Download Honors Geometry

Honors Geometry
Unit 1 – Introduction to Geometry
Day
Day 1
1.1
Thursday
8/27
Day 2
1.2
Monday
8/31
Day 3
Wednesday
9/2
Day 4
1.3 - 1.4
Friday
9/4
Day 5
1.5-1.6
Wednesday
9/9
Day 6
Objective
Packet Pages
Define basic terms of geometry.
Apply basic facts about points,
lines, and planes.
Pages 2-5
Use length and midpoint of a
segment. Construct midpoints and
congruent segments.
Pages 6-9
Quiz 1.1-1.2
Homework
Page 10
Start Angles
Pages 10-12
Name and classify angles.
Measure and construct angles and
angle bisectors.
Pages 13-16
Review Angles
Quiz 1.3-1.6
Review
Pages 17-19
Friday
9/11
Day 7
Unit 1 Test
NO HOMEWORK!
Tuesday
9/15
1
Honors Geometry
Day 1 – Basic Definitions
Term
Point
Definition
Labeling
Dot
Place in space
Line
Series of points that extends in
two directions forever
Plane
Flat and infinite surface with
no thickness
Collinear
points
Points that lie on the same line
Coplanar
Points that lie in the same
plane
Segment
Part of a line consisting of two
points and all points between
them
Endpoint
Point at one end of segment or
starting point of a ray
Ray
Part of a line that starts at an
endpoint and extends forever
in one direction
Opposite
Rays
Two rays that have a common
endpoint and form a line.
2
Picture
4 Parts of the mathematical system:
1.) Postulates: statements assumed to be true (accepted without proof)
(axiom)
2.) Theorems: statements that are proven true (can be proved true)
3.) Defined terms: terms that have a definition
4.) Undefined terms: can be explained by using examples and descriptions
IMPORTANT POSTULATES AND THEOREMS ABOUT POINTS, LINES, AND PLANES
1) Postulate: Through any two points, there is exactly one ____________.
2) Postulate: Through any three non-collinear points, there is exactly one ______________.
3) Postulate: If two points lie in a plane, then the __________________ containing those
two points lies entirely in the plane.
4) Postulate: If two planes intersect, then their intersection is a _____________.
_________________________________
5) Theorem: If two lines intersect, then their intersection is exactly one ______________.
6) Theorem: Given a line and a point that is NOT on that line, there is exactly one
______________, which contains both the point and the line.
7) Theorem: If two lines intersect, there is exactly one _____________, which contains the
two lines.
ALWAYS, SOMETIMES, NEVER
1) Two points lie in exactly one line.
Circle the letter(s) that apply:
8) Three different points:
a) always determine a plane
b) can be collinear
c) can be non-collinear
d) always lie in at least one plane
2) Three points lie in exactly one line.
3) Three collinear points lie in exactly one plane.
4) Two intersecting planes intersect in a segment.
9) A plane is determined by:
a) a line and a point
b) two intersecting lines
c) any three points
d) a line and a point not on the line
5) Three points determine a plane.
6) Two intersecting lines determine a plane.
7) Two non-intersecting lines determine a plane.
10) Three points are _____________ coplanar. (A/S/N)
3
4
Draw each of the following.
1.
2. Line l intersects
plane M in point P.
AB and point C are
coplanar, but A, B, and C
are non–collinear points
M.
4. Points W, X, Y, and Z
are coplanar, but points
W, X, Y, and P are not.
8.
3. Plane M and plane
N intersect in
PX lies in plane
5. Plane M contains
AB and
XY , but AB and XY do not
AB .
6.
AB Intersects plane M in point X.
PQ is contained in plane M, but
intersect each other.
AB and PQ do not intersect each other.
Plane M and plane N intersect in
7.
XY . Points X, Y, and Z are coplanar
but not collinear. Points X, Y, Z, and W
are neither coplanar nor collinear.
5
Line l intersects plane M in
point A. Line k intersects line l at
point B. Line k does not intersect
plane M.
Day 2 – Segments and lengths
Ruler Postulate: points of a line can be put into a one-to-one correspondence with the set of
real numbers so that no two points are paired with the same coordinate.
letters- points (top)
A

-4
“
coordinates- numbers (bottom)
C

-2
B

-3
D

-1
E

0
AB ” refers to “segment AB”
F

1
G

2
H

3
I

4
L

7
K

6
J

5
M

8
“AB” refers to the length of segment AB
CJ = the distance between points C and J = number value which can never be negative!
Distance: between any two points is the absolute value of the difference of the coordinates.
Distance Formula:
Length: distance between two points
One dimensional distance: To find the distance you subtract the coordinates and then take
the absolute value to ensure the answer is positive!
BD = _____
AM = _____
EK = ______
HM = ______
Congruent Segments: segments that are equal in length. (have the same lengths)
T or F
Use the number line at the top.
1. BG  DK
2.
HK  AD
Midpoint: a point that divides a segment into 2  segments.
1.) If V is the midpoint of AG , then _____  _______.
2.)
C is the midpoint of
3.) Y is the midpoint of
a) coord. Of X is –8
coord. Of Z is 6
coord. Of Y = ____
AB . AC = 2x + 1 and CB = 3x – 4 . Find x, AC, CB, and AB.
XZ . Find the coordinate of the 3rd point for each example.
b) coord. of X is 11
coord. of Y is 15
coord. of Z = _____
6
c) coord. of Y is -8
coord. of Z is -19
coord. of X = _____
4) Find the midpoint of AG if A(4, -7) and G(-9, 4). ______________
5) Find B if M (1, 11) is the midpoint of BT and T(7,-14) _________________________
6) Use the Distance Formula to find the distance between K(-7, -4) and L(-2, 0)._______________
7)Find the length of F(9, 5) and G(–2, 2). _____________________________
Segment Bisector: A segment, line, or ray that intersects a segment at its midpoint.
Line a bisects MS at point D.
What is the first conclusion you can make?
____________________________
What is the second conclusion? ________________________
H

HL and KT intersect at the midpoint of HL .
True or False.
1. KM = MT
2. KT is a bisector of
T
M
LH .
3.
MT bisects LH .
4. HL is a bisector of KT .
5. M is the midpoint of KT .
K
L
In order for you to say that point B is between two points A and C, all three points must lie on
the same line and AB + BC = AC.
SEGMENT ADDITION POSTULATE (Def. of Between)
If R is between S and I, then SR + RI = SI
**note the segment symbols are not used**
Ex 1.
a) If SR = 15 and RI = 22, find SI.
●
●
S
R
b) If SR = 5 and SI = 18, find RI.
c) If SI = 60, SR = 2x – 8 , RI = 3x – 12 , find x and then find SR and RI.
d) If SR = x2 + 7, RI = 4x – 5 , and SI = 34, find x.
7
●
I
1. What set of numbers does the Ruler Postulate put in one-to-one correspondence with the set of
points on a line?
2. If 6 is the coordinate of point X on
AB , can any other point on AB
have 6 as a coordinate? Why?
3. If c is a real number, is there a point on line m paired with c?
4. How many points are contained in a line? Why?
5-8 use the number line:
A

-4
C

-2
D
0
E
1
F
2
G
4
H

5
I
6
5. Name two points whose distance from E is 3.
6. Name two points whose distance from C is 2.
7. What does CH represent? Find CH.
8. Find IA.
In 9 – 14, the numbers given are the coordinates of two points. Fin the distance between the points.
9. -8 and 4
10. -13 and -14
11. 0 and 7.5
12. 12 and 37
13. -5.6 and 7.4
14. a and b
In 15 – 20, Points R,S, and T are distinct and collinear. In each case draw a segment representing the situation given. Tell
which point is between the other two.
15. RS = 7, ST = 3, RT = 10
16. RS = 6, ST = 14, TR = 8
17. ST = 6.3, RT = 4.7, RS = 1.6
18. TR = 1.2, RS = 7, TS = 5.8
Make a number line representing the situation. Find the missing coordinate.
19. RS = 10 and RT = 14. The coordinate of S is 7. The coordinate of T is 3.
20. TS = 7 and TR = 9. The coordinate of S is -2. The coordinate of R is 0.
8
For the next examples, draw a picture, give the formula, and find a solution.
Example 1 -In the figure below, B is the midpoint of . AB = 2x + 12 and BC = 5x + 10. Draw the
example, come up with an equation, and find x and AC.
Example 2- In the figure, RS = 2x – 4, ST = 3x + 7, and RT = 43. Draw the example, come up with an
equation, and find x, RS, and ST.
Example 3 -In the figure below, C is the midpoint of
come up with an equation, and find x and AC.
. AC= 5x-6 and CB = 2x. Draw the example,
Example 4 -In the figure, RS = 3x – 12, and ST = 2x – 8, and RT = 60. Draw the example, come up
with an equation, and find x, RS, and ST.
Example 5 G is the midpoint of
equation, and find x and EG.
. EG = 3y, and GF = 36 – y. Draw the example, come up with an
Example 6. If SR = x2 + 7, RI = 4x – 5 , and SI = 34, find x. Draw an example.
9
Day 4 – Angles and Angle Bisectors
Angle: the union of 2 non-collinear rays with a common endpoint
Vertex:
B
1
Sides:
A

Name this angle in 4 different ways:

C
Interior of an angle:
Exterior of an angle:
Name three of the angles.
Measure: of an angle is usually given in degrees.
Protractor Postulate: Given line AB and a point O on AB, all
rays that can be drawn from O can be put into a one-to-one
correspondence with the real numbers from 0 to 180.
Acute-
Obtuse-
Right-
StraightPerpendicular –
Example - Use the diagram to find the measure of each angle. Then classify each as acute, right, or
obtuse.
a) <BOA
b) <DOB
c) < EOC
d) <EOB
e) <ABD
10
Congruent Angles: angles that have the same measure.
Angle Bisector: a ray that divides an angle into two congruent angles.
Example 1 - Draw a picture. If CD bisects  ACB,
Then _______  _______.
Example 2 –Draw a picture and solve. m<DEG = 115 and m<DEF = 48. Find m<FEG.
Example 3- Draw a picture for the following and write an equation to solve for x. RQ bisects  PRS. If
m  PRQ = x + 40 and m  QRS = 3x – 2, solve for x,
Example 4 - Write an equation to solve for x.
x = _____________
m BAD = 122, x = _______
Example 5 - Draw a picture for the following problem. E is on the interior of <SRP. mSRP = 157 m 
ERP = 98
Find mPRS = _________and
m SRE = _________
Example 6- Draw a picture for the following and write an equation to solve for x. If SU bisects <RST, and
m<RSU = 2x - 11, and m<RST = 3x+23, find
x = ___________
m<TSU = ______________ m<RST = ________________
11
Example 7 - Draw a picture for the following and write an equation to solve for x. <DEF is a right angle.
m<DEG = (3x + 8) ° and m<GEF = (6x – 17) °. Find the m<DEG.
Write the equation for #s 9-13.
12
Pairs of Angles
*Two angles are ________________ if their sides form two pairs of opposite rays.
2
1
Examples:
3
4
vertical angles are always ______________
*Two angles are ________________ if they have a common side, a common vertex, and no common
interior points.
1
2
3
Examples:
*Two angles are ____________________ if the sum of their measures is 90.
Each angle is a complement of the other.
A
35
Examples:
55
C
B
*Two angles are ____________________ if the sum of their measures is 180.
Each angle is a supplement of the other.
Examples:
E
Linear Pair: _____________angles whose non-common sides form a _________
linear pair is always _____________and ________________
m<1 + m<2 = ________
1
13
2
D
Use this figure for 1 - 6
1. Name all angles shown.

G
2. Name 1 in 2 other ways.
H
1

D
3. Name all angles which have side EF .
2

E
3

F
4. Name a point in the interior of GEF.
5. Name a point in the exterior of 2.
6. Name 2 pairs of adjacent angles.
State whether the numbered angles shown in 9 – 16 are adjacent or not. If NOT, explain.
7.
8.
2
1
12
1
D
2
B
1
2
11.
A
10. ABC, ABD
9.
C
12.
1
2
13.
14.
1
2
1
2
Verifying Angle Relationships-Label the diagram below so that m∠BGC=33 and m∠DGE=57
1. ∠FGA ≅ __________2. m∠CGD = _________
5. ∠EGC and ______________are supplementary
3. ∠BGF and _________ are supplementary
4. m∠AGF = _________
6. m∠AGB = __________
F
A

7. m∠AGC = ________
8. m∠BGD= _________
9a. Are ∠AGF and ∠CGD supplementary?________why/why not?
B
G

Solve the following for the indicated variable. Set up an equation using the angle relationships
1.
2.
3x – 5
6x – 23
x + 16
14
E

D

C
b. Are they a linear pair?________why/why not?

2x – 16
3.
4.
x
3x – 8
50
2y – 17
3x – y
x
2x – 36
More Angles
1. The supplement of a right angle is a ____________angle.
2. The supplement of an obtuse angle is a _____________angle.
3. The supplement of an acute angle is a ______________angle.
4. Vertical angles are ____________.
5. ________________lines form right angles.
6. Congruent supplementary angles each have a measure of ___________.
7. Congruent complementary angles each have a measure of __________.
8. The angles in a linear pair are ___________________and_____________________.
9. m<1 = 40. What is the measure of its complement? _________Its supplement? ________
10. m<2 = 120. What is the measure of its complement? _________ Its supplement?_________
11. m<A = x. What is the measure of its complement? __________Its supplement? __________
Complete with Always, Sometimes, or Never
1. __________ Two <’s that are supplementary ___________form a linear pair.
2. __________ Two <’s that form a linear pair are ____supplementary.
3. __________Two congruent angles are _________right.
4. __________Two right angles are ___________congruent.
5. __________ Two angles that are vertical are _________adjacent.
6. __________ Two angles that are non-adjacent are _________vertical.
7. __________ Two angles that form a linear pair are ___________congruent.
8. __________Two angles that form a right angle are ______complementary.
9. __________Two angles that form a right angle are ______ supplementary.
10. _________ Two angles that are supplementary are _______ congruent.
11. __________ Vertical angles ________ have a common vertex.
12. ___________ Two right angles are _________ complementary.
13. __________ Right angles are __________ vertical angles.
14. __________ Angles A, B, and C are __________ supplementary.
15. __________ Vertical angles __________ have a common supplement.
15
Given: ∠1 is a right angle, m∠5=30, and ∠2 ≅ ∠3. Find the following
1
2
3
4
m∠1=______ m∠2=_____
7
●
6
5
m∠3=_____
m∠4=_____
m∠6=_____
m∠7=_____
m∠3 + m∠2 +m∠1 = ___________
EOA is
a.
b.
c.
d.
e.
f.
Acute
Right
Obtuse
Linear Pair
Congruent
Vertical
________
EOA and DOB are
________
COB is
_________
EOA and AOB are
__________
EOD and AOB are
__________
DOC is
__________
DOC and <AOC
g.
h.
i.
Complementary
Supplementary
Adjacent
_________
DOC and COB are
__________
COB and BOA are
___________
Incorporating Algebra into Geometry
I. Define variables and write an equation for each problem, then solve and circle final answers.
1. Two angles are complementary and one’s measure is twice the measure of the other. Find the measures of the two angles.
2. Two angles are complementary and the measure of one angle is 36 less than the other. Find the measures of
the two angles.
3. The supplement of an angle exceeds five times the measure of the angle by 6. Find the measures of the
supplementary angles.
4. The supplement of an angle exceeds five times the measure of the complement of the angle by 10. Find the
measure of the angle.
5. An angle is six less than two times the measure of its complement. Find the measure of the angle.
6. The measure of a supplement of an angle is 15 more than 2 times the complement. Find the measures of the
angle, the complement, and the supplement.
16
Honors Geometry
Review Sheet Unit 1
I. True/False
Name______________________
________1. EF is in plane Y.
________2. Plane X intersects plane Y in CD .
________3. C, G, D, and H are non-coplanar points.
X
________4. Point D is in planes X and Y.
________5. GH lies in both planes.
G

A

E
________6. EF and GH intersect each other.
________7. AB and GH intersect each other.

M
C

F

________8. Points A, M and D are non-coplanar.
D

B
Y
J
H
________9. Points C, M and D are collinear.
________10. Plane X has four edges.
________11. Points A and H are collinear.
________12. CD is in both planes X and Y.
________13. AJ lies in plane Y.
________14. It is possible for a line and a point not on the line to be in more than one plane.
________15. It is possible for 3 collinear points to be in many different planes.
II. What kind and how many points does it take to determine:
____________________________________One line
____________________________________One plane
____________________________________Space
Q
R
III. True/False Use the diagram at the right.
_________1. Point U is the endpoint of QS .
U
_________2. QS bisects RT .
_________3. RT is a bisector of QS .
_________4. QU + US = SQ.
T
S
_________5. UQ and US are opposite rays.
_________6. QU and TS do not intersect, but QU and TS would.
_________7. If RT was a bisector of QS , then U would be the midpoint of RT .
_________8. TU  RU
J
IV. Use the number line to answer 1 – 3.

_________1. JM = __.
-10
_________2. Find x if M is the midpoint of LN .
_________3. What is the coordinate of the midpoint of KM ?
17
K

-1
L

3
M

18
N

x
V. Fill in the blank or True or False using the line to the right:
_____1. EO and OE are opposite rays.
G
|
|
E
|
O
|
M
|
T
|
R
_____2. Point O lies on GE .
_____3. GM and GR name the same ray.
_____6. The ray opposite of TG is ____.
A
Use the diagram to the right for 7 – 11.

D
_____7. FA and FB are opposite rays.
_____8. F is the midpoint of DC .
_____9. DC is a segment bisector.
_____10. F is the midpoint of AB .

F
C

B
_____11. BA and AB name the same set of points.
VI. Use the number line to the right for 1 – 3.
1. TA = _________; RK = _________
2. TR  ______
3. A is the midpoint of _______.
T
|
-4
|
R
| |
-1
|
A
C K
1 2
3
VII. Draw a diagram to answer the following:
_________1. V is the midpoint of TM , VM = 8, TV = __.
_________2. M is the midpoint of XY , XM = 8 – x , MY = 5x – 10. Find XY.
_________3. The coordinate of T is -9, the coordinate of P is 4. Find the coordinate of the midpoint of TP .
_________4. M is the midpoint of AB . The coordinate of A is 5, and M is -1. Find the coordinate of B.
VIII. Answer the following:
_________1. A(-4,0) and B(-2,4). Find the coordinates of the midpoint of AB .
_________2. Find AB in #1.
_________3. The midpoint of RS is U. If U(-3, 5) and S(2, 13) Find the coordinates of R.
_________4. Find the points of intersection for: a) y = 2x and 3x + y = 10
b) 4x + 2y = -10 and 3x + y = -6
________
IX. True or False
_______1.
_______3.
_______4.
_______5.
_______6.
Three points determine a plane. _______2. If a point P of PQ lies in plane X, then so does point Q.
If 2 lines intersect, then they intersect in exactly one point.
Two intersecting lines are always contained in exactly one plane.
Three non-collinear points can lie in each of 2 different planes.
Horizontal planes will never intersect.
Answer the following:
1. In order to determine a line, what is required?
2. In order to determine a plane, what is required? (There are three possible answers.)
3. Why does a three-legged support (tripod) work better than on with 4 legs?
18
4. If 3 non-collinear points determine a plane, how many points determine space?
5. How many planes may contain three collinear points?
II. True or False?
1. _____________A given triangle can lie in more than one plane.
2.______________Any two points are collinear.
3.______________Two planes can intersect in only one point.
4.______________Two lines can intersect in two points.
III. Use the diagram below to answer the following questions.
1. How many lines contain both of the points A and E? Why?
2. List 3 points that determine Plane K.
3. Could both of points A and C be contained in the intersection of
AC and AB
? Explain.
4. Why do points B, C, and E determine a plane?
5. Is
FC
6. Why do
contained in Plane K ? Why/ why not?
AD
and
CD have only one point in common?
7. Consider the planes determined by A, E, and B and D, C, and E. Do they have more than one point
in common? Why/why not?
8. Why do
DE
and point F determine a plane?
9. How many planes contain
AC and CE ?
Why?
10. Name 2 lines contained in the plane determined by A, E, and B.
19