5 1
2
3
C
12
15
14
F
13
11 6 7
10 9 8G
E
D
Given
a) ∠ 2 = ∠ 8
b) ∠ 1+∠ 2=∠ 7+∠ 8
c) ∠ 3 + ∠ 13 = 180°
d) ∠ 8 = ∠ 15
e) ∠ 3 = ∠ 14
f) ∠ 3+∠ 10+∠ 11=180°
g) ∠ 1 + ∠ 11 = 180°
h) ∠ 2 = ∠ 11
Parallel
segments
AB//EG
Reason
corr. angles
– 11 –
This copy is for me, Peter Lin, only.
Honors Geometry
Issue 1
40. A triangle has two congruent sides, and
Standard Test
the measure of one angle is 40°. Which
37. A square is a special case of all of the
of the following types of triangles is it?
following geometric figures EXCEPT a
(A) isosceles
(A) parallelogram
(B) equilateral
(B) rectangle
(C) right
(C) rhombus
(D) scalene
(D) trapezoid
41. A triangle has one 30° angle and one
60° angle. Which of the following types
of triangles is it?
(A) isosceles
(B) equilateral
(C) right
(D) scalene
38. A polygon is a plane figure composed
of connected lines. How many
connected lines must there be to make
a polygon?
(A) 3 or more
(B) 4 or more
(C) 5 or more
(D) 6 or more
42. A triangle has angles of 71° and 62°.
Which of the following best describes
the triangle?
(A) acute scalene
(B) obtuse scalene
(C) acute isosceles
(D) obtuse isosceles
39. Which of the following statements is
true?
(A) Parallel lines intersect at right
angles.
(B) Parallel lines never intersect.
(C) Perpendicular lines never intersect.
(D) Intersecting lines have two points
in common.
– 12 –
This copy is for me, Peter Lin, only.
Honors Geometry
Issue 1
46. If pentagon ABCDE is similar to
43. Which of the following does NOT
pentagon FGHIJ, and AB = 10, CD =
have parallel two pairs of line
5, and FG = 30, what is IH?
segments?
(A) 3.5
(A) a rhombus
(B) 5
(B) a square
(C) 15
(C) a trapezoid
(D) 30
(D) a rectangle
47. What is the greatest area possible
enclosed by a quadrilateral with a
perimeter of 24 feet?
(A) 6 square feet
(B) 24 square feet
(C) 36 square feet
(D) 48 square feet
44. In a triangle, angle A is 70° and angle B
is 30°. What is the measure of angle C?
(A) 90°
(B) 70°
(C) 80°
(D) 100°
45. What is a quadrilateral with two
parallel sides and an angle of 54°?
(A) triangle
(B) rectangle
(C) square
(D) parallelogram
48. What is the difference in area between a
square with a base of 4 feet and a circle
with a diameter of 4 feet?
(A) 16 - 2π square feet
(B) 16 - 4π square feet
(C) 8π -16 square feet
(D) 16π - 16 square feet
– 13 –
This copy is for me, Peter Lin, only.
Honors Geometry
Issue 1
49. What is the difference in perimeter
52. What is the perimeter of the regular
between a square with a base of 4 feet
hexagon shown below?
and a circle with a diameter of 4 feet?
(A) 8 - 2π feet
5
(B) 16 - 2π feet
(C) 16 - 4π feet
(D) 16 - 8π feet
(A) 20
(B) 24
(C) 28
(D) 30
50. A rectangle’s topmost side is 3 times
that of the leftmost side. If the leftmost
side is A inches long, what is the area of
the rectangle?
(A) 3A
(B) 6A
(C) 3A2
(D) 6A2
53. What is the perimeter of the following
figure?
a
45o
(A) a2 ÷ 2
(B) 2a + 2a2
(C) (2 + 2 )a
(D) 4a
51. What is the perimeter of the triangle
shown below?
8
10
(A) 24
(B) 28
(C) 16
(D) 14
– 14 –
This copy is for me, Peter Lin, only.
Honors Geometry
54. The perimeter of a rectangle is 148 feet.
Its two longest sides add up to 86 feet.
What is the length of each of its two
shortest sides?
(A) 31 feet
(B) 42 feet
(C) 62 feet
(D) 72 feet
Issue 1
55. How many feet of ribbon will a
theatrical company need to tie off a
performance area that is 34 feet long
and 20 feet wide?
(A) 54
(B) 68
(C) 88
(D) 108
56. What is the outer perimeter of the
doorway shown below?
10
4
(A) 12
(B) 24
(C) 20 + 4π
(D) 24 + 2π
– 15 –
This copy is for me, Peter Lin, only.
Honors Geometry
Issue 1
Answer Key
11. 120°
Divide 180° into 2+1 parts. Each part
gets 60°. The angle in the question
should get two parts, thus it has 120°.
Supplementary and Complementary
Angles
1. x°+50° = 180°
x = 130°.
12. 45°
The complementary angle for 45° is
still 45°.
Vertical Angles
2. vertical: ∠4
adjacent: ∠2
13. The complementary angle of 35° is 55°.
The supplementary angle of 35° is 145°.
3. Vertical angle pairs: (∠1, ∠4), (∠2, ∠5),
(∠3, ∠6).
Adjacent pairs (∠1, ∠6), (∠6, ∠5), (∠5,
∠4), (∠4, ∠3), (∠3, ∠2), (∠2, ∠1).
14. 100°
The complementary angle of 40° is 50°.
The angle is two times 50°, so it must
be 100°.
4. 75°
180° - 105° = 75° since ∠1 and ∠2 are
linear pair with respect to line M.
15. ∠1 = ∠4, (vertical angle)
∠4 = ∠3 (given)
∠3 = ∠2 (vertical angle)
∠1 = ∠2 (transitivity)
5. 105°
180° - 75° = 105° since ∠2 and ∠3 are
linear pair with respect to line L.
16. 140°
∠AOB + 40° = 90° (Given)
∠AOB = 50°
∠AOD = ∠AOB + BOD (angle
addition property)
= 50° + 90°
= 140°
6. vertical angles
congruent
7. 75°
∠4 = ∠2 = 75° for vertical angles.
8. Yes, it is always true.
17. ∠1 = 90° + 45° = 135° since the
bisector divides the right angle to two
angles with 45°.
9. 150°
The angle is 30° since 90-60=30.
Therefore, its supplementary angle is
180°-30° = 150°.
18. 60 - x = 25 + 3x + 15
60 - x = 40 + 3x
20 = 4x
x=5
∠AOB = (60 - x)° = 55°
10. 60°
The complementary angle for 60° is
30°.
– 16 –
This copy is for me, Peter Lin, only.
Honors Geometry
Issue 1
Parallel Lines and Angles
Angle Bisectors
36. The answer is listed in the following
table.
19. x = 50°
x+40°=90° ⇒ x = 50°
Given
a) ∠ 2 = ∠ 8
b) ∠ 1 + ∠ 2 = ∠ 7 + ∠ 8
c) ∠ 3 + ∠ 13 = 180°
d) ∠ 8 = ∠ 15
e) ∠ 3 = ∠ 14
f) ∠ 3+∠ 10+∠ 11=180°
g) ∠ 1 + ∠ 11 = 180°
h) ∠ 2 = ∠ 11
20. x = 60° since 180°÷3 = 60°.
21. x = 120°
22.
2x - 5 + 3x + 5 = 90°
5x = 90°
x = 18°
Parallel segments
Reason
AB//EG
corr. angles
BF/CD
corr. angles.
AE//BG
consec. int.
None
AE//BG
corr. angles
BF//CD
consec. int.
None
AB//EG
alt. int. angles
Standard Test
23. Linear pair
37. D
24. complementary
38. A
25. supplementary
39. B
26. ∠2 and ∠4 are supplementary to ∠1
40. A
27. 45°
41. C
28. Yes. Straight angle has 180° and a right
angle is 90°.
42. A
29. ∠AOC = 180° - 40° = 140°
∠2 = 1∠AOC = 1(140°) = 70°
43. C
30. ∠AOB + ∠BOC = 180° (linear pair)
∠1 = 1∠AOB (bisector)
∠2 = 1∠BOC (bisector)
∠1 + ∠2 = 1(∠AOB + ∠BOC) = 90°
45. D
31. x = 180° - 2(30°) = 120°
48. B
32. ∠CBA = 80°, therefore, x = 1(80) =
40.
49. C
44. C
46. C
47. C
50. C
33. Yes, since x = y = 140°.
51. A
34. ∠COE = 40°. ∠AOC=140°. Thus, x
= 1(140) = 70.
52. D
5×6 = 30
35. 90°
5
5
5
5
5
5
5
5
53. C
– 17 –
This copy is for me, Peter Lin, only.
54. A
Honors Geometry
56. D
Issue 1
55. D
– 18 –
This copy is for me, Peter Lin, only.