Download Geometry Review

SEMESTER 1 EXAM REVIEW
• This review was put together for January 2010. It
covers chapters 1-5 and of course emphasizes
what I felt was important which may be different
from you. Feel free to modify or use as you wish.
This will be the class review-and of course it will
come with the reward of Jolly Ranchers for
correct responses!!
• If you have any questions you can ask me-if you
can find me!!!!
• Diane Bilyeu
Chapter 1
True-False
• 1. Two points are needed to determine a line.
– True
A
B
• 2. Two points are needed to determine a
plane
– False- a plane needs three points to establish the
plane.
A
B
C
REFER TO DIAGRAM 1
•
•
•
•
•
•
•
3. NAME 3 COLINEAR POINTS
4. NAME A PLANE
5. NAME THREE NON COLINEAR POINTS
6. NAME TWO LINES
ANSWERS
3. CDE
4. T or S
5. Various
6. x, or y, or z
True or False
•
•
•
•
•
•
7. A line has two endpoints
8. A segment has two endpoints
9. A ray has two endpoints
10. Identify the figure on diagram 2 A
11. Identify the figure on diagram 2 B
12. The intersection of two lines is a segment.
Answers
7. False
8. True
9. False
10. Ray XY 11. Segment LM 12. False
A
B
C
13. If AC is 8, AB is 2, then BC= __________
14. If AB is 87, BC is 40, then AC = ________
15. If BC is 104, AC is 214, then AB = ______
16. What is a “tick mark?”
Answers
13. 6
14. 127
15. 110
16. Shows congruence between figures.
Naming Angles
•
A
•B
C
D
17. Name 3 different angles in the diagram.
<ABC, <CBD, <ABD
Why not <B?
Classifying Angles
•
•
•
•
•
•
•
• Acute, Right, Straight, Obtuse
18. An angle with a measure of 180 degrees
19. An angle with a measure of 90 degrees
20. An angle with a measure of greater than 90
degrees but less than180 degrees
21. An angle with a measure of less than 90
degrees
ANSWERS
18. Straight
19. Right
20. Obtuse
21. Acute
Angle Measures
A
•
B
C
D
22. If m <ABC = 42, and the m < CBD = 12, what is the
m<ABD?_____________
23. If the m < ABD is 67, and the m < ABC is 50, what is
the m < CBD?________________
24. If the m<ABD is 87 and the m<CBD is 42, what is the
m>ABC?__________________
ANSWERS
22 54
23. 17
24. 45
Chapter 2
A
1.
2.
3.
4.
5.
6.
7.
B
C
If AC = 12, AB = 6, then BC= ____________
If AB = 19, BC=10, then AC= _____________
If AC= 94, BC=18, then AB = _____________
If AB =6x, BC=4x, and AC =200. Solve for x
If AB= 14x, BC =24 and AC=20x. Solve for x
AC=24. if B is the midpoint, what is AB?___
B is a bisector of AC. If AB=72,and BC is 9x, what is the
value of x
ANSWERS
•
•
•
•
•
•
•
•
1.
2.
3.
4.
5.
12-6 = 6
19 + 10 = 29
94-18 = 76
6x + 4x = 200 10x = 200 x=20
14x + 24 = 20x
24 = 6x x=4
6. 12
7. 72 = 9x x=8
Midpoint Formula (55)
• 8. Find the midpoint of A ( 1,2) and B (7,4)
1+7
2
4+2
2
8
2
6
2
(4,3)
9. Find the midpoint of C (-2, 3) D (5, -1)
5 + (-2) (-1 + 3) 3 2 ( 3/2, 1)
2
2
2 2
Bisectors
• Refer to diagram 3
10. Ray AC bisects<BAD. The measure of m<BAD is
102. What is the measure of <BAC?
11. Ray AC bisects <BAD. m<BAD= 150 degrees and
the m< CAD = 10x. Solve for x.
ANSWERS
10. ½ X 102 = 51
11. M<CAD= ½ x 150 =10x 75 =10x 7.5 = x
Bisectors
12. AC is an angle bisector. m< BAC=60 degrees,
m<CAD = 5x. Solve for x.
Refer to diagram 4. BC is a bisector of <ABD
13. Solve for x.
• ANSWERS
• 12. 60 =5x 12=x
• 13. 8x + 3 = 9x 3=x
Supplementary/Complimentary Angles
14. What is the compliment of a 78 degree angle?
15. What is the supplement of a 129 degree angle?
16. What is the supplement of a 46 degree angle?
17. What is the compliment of a 46 degree angle?
ANSWERS
14.12
15. 51
16. 134
17. 44
VERTICAL ANGLES
• Refer to diagram 5.
• 18. If the m< 1 = 113 find the measures of < 2,
<3, <4.
• m<2 = 67, m<3 =113, m<4= 67
• 19. If the m<1 = 120 and the m< 4=6x. Solve
for x.
• 120 + 6x = 180 6x = 60 x=10
Vertical Angles
Refer to Diagram 5
• 20. If the m< 4 = 75 and the m<2 =3x. Solve
for x.
• 75 =3x 25 = x
• 21. If the m<3 =97, find the measures of <1,
<2, and <4.
• m>1 =97, m<2 = 83, m<3 = 83
Angle Measures
Refer to diagram 6.
22. Find the measures of the angles.
m<1=40, m<2=140, m<3= 40
Refer to diagram 7.
23. Find the measures of the angles.
m<1=64, m<2= 93, m<3 =23, m<4 =93
Refer to diagram 8
24.Find the value of x and the measures of the
angles.
144+6x =180 6x=36 x=6 m<1=144, m<2=36
Properties of Equality and Congruence
• Reflexive Property???
• Symmetric Property????
• Transitive Property????
Chapter 3
• Refer to diagram 9.
• 1. Name 2 lines that are parallel
• 2. Name 2 lines that are perpendicular
• True or false
• 3. All right angles are congruent.
• 4. If two lines are parallel, then they intersect to
form four right angles.
• 5. Perpendicular lines form right angles.
• 1. c, d 2. b, d 3. True 4. False
5. True
Practice
•
•
•
•
•
•
•
•
•
•
•
Refer to page 116 in your book.
6. Do problem 7
X=18
7. Do problem 8
X=6
Refer to page 118 in your book.
8. Do problem 23 -Read directions carefully.
x=10, m<CBD =70
Refer to page 120 in your book
9. Do problem 7.
x=34
Angles formed by Transversals
•
•
•
•
•
•
•
Refer to diagram 10.
10. List 4 pairs of corresponding angles.
11. List 2 pairs of alternate interior angles.
12. List 2 pairs of alternate exterior angles.
13. List 2 pairs of same-side interior angles.
14. List 4 pairs of vertical angles.
15. List 2 pairs of supplementary angles.
Answers to 10-15
•
•
•
•
•
•
10.
11.
12.
13.
14.
15.
1+5, 3+7, 2+6, 4+8
3+6, 4+5
1 +8, 2+7,
3+5, 4+6
1+4, 2+3, 5 +8 6+7
5+6, 5+7, 1+3, 2+4, etc…..
True or False
•
•
•
•
•
•
•
•
•
16. Corresponding angles are congruent.
17. Same side interior angles are congruent.
18. Alternate exterior angles are congruent.
19. Alternate interior angles are congruent.
Refer to page 131 in your book.
20. Do problem 16.
21. Do problem 17.
Refer to page 133 in your book.
22. Do problem 33
Answers to 16-22
•
•
•
•
•
•
•
16.
17.
18.
19.
20
21.
22.
True
False
True
True
x=85
x=104
x=23
Chapter 4
Triangles
•
•
•
•
•
•
•
•
•
Equilateral, Isosceles, Scalene
1. A triangle with 2 equal sides_________
2. A triangle with no equal sides_________
3. A triangle with 3 equal sides__________
Equiangular, Acute, Obtuse, Right
4. A triangle with all angles less than 90 degrees.
5. A triangle with 1 angle of exactly 90 degrees.
6. A triangle with one angle of over 90 degrees.
7. A triangle with all angles of equal measure (60
degrees)
Answers to 1-7
•
•
•
•
•
•
•
1.
2.
3.
4.
5.
6.
7.
Isosceles
Scalene
Equilateral
Acute
Right
Obtuse
Equiangular
Practice
• Refer to page 176 in your book
• 8. Do the following problems
–
–
–
–
–
–
–
–
–
–
–
–
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
Answers to 11-22
• 8. Do the following problems
–
–
–
–
–
–
–
–
–
–
–
–
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
Scalene
Isosceles
Equilateral
Isosceles
Scalene
Isosceles
Acute
Acute
Right
Acute
Right
Equilateral
What is the Triangle Sum Theorem?
If you add the measures of the angles of a triangle they total 180
What is the Exterior Angle Theorem?
The sum of an interior angle and it’s adjacent exterior angle total 180
• Refer to page 182-183 in your book.
• 9. Do the following problems
–
–
–
–
–
6.
11.
12.
18.
20.
Answers to Questions
–
–
–
–
–
6.
11.
12.
18.
20.
71
45
148
33
x=48, y= 32
True or False
• 10. The congruent sides of an isosceles triangle
are called the legs.
• 11. The third side of an isosceles triangle is called
the base.
• 12. If two sides of a triangle are congruent, the
opposite angles are also congruent.
• 13. If two angle of a triangle are congruent the
opposite sides are also congruent. True
• 14. If a triangle is equilateral it is also
equiangular.
• ALL ANSWERS ARE TRUE
Practice
• Refer to page 188-189 in your book.
• 15. Do the following problems in your book.
Find the value of x
– 7.
– 9.
– 10.
– 14.
– 18.
– 25
ANSWERS TO QUESTIONS
– 7. x= 55
– 9. x =45
– 10. x = 7
– 14 5x + 7 = 52 5x =45 x=9
– 18. 80
– 25. 8y – 10 = 4y + 2 4y =12 y=3
What is the Pythagorean Theorem?
Leg squared + leg squared = hypotenuse squared
What is the Distance Formula?
See page 194
• Refer to page 195 in your book.
16. Do the following problems in your book.
8.
10.
14.
15.
ANSWERS
8. 15
10. 97
14. 7
15. 80
DISTANCE FORMULA
17. Find the distance between the points
A (1, 4 ) B (3, -2)
18. Find the distance between the points
C (-2, 2) D (-3, -3)
Classifying Triangles
Pythagorean Theorem
• Refer to chart on page 201 in your book.
• Classify the triangles based on the length of
their sides.
• 19. 4, 6, 7
• 20. 12, 35, 37
• 21. 2, 5, 6
– ANSWERS 19. ACUTE 20. RIGHT 21. OBTUSE
Medians of a Triangle
• Median of a triangle is a segment from a vertex to
the midpoint of the opposite side.
• The intersection of the 3 medians of a triangle is
called the centroid. The centroid is a point that
is 2/3 the distance from the vertex to the
opposite side. (see page 208)
• Refer to page 209 in your book.
• 22. Do the following problems in your book.
–
–
–
–
4.
6.
7. PT =
8. BD=
ST =
ED=
ANSWERS TO QUESTIONS
– 4.
– 6.
– 7.
– 8.
4
3
PT= 22, ST= 11
BD= 18 ED= 6
Chapter 5
• Refer to diagram 11
• 1. List 3 pairs of congruent angles and 3 pairs
of congruent sides.
• 2. Write a congruence statement for the two
triangles
• ANSWERS
• 1. D+ T, C+S, E+T, CD=ST, TW=DE, CE=SW
• 2.
CDE = STW
• Refer to diagram 12
• 3. Write a congruence statement for the two
triangles.
• 4. Why is m<LJK congruent to m< HJG?
• 5. Why is m<JLK congruent to m <JGH
• ANSWERS
• 3.
JKL =
JHG (VARIOUS)
• 4. Vertical angles
• 5. Corresponding parts of congruent triangles.
Proving Triangles Congruent
SSS, SAS, ASA, AAS, HL
Identifying Congruent Triangles
• Writing a Proof
– List the given information first
– List the information shown in the diagram
– Give a reason for every statement
– Use definitions, postulates, theorems etc.
– End the proof with what you are trying to prove.
L
i
s
Write a two column proof
(Refer to diagram 13)
• Given:
c JL is congruent to NL
o L is the midpoint of KM
•
n
Prove: g JKL is congruent to
NML
r
Statement
u
e
1. JL is congruent
to NL
n
2. L is the
t midpoint to KM
3. <JLK is congruent to <NLM
t
4. KL is ocongruent to ML
5. JKLN is congruent to
NML
M
L
6. Page 243 in book.
Reason
1.
2.
3.
4.
5.
PROOF ANSWERS
• Given: JL is congruent to NL
•
L is the midpoint of KM
Prove:
JKL is congruent to
NML
Statement
1. JL is congruent to NL
2. L is the midpoint to KM
3. <JLK is congruent to <NLM
4. KL is congruent to ML
5. JKL is congruent to
NML
1. Page 243 in book.
Reason
1. Given
2. Given
3. Vertical Angles
4. Def. of midpoint
5. SAS
More Proofs
•
•
•
•
•
•
•
•
•
•
•
•
Refer to page 247-248 in your book
7. Do problem 34 in the book
8. Do problem 35 in the book
9. Do problem 36 in the book
Refer to page 255-256 in your book
10. Do problem 34 in the book
11. Do problem 35 in the book
Refer to page 262 in your book
12. Do problem 32 in your book
Refer to page 270 in your book
13. Do problem 19 in your book
14. Do problem 20 in your book
Using overlapping triangles
Refer to page 269 in your book. Sketch triangles
separately.
15.Do problem 10 in your book
16.Do problem 17 in your book Set up as a
proof.
Answers
15.SAS
16.AAS
Using Bisectors and Perpendicular
Bisectors
•
•
•
•
•
•
•
•
Refer to page 277-278 in your book
17. Do problem 10 in your book
18. Do problem 11 in your book
19. Do problem 14 in your book
20. Do problem 19 in your book
Answers
17. 10
19. 38
18. x=6
20. AD= 12, BC = 16