Khan Academy
Introduction to Geometry Summer Package
Table of Contents
Go to www.khanacademy.com. Under “Learn” select “Math”. Then select
“Geometry” and finally, “Introduction to Euclidean Geometry” or “Angles and
Intersecting Lines”. Within these sections will be a combination of lessons and
quizlets. The Lessons have an arrow beside the title and the Quizlets have an astric.
As you watch the lessons, fill in the answers to the questions in the form with the
same title. After watching the lessons, you are to complete the quizlets and fill in the
form provided. Remember, a parent is to sign the quizlet.
I.
Introduction
A.
→Euclid as the Father of Geometry
p. 4
B.
→Language and Notation of basic geometry
p.5
C.
→Lines, line segments, and rays
p.6
D.
*Recognizing rays, lines and line segments
p.6
E.
→Drawing lines (exercise example)
p.7
F.
*Drawing lines
p. 7
G.
→Specifying planes in 3-D
p. 8
H.
*Points, lines and planes
p. 9
I.
→Language and notation of the circle
p. 9
J.
→Golden Ratio
p. 10
K.
→Identifying Rays
p. 11
1
I.
Angles and Intersecting lines
A.
B.
Angle basics
1) →Angle basics
p. 11
2) →Measuring angles in degrees
p. 12
3) →Using a protractor
p. 12
4) →Measuring angles (watch lesson only)
p. 13
5)
* Measuring Angles Quizlett
p. 13
6)
→Acute, right and obtuse angles
p. 14
7) *Angle types
p. 14
8) →Vertical, adjacent and linearly paired angles
p. 15
9) *Exploring angle pairs
p. 16
Angles between intersecting and parallel lines
1.
→Introduction to vertical angles
p.17
2.
*Vertical angles
p.17
3.
→Using alg. to find the meas. of vert. angels
p.18
4.
*Vertical angles – 2
p. 18
5.
→Proof: Vertical angles are Congruent
p. 19
6.
→Angles formed by parallel & transversals
p. 19
7.
→Identify Parallel and Perpendicular lines
p. 20
8.
→Figure out the Angle between Transversals and Parallel
Lines
p.20
9.
*Congruent Angles
p. 21
2
10.
*Parallel lines-1
p. 21
11. →Use Algebra to find the measure of angles from transversal
parallel lines -2
p. 22
12.
*Parallel lines -2
p.22
13.
CA Geometry: Deducing Angle measurements (optional)
3
Khan Academy
Euclid as the Father of Geometry
1. Who was the _______________ philosopher who lived ________ years before Christ considered to be the
___________________ of _____________? __________________________________
2. Write the Quote from this philosopher which is at the top of the screen.
______________________________________________________________________________
3. Why was Euclid considered the Father of Geometry? ___________________________________
______________________________________________________________________________.
4. This book which is a _________ volume “textbook”, was written in Greek about ______________ and
translated into English in ____________.
5. Euclid began with basic ___________________ called _______________________ or
___________________________ and proved other statements called _____________________.
6. For the next 2,000 years if you did not read and understand Euclid’s Elements you were considered
_____________________.
7. This _________________ textbook was second only to the ________________ in the number of printings
until the late 1700’s early 1800’s.
8. Which American President studied Euclid’s Elements? __________________________________
9. Why did he study Euclid’s Elements? ________________________________________________
______________________________________________________________________________
4
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Language and Basic Notation
1. The word “geometry” is broken down into two parts, each with its own definition. What are the two
parts and what are their meaning?
Part 1:
Meaning:
Part 2:
Meaning:
2. A point is just a __________________________. You cannot _____________ on a point in any direction.
To refer to points we use ___________________________.
3. If we start at point A and connect all points to point B, we draw a ____________________ and can
name/label it _______________. A line segment is labeled with its two _______________________.
4. How many dimensions does a point have? __________________
5. How many dimensions does a line segment have? __________________
6. Label line segment AB correctly in the space provided ________________
7. What is the difference between ray AD and ray DA? __________________________________
______________________________________________________________________________
______________________________________________________________________________________
______________________________________________________________________
8. In ray AD, endpoint A is also called the ___________________ of ray AD.
9. What are collinear points? ________________________________
10. If point Z is half way between point X and point Y, point Z is called the _________________________ of
line segment XY.
11. Things which are two dimensional such as a computer screen or a piece of paper but which also
continue/extend in both directions forever and are called ___________________ or
___________________.
5
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Lines, Line Segments and Rays
1. A line segment has starting ________________ and an _______________ __________________.
2. A line has no ___________________ _____________________ and no ___________________
____________________ but continues in both directions forever.
3. A ray has a starting _________________ but not ________________________, it _______________ in
that direction forever.
4. What do we draw at the end of a ray or at the ends of a line to indicate that it continues forever?
________________________
Khan Academy
Recognizing Lines, Line Segments and Rays
This quizlet is online at Khanacademy.com under Geometry/Introduction to Euclidian Geometry/Recognizing
Lines Segments and Rays.
Directions: You are to successfully answer 5 questions in a row for the quizlet to give you your score. If you
cannot answer five in a row the first time, continue to answer questions until you complete 5 in a row. Record
your scores in the upper chart. If you struggle with this quizlet and answer a total of 30 questions, fill in the lower
chart.
Upper Chart
Points Earned ____________ Bonus Points For Completion ______________ Total Score __________
__________ questions answered
-
__________ Percent
Or –
Lower Chart
__________ questions out of 30 answered correctly.
Parent’s signature __________________________________
6
Khan Academy
Drawing Lines Quizlet
Watch “Drawing Lines Example” prior to taking the quizlet.
This quizlet is online at Khanacademy.com under Geometry/Introduction to Euclidian Geometry/Drawing lines.
Directions: You are to successfully answer 5 questions in a row for the quizlet to give you your score. If you
cannot answer five in a row the first time, continue to answer questions until you complete 5 in a row. Record
your scores in the upper chart. If you struggle with this quizlet and answer a total of 30 questions, fill in the lower
chart. Be sure to scroll down in order to see the full problem.
Upper Chart
Points Earned ____________ Bonus Points For Completion ______________ Total Score __________
__________ questions answered
-
__________ Percent
Or –
Lower Chart
__________ questions out of 30 answered correctly.
Parent’s signature __________________________________
7
Khan Academy
Specifying Planes in 3-D
1. You can view a plane as _____________ __________________ that exist in ____ dimensions which goes
off in every ___________________.
2. Why can you not specify a plane by using one point or two points (or a line)?
______________________________________________________________________________
______________________________________________________________________________
3. How do you specify a plane? _______________________________________________________
4. There is only ____ plane which contains ______ non-collinear ____________
5. How did the video specify the plane? Give two examples
a. ________________ (one letter)
b. ________________ (more than one letter)
Khan Academy
Points, Lines and Planes Quizlet
This quizlet is online at Khanacademy.com under Geometry/Introduction to Euclidian Geometry/Points, Lines and
Planes.
Directions: You are to successfully answer 5 questions in a row for the quizlet to give you your score. If you
cannot answer five in a row the first time, continue to answer questions until you complete 5 in a row. Record
your scores in the upper chart. If you struggle with this quizlet and answer a total of 30 questions, fill in the lower
chart.
Upper Chart
Points Earned ____________ Bonus Points For Completion ______________ Total Score __________
__________ questions answered
-
__________ Percent
Or –
Lower Chart
__________ questions out of 30 answered correctly. Parent’s signature __________________________________
8
Khan Academy
Language and Notation of Circles
1. The set of all points which are a _____________, ______________ or ___________ distance from a center
point forms a _________________.
LINES and SEGMENTS of CIRCLE A
2. A line segment which joins the center of circle A to point B on the circle is called a ________________.
(the plural form of this word is ______________)
3. A line which intersects a circle at exactly one point is called ____________________. Line ____ is
_____________ to circle A at point ________.
4. A line which intersects circle A at 2 points (F and G) is called a ________________ or _____________
_______________.
5. The line segment FG is called a ________________ of Circle A. It connects ______ points on Circle A.
6. A cord which goes through the center of the circle is called a _________________. Line Segment FH starts
at ________ goes through _________ and ends at ___________.
7. The diameter of a circle is made up of ____ _______________. So the length of the diameter is ________
the ______________.
PARTS of a CIRCLE using CIRCLE B
8. The length of a circle that goes through two points on the circle is called a(n) ________________.
9. The shortest distance between point J and point K is called the ________________________ and is
written as ________________
10. The longest distance between point J and point K is called the _______________________ and is written
in three ways ______________, ____________________, _____________________.
9
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The Golden Ratio
1. What formula was used to solve for phi? ______________________________________
2. What value was given for phi? (round to the nearest thousandth) __________________
3. Give three places/examples where the golden ratio is seen or used. ___________________,
_____________________, ______________________
4. Give the common name for the pentagram used as an illustration for where the golden ratio occurs
numerous times. _______________________
5. In a pentagon, the ratio of one of the ____________________ to one of the _________________ is the
golden ratio.
10
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Identifying Rays
1. A ray ____________ at some point and continues ______________ in some _________________. In
order to define a ray you need the _____________ ______________ x and another point y which
________________________________________________. You need the second point to specify
________________________.
2. Name the six rays in the image on the video. Remember to use proper notation.
a. ___________________
b. ___________________
c. ___________________
d. ___________________
e. ___________________
f.
___________________
3. Which two pairs of rays are essentially the same rays? Remember to use proper notation.
a. ___________ and ___________
b. ___________ and ___________
c.
Khan Academy
Angle Basics
1. Ray AB and Ray AB have the same ________________. Any time two rays share the same _____________
you create a(an) ________________, which is Latin for ______________.
2. A is the ______________ of Ray AB, Ray AC and ____________________________.
3. How do you specify an angle? ________________ letters with the _______________ in the middle.
4. We measure angles based on how ____________ or _____________ the angles are.
11
Khan Academy
Measuring Angles in Degrees
1. What tool do we use to measure angles? __________________________
2. This tool is divided into __________ sections.
3. When measuring an angle, we place the ___________ of the angle at the _____________ of the
protractor and one ___________ along the ____________________________.
4. Angles are measured in ________________ or _________________. We will study the second
measurement in Trigonometry.
5. A zero degree angle is a ______________ angle or otherwise called a(n) _________________.
6. A 90° angle is called a _______________angle.
7. A 180° angle forms a ________________. (In class we will also call it a straight angle).
8. An angle which measures less than 90° is called a(n) ______________ angle.
9. An angle which measures more than 90° is called a(n) _______________ angle.
10. Acute means ______________ or ________________.
Khan Academy
Using a Protractor
Put the __________ of the angle at the ____________ of the _________________. Rotate either the
_____________________ or the __________________ so that the ___________ mark is at one ____________ of
the angle and the other side/ray is _____________ the protractor.
12
Measuring Angles
Watch “Measuring Angles” prior to taking the quizlet. The lesson will have an arrow beside it.
This quizlet is online at Khanacademy.com under Geometry/Angles and Intersecting Lines/Angle Basics and
Measurements/Measuring Angles. The quizlet will have a star beside it.
Directions: You are to successfully answer 5 questions in a row for the quizlet to give you your score. If you
cannot answer five in a row the first time, continue to answer questions until you complete 5 in a row. Record
your scores in the upper chart. If you struggle with this quizlet and answer a total of 30 questions, fill in the lower
chart. Be sure to scroll down in order to see the full problem.
Upper Chart
Points Earned ____________ Bonus Points for Completion ______________ Total Score __________
__________ Questions answered
-
__________ Percent
Or –
Lower Chart
__________ Questions out of 30 answered correctly.
Parent’s signature __________________________________
13
Khan Academy
Acute, Right and Obtuse Angles
1. __________ Angles have a measure smaller than a ________________________ or less than _________.
2. __________________ Angles measure exactly _______.
3. _______________ Angles measure greater than a ______________ or more than ____________.
4. Obtuse means _________ __________ __________ or _________ ____________ _______________.
5. Acute means ____________ ________________ or ____________________ ______________________.
Khan Academy
Angle Types Quizlet
Directions: You are to successfully answer 5 questions in a row for the quizlet to give you your score. If you
cannot answer five in a row the first time, continue to answer questions until you complete 5 in a row. Record
your scores in the upper chart. If you struggle with this quizlet and answer a total of 30 questions, fill in the lower
chart. Be sure to scroll down in order to see the full problem.
Upper Chart
Points Earned ____________ Bonus Points for Completion ______________ Total Score __________
__________ Questions answered
-
__________ Percent
Or –
Lower Chart
__________ Questions out of 30 answered correctly.
Parent’s signature __________________________________
14
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Vertical, Adjacent and Linearly Paired Angles
1. Adjacent angles have ________________________ in common. Name 4 rays adjacent to angle BGD.
_________________________, _______________________, ___________________________,
________________________
2. Vertical Angles: Imagine _________ lines crossing or intersecting at a _________________. They form
____________ sets of vertical angles. A vertical angle is on the ____________ side of the _______________
and is not one of the __________________ angles.
3. Imagine the intersection on line DA and line EB. Then the nonadjacent angle to angle EGA is
______________________________.
4. Linear Pairs will be ______________________________angles where the two ___________________rays
combine to form a _____________________.
5. Angle DGF and angle __________ form line _____________ )not the same line as in #5).
6. Vertical angle to angle FGB is _________________________.
7. Vertical angle to angle FGB is ____________________.
15
Khan Academy
Exploring Angle Pairs Quizlet
Directions: You are to successfully answer 5 questions in a row for the quizlet to give you your score. If you
cannot answer five in a row the first time, continue to answer questions until you complete 5 in a row. Record
your scores in the upper chart. If you struggle with this quizlet and answer a total of 30 questions, fill in the lower
chart. Be sure to scroll down in order to see the full problem.
Upper Chart
Points Earned ____________ Bonus Points for Completion ______________ Total Score __________
__________ Questions answered
-
__________ Percent
Or –
Lower Chart
__________ Questions out of 30 answered correctly.
Parent’s signature __________________________________
16
Khan Academy
Introduction to Vertical Angles
1. We know that the measure of angle BED = 70°. We also know that angle BED + angle CEB are
__________________ and that angle BED + angle CEB = angle _________ which is a ______________
angle. Therefore they are also ____________________ which means the measure of angle BED +
measure of angle CEB = ______________.
2. If the measure of angle BED = 70°, then the measure of angle CEB = __________________.
3. Using the same logic we find the measure of angle CEA = _______________ and the measure of angle AED
= _____________________.
4. Angle CEA and angle BED are ________________ angles or ________________ angles which means they
are across a(n) _______________________ from one another. Angle CEB and angle AED are also
_______________________ angles.
5. If you go all the way around the circle you find the measure of all the angles add up to ____________.
Khan Academy
Vertical Angles Quizlet
Directions: You are to successfully answer 5 questions in a row for the quizlet to give you your score. If you
cannot answer five in a row the first time, continue to answer questions until you complete 5 in a row. Record
your scores in the upper chart. If you struggle with this quizlet and answer a total of 30 questions, fill in the lower
chart. Be sure to scroll down in order to see the full problem.
Upper Chart
Points Earned ____________ Bonus Points for Completion ______________ Total Score __________
__________ Questions answered
-
__________ Percent
Or –
Lower Chart
__________ Questions out of 30 answered correctly.
Parent’s signature __________________________________
17
Khan Academy
Use Algebra to Find the Measure
If we have ___ intersecting lines and we know the measure of one angle is 7x+182 and its vertical angle is
____________________, what is the measure of each angle?
X= _______________
7x+182 = ____________
Khan Academy
Vertical Angles- 2 Quizlet
Directions: You are to successfully answer 5 questions in a row for the quizlet to give you your score. If you
cannot answer five in a row the first time, continue to answer questions until you complete 5 in a row. Record
your scores in the upper chart. If you struggle with this quizlet and answer a total of 30 questions, fill in the lower
chart. Be sure to scroll down in order to see the full problem.
Upper Chart
Points Earned ____________ Bonus Points for Completion ______________ Total Score __________
__________ Questions answered
-
__________ Percent
Or –
Lower Chart
__________ Questions out of 30 answered correctly.
Parent’s signature __________________________________
18
Khan Academy
Proof: Vertical Angles are Equal
1. If it can be proved that angle CBE is always equal to ___________, then vertical angles are __________
equal.
2. What proof are we determining: ________________ = _________________
3. What do we know:
i)
Angle _________ and angle _______ are _______________________
ii) Which means angle ___________ + angle __________ = ___________
iii) Angle _________and angle _________ are also __________________
iv) Which means angle ___________+ angle ____________ = ______________
4. Therefore angle CBE = 180-angle DBC and angle DBA = 180 – angle DBC. Therefore, angle _________=
angle _____________. Vertical angles are always ____________.
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Angles Formed by Parallel Lines and Transversals
1. Line AB and Line CD sit on the same ______________ and never ________________. They are
_______________. They have the same ____________ but different ______________________.
2. How do angles relate to parallel lines? If we draw line l which intersects both lines which are parallel, line
l is called a _____________________.
3. First we know that vertical angles are ________________. Next we find that _________________angles
are also _____________________
4. Angle b and able c are vertical angles. Angle b and angle f are cooresponding angles. Angle f and angle g
are vertical angles. Angle g and angle c are cooresponding angles.
5. Alternate interior angles are on the interior of the parallel lines but on _________________ sides of the
transversal.
19
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Identifying Parallel and Perpendicular Lines
Two lines are ______________ if they are in the same __________ and never _______________. If two lines that
are intersected by a third line at the same angle (thus their cooresponding angles are the same) then the two lines
are parallel. Line ST parallel to line _________.
Perpendicular lines are lines that intersect at a __________________________.
Line ST is perpendicular to line _________
Line UV is Perpendicular to line ________
Line AB is perpendicular to line ________
Khan Academy
Figure out the angles between Transversals and Parallel Lines
Watch the video (2 min 16 sec) before taking Quizlett Parallel Lines 2
20
Khan Academy
Congruent Angles Quizlet
Directions: You are to successfully answer 5 questions in a row for the quizlet to give you your score. If you
cannot answer five in a row the first time, continue to answer questions until you complete 5 in a row. Record
your scores in the upper chart. If you struggle with this quizlet and answer a total of 30 questions, fill in the lower
chart. Be sure to scroll down in order to see the full problem.
Upper Chart
Points Earned ____________ Bonus Points for Completion ______________ Total Score __________
__________ Questions answered
-
__________ Percent
Or –
Lower Chart
__________ Questions out of 30 answered correctly.
Parent’s signature __________________________________
Khan Academy
Parallel Lines - 1 Quizlet
Directions: You are to successfully answer 5 questions in a row for the quizlet to give you your score. If you
cannot answer five in a row the first time, continue to answer questions until you complete 5 in a row. Record
your scores in the upper chart. If you struggle with this quizlet and answer a total of 30 questions, fill in the lower
chart. Be sure to scroll down in order to see the full problem.
Upper Chart
Points Earned ____________ Bonus Points for Completion ______________ Total Score __________
__________ Questions answered
__________ Percent
- Or –
Lower Chart
__________ Questions out of 30 answered correctly.
Parent’s signature __________________________________
21
Khan Academy
Using Algebra (2) to find Measures of Angles Formed from Transversals
We are given two parallel lines cut by a transversal. WE are told one angle equals 9x +88 and the other angle is
__________________. How do we find x and the measure of each angle?
1. We find the _____________________________ angle to the angle whose measure is 6x + 182.
2. Then 6x+182+9x+88= 180 because the two angles are ______________________.
3. After solving, we find x = _______
4. We can substitute the value of x into the orange angel (9x+88) and thus the orange angle = ___________
5. The blue angle is (6x-182) is ____________- ______________ = ______________.
Khan Academy
Parallel Lines - 2 Quizlet
Directions: You are to successfully answer 5 questions in a row for the quizlet to give you your score. If you
cannot answer five in a row the first time, continue to answer questions until you complete 5 in a row. Record
your scores in the upper chart. If you struggle with this quizlet and answer a total of 30 questions, fill in the lower
chart. Be sure to scroll down in order to see the full problem.
Upper Chart
Points Earned ____________ Bonus Points for Completion ______________ Total Score __________
__________ Questions answered
-
__________ Percent
Or –
Lower Chart
__________ Questions out of 30 answered correctly.
Parent’s signature __________________________________
22