Download Lesson 1.1: Points, Lines, and Segments - Math Site

Name:_____________________________________________________ Date:______________________
Period:_______
Geometry, Unit 1: Introduction to Geometry – Student Guide
Lesson Title
Content Objectives
Language Objectives
Lesson: 1.1
Points, Lines,
and Segments
*remedial
Students will learn, draw,
and label the following
vocabulary words: point,
line, plane, collinear,
segment, endpoint and
ray.
Lesson 1.2:
Congruency
and Midpoints
Students will learn, draw,
and label the following
vocabulary words:
congruent, midpoint, and
bisects.
Students will be able to
label, measure and
determine congruency of
angles and their parts
SWBAT Define and
categorize types of lines
and angles and angle
relationships
Students will be able to
find the measure of
missing angles or
variables using the
properties of special
angles on parallel lines.
Students will be able to
define, classify and
determine congruency of
polygons
Lesson: 1.3
Angles and
their Parts
Lesson: 1.4
Special Angles
and Angle
Relationships
Lesson: 1.5
Special Angles
on Parallel
Lines
Lesson: 1.6
Polygons and
their Parts
Lesson 1.7:
Special
Triangles and
Special
Quadrilaterals
1
Students will be able to
define and classify
triangles, quadrilaterals
and their related parts.
Learning
Standards
DP
Competencies
Time
(Days)
SWBAT complete a concept
map to compare and
contrast the following
terms: point, line, plane,
collinear, segment,
endpoint, ray, collinear,
coplanar, and endpoint.
SWBAT complete a concept
map to compare and
contrast the following
terms: congruent, bisect,
and midpoint.
SWBAT complete a Frayer
Model vocabulary diagram
for the word bisect.
G-CO.1
G-GPE.6
M05
1
G-CO.1
G-GPE.6
M05
1
G-CO.1
M05
2
SWBAT match all
vocabulary words covered
in this unit so far to
diagrams.
SWBAT write a deductive
argument explaining why
the Alternate Exterior
Angles Conjecture is true.
G-CO.9
M05
2
G-CO.9
M01
M06
2
SWBAT explain in writing
how equiangular,
equilateral, regular,
concave, and convex
polygons are all related.
SWBAT answer critical
questions about how
polygon classifications are
related to each other in
writing.
G-SRT.5
M04
2
G-SRT.5
M04
2
Lesson 1.1: Points, Lines, and Segments
Content Objective: Students will learn, draw, and label the following vocabulary words: point, line, plane, collinear,
segment, endpoint and ray.
Diploma Plus Competencies: M05. Communication
Geometry has its very own sets of words and definitions, so this first section is all about getting comfortable with
the basic words that Geometry is based on. Try to commit these ideas to memory as best you can!
Definitions:
Read pages 14 -15 in Holt McDougal’s Geometry book. Fill in the following table and define the following:
Name
Point
Picture (with labels)
Notation
How would you say these out loud?
Line
Plane
Segment
Ray
Collinear:
Coplanar:
Endpoint:
Language Objective: SWBAT complete a concept map to compare and contrast the following terms: point, line, plane,
collinear, segment, endpoint, ray, collinear, coplanar, and endpoint.
That was a bunch of definitions all in a row! Let’s make sure you fully understand them. Create a concept map in
the space below. Using all the definitions above, connect the ones you think are related or not related, and write
how they are similar or different on each connection. I have provided you with an example to get you started!
CONCEPT MAP
LINE
2
A ray is like a line,
but it only extends
forever in one
direction
RAY
Practice:
Complete all problems identified below on pages 17-18 in the Holt McDougal Geometry Textbook. Attach work
below.
1.
3.
5.
7.
9.
11.
13.
15.
17.
19.
21.
23.
25.
27.
29.
31.
33.
35.
Complete all problems identified below on page 19 in the Holt McDougal Geometry Textbook. Attach work below.
63.
64.
65.
1. Name the ray below in two different ways. Be careful to remember how we’re supposed to name rays!
̅̅̅̅. With three collinear points A, B,
2. Given two points A and B, there is only one segment that you can name: 𝐴𝐵
̅̅̅̅
̅̅̅̅
̅̅̅̅
and C, there are three different segments you can name: 𝐴𝐵, 𝐴𝐶 , 𝐵𝐶 . With five collinear points A, B, C, D, and
E, how many different segments can you name?
Lesson 1.2: Congruency and Midpoints
Content Objective: Students will learn, draw, and label the following vocabulary words: congruent, midpoint, and
bisects
Diploma Plus Competencies: M05. Communication
Vocabulary
Congruent: Two segments (angles, shapes, etc.) are congruent if and only if they have the same measure or length.
The math symbol for congruent is an “=” with a “~” on top: ≅ .
IMPORTANT: Distances of line segments are
equal to each other, but the actual line
segments are congruent to each other
3
IMPORTANT: To mark segments as
congruent, you only need the same
number of “tick marks”:
Midpoint: The midpoint of a line segment is the point that is the same distance from both endpoints. The point
that’s in the exact middle!
Bisect: A midpoint bisects a line segment into two congruent segments. “Bi” means two (like BI-cycle, meaning
“two wheels”), “sect” means to split or cut. Bisect simply means to split into two congruent parts!
IMPORTANT: R is also the midpoint of line
segment ̅̅̅̅
𝑃𝑄
Vocabulary Break Down:
Vocabulary word
Your own definition
Real life example
(Ex. The half court line bisects the
basketball court)
Congruent
≅
Midpoint
Bisect
Practice:
1. What are three different ways you can describe the relationship shown in the figure below?
Using the following Description using figure
vocabulary words
Congruent
Midpoint
Bisect
2. What is the difference between…?
̅̅̅̅
⃖𝐴𝐵
𝐴𝐵
4
𝐴𝐵
⃖𝐴𝐵
AB
3. Complete all problems identified below on pages 32-33 in the Holt McDougal Geometry Textbook. Attach
work below.
22.
24.
38.
39.
4. Which segment has a measure of 7?
𝑃𝑁
b.
c.
d.
̅̅̅̅
𝑁𝑄
̅̅̅̅̅
𝑀𝑃
̅̅̅̅̅
𝐾𝑁
5. Find the coordinate of the midpoint of ̅̅̅̅
𝐺𝐻.
a. 1.5
b. 0.5
c. -0.5
d. -1.5
6.
Find FG on the number line shown.
To reach Proficient or Advanced in M05. Communication:
 Draw two segments that have the same midpoint. Mark your drawing to show congruent segments (this
proves the point is a midpoint).
̅̅, SP = PT, and T is the midpoint of ̅̅̅̅
3. Draw and mark a figure in which M is the midpoint of ̅̅
𝑆𝑇
𝑃𝑄 .
5
Language Objective: SWBAT complete a concept map to compare and contrast the following terms: congruent,
bisect, and midpoint.
Create a concept map using the following terms: congruent, bisect, midpoint, segment, and angle. Connect each
word to another at least two times.
Lesson 1.3: Angles and Their Parts
Content Objective: Students will be able to label, measure and determine congruency of angles and their parts
Diploma Plus Competencies: M05. Communication
Definitions:

Vertex: The common endpoint of an angle.

Sides: The rays, lines or segments that make the sides of an angle.

Angle: The space between two sides that meet at a vertex.
IMPORTANT: An angle can also be named by just the vertex point if there’s no confusion
with which angle you are naming. For instance, you could name ∠T below but not ∠U,
because you could be referencing ∠TUV or ∠VUR. Angle ∠VUR could also be called ∠1
because there is a 1 labeling the angle region.

Congruent Angles: Two angles that have the same measure
This m means the
measure of the angle

Angle Bisector: A ray that divides one angle into two congruent angles.
6
Practice:
7
7. Fill in the blanks in the following statements using the information provided in the figure:
8
8. Mark the following information on the figures below:
Language Objective: SWBAT complete a Frayer Model vocabulary diagram for the word bisect.
Complete the following diagram for the word bisect. Draw examples of angles and segments that have been
bisected.
Bisect
Lesson 1.4: Special Angles and Angle Relationships
Content Objective: SWBAT Define and categorize types of lines and angles and angle relationships
Diploma Plus Competencies: M05. Communication
Angle Vocabulary:
Looking at the angles that are, and are not right angles, we
can establish the following definition:
Right Angle: An angle that measures 90°.
9
The other angles that are not right angles fall into the
following definitions:
(IMPORTANT: The square symbol means 90°!)
Acute Angle:An angle that measures less than 90°.
Obtuse Angle: An angle that measures more than 90°, but less than 180°.
QUICK PRACTICE: Sketch an example of each of the following angles, and use a protractor to measure it. Check in with
your group members or your teacher when you’ve completed these three problems!
a. Obtuse Angle
b.
Acute Angle
c. Right Angle (do you need to measure this?)
Vertical Angles: Vertical angles are angles that are formed
by two intersecting lines; they share a common vertex but
not a common side.
Linear Angles: Linear angles share a vertex and a common
side; their non-common sides form a line.
Complementary Angles: Complementary angles
sum (add) to 90°.
Supplementary Angles: Supplementary angles sum
(add) to 180°.
QUICK QUESTION: Why are ∠1 and ∠2 supplementary??
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
10
Angle Vocab Review:
Complete the following table using the vocabulary words listed above. Write the vocabulary definition, create a
geometry sketch that represents it, and explain/sketch a real-life example. Be creative with your real-life example,
there’s no wrong answer!
Term
Definition
Sketch
Real-Life Example
Right Angle
Acute Angle
Obtuse Angle
Vertical Angle
Linear Pair
Supplementary
Angle
Complementary
Angle
Line Vocabulary:
We discussed a number of angle definitions, but we can’t finish our vocab discussion without two very important
line definitions and their related geometry markings!
Parallel Lines: Parallel lines are lines in the same plane that never cross.
VERY IMPORTANT: The two arrow markings on lines i and j to the right
are how we show two lines are parallel in a sketch. To show two lines
are parallel, you only need the same number of arrow marks on each
line. The statement “i || j” reads “line i is parallel to line j,” because the
symbol “||” is how we write parallel. Be sure to keep these differences
between writing and sketching in mind!
11
Perpendicular Lines: Perpendicular lines are lines that cross at exactly a 90° angle
(right angle).
VERY IMPORTANT: Remember that the square symbol on the sketch of any angle means
it is a 90° (right) angle. To write perpendicular using a geometry symbol, we use the
symbol “⏊” which is read “perpendicular.” Therefore, the statement to the left “k ⏊ l”
would be read “line k is perpendicular to line l.”
RECAP: Provide an example of a sketch and a written statement for two parallel and perpendicular lines in the
spaces provided below:
Perpendicular Lines
Parallel Lines
Sketch (with labels):
Written:
Sketch (with labels):
Written
Practice:
12
13
For problems 1-2, sketch, label and mark a figure showing the following geometry relationships:
1.
(ℓ1 simply means “line 1.” You should just use the same name to label your sketch)
2.
(Be careful to pay attention to what the symbols above PQ and PR mean!)
3. Give two examples of parallel lines in real-world situations.
14
4. Give two examples of perpendicular lines in real-world situations.
Language Objective: SWBAT match all vocabulary words covered in this unit so far to diagrams.
Lesson 1.5: Special Angles on Parallel Lines
Content Objective: Students will be able to find the measure of missing angles
or variables using the properties of special angles on parallel lines.
Diploma Plus Competencies: M01. Problem Solving and M06. Mathematical
Connections
15
There a few more conjectures that are important to know.
A line intersecting two or more other lines in the plane is called a transversal. A transversal creates different types
of angle pairs. The three types of angles pairs are:
Read page 127 in Discovering Geometry to fill in the chart below.
Conjecture (written out)
Corresponding Angles
Drawing
Alternate Interior Angles
Alternate Exterior Angles
Parallel Lines Conjecture (Fill in the blank)
If two parallel lines are cut by a transversal, then corresponding angles are _________________________, alternate interior
angles are ___________________________, and alternate exterior angles are ___________________________________.
Converse of the Parallel Lines Conjecture: (Fill in the blank):
If two lines are cut by a transversal to form pairs of congruent corresponding angles, congruent alternate interior
angles, or congruent alternate exterior angles, then the lines are _____________________________________.
Practice:
16
17
18
To reach Proficient or Advanced in M01.Problem Solving and M06. Mathematical Connections:
19
Language Objective: SWBAT write a deductive argument explaining why the Alternate Exterior Angles Conjecture is
true.
1. In your notebook, write a paragraph explaining why the Alternate Exterior Angles Conjecture is true. In
your explanation, you can assume the Vertical Angles Conjecture and Corresponding Angles Conjecture is
true.
Lesson 1.6: Polygons and Their Parts
Content Objective: Students will be able to define, classify and determine congruency of polygons
Diploma Plus Competencies: M04. Mathematical Representations
Vocabulary:
Polygon:
Polygons also have different names based on their number of sides. Here are the basic names for polygons
with up to eight sides:
IMPORTANT: Notice the beginning of each word (the prefix)
can be found in other words you might know! Tri means
three, like a Tricycle, which has three wheels. Quad is like
the Spanish word “Quatro,” which means the number 4. Oct
means eight, like an Octopus that has eight arms!
Diagonal:
Convex:
Concave:
20
Nonconsecutive vertices means “vertices that aren’t
next to each other” (vertices is the plural of vertex)
A good way to remember these two words is conCAVE
means that the polygon has a “cave” in it. Convex then
means the opposite, that there’s no “cave.”
Equiangular:
Equilateral:
Notice the markings on this polygon’s sides
Notice the markings on this polygon’s angles
Regular:
Notice the markings on
this polygon’s sides
AND angles
Vocabulary Review: Complete and fill in the following graphic organizer with the above Geometry definitions,
according to which definitions include others. (For instance, the circle for the definition concave would be inside
the circle for polygon, because in Geometry something that is concave must be a polygon. To complete it, you will
likely need to add your own circles/graphic organizations)
POLYGONS
Practice:
1. Please fill in the following table with the name and sketch of a polygon with the given number of sides. Try
to do this from memory, without directly copying from the definitions!
# of Sides
Name
Sketch
21
3 Sides
4 Sides
5 Sides
6 Sides
7 Sides
8 Sides
2. Name the following polygon in more than one way:
____________________
3. Sketch the concave pentagon TIGER.
4. Sketch the convex hexagon HOCKEY.
5. Sketch the quadrilateral ABCD and show it is equiangular.
6. Sketch the pentagon GHJKL and show it is regular.
7. Sketch the concave heptagon PSULION that has exactly three outside diagonals.
Language Objective: SWBAT explain in writing how equiangular, equilateral, regular, concave, and convex polygons
are all related.
1. Sketch an equilateral, concave pentagon. Then, attempt to sketch an equiangular, concave pentagon. Is
it possible? Show it is, or explain why it is not.
2. Explain how equiangular, equilateral, and regular polygons are all related.
22
Lesson 1.7: Special Triangles and Special Quadrilaterals
Content Objective: Students will be able to define and classify triangles, quadrilaterals and their related parts.
Diploma Plus Competencies: M04. Mathematical Representations
Definitions:
Notice that each side has a different number of tick
marks, showing that none of the sides are congruent
Notice the two single tick marks on this figure show two
Fun Fact: Acute means “sharp,” which is helpful for
congruent sides
remembering this definition, as acute angles are sharp: less
than 90 degrees. (You can say someone has an “acute mind”
if they’re smart, like saying someone is really “sharp.”)
Notice that all three sides of this triangle are marked as
Fun Fact: Obtuse can mean “dumb,” so if your friend is acting
congruent because each has the same number (one)
a fool, you can say: “Don’t be obtuse!”
Vocabulary Review:
Use the following organizer to review the above special triangle definitions.
Name and Picture
It’s this kind of special triangle
because…
23
This is related to what else in
geometry?
More Definitions:
Trapezoid:
Parallelogram:
Notice that this definition only requires a quadrilateral to
Have one pair of parallel sides.
The name of a parallelogram pretty much gives away
its special properties – both sets of sides are parallel!
Rhombus:
Kite:
This means all its sides are the same and has two
sets of parallel sides. Notice this marked in the sketch!
Consecutive means following, so this means the sides
directly next to each other. Distinct means different
from others, so each pair of sides are different lengths.
Rectangle:
Square:
Remember that
regular means
congruent sides and
angles.
Quick Review:
Please create a concept map for the 6 special quadrilaterals above. Sketch each and show how it connects to
different common properties. The first special quadrilateral (a rectangle) has been done for you below as an
example:
2 Pairs of Congruent Sides
All Sides Congruent
Right Angles
Rectangle
No Parallel Sides
1 Pair of Parallel Sides
2 Pairs Parallel Sides
Practice:
1. Classify the following triangles by their sides and angles. (For instance, a triangle might be an equilateral
24
acute triangle because all its sides are congruent and all its angles are less than 90°):
2. Classify the following special quadrilaterals:
3. Sketch the following:
a. An isosceles right triangle
b.
A scalene acute triangle
For problems 4-7, sketch the following:
4.
5.
6.
7.
8.
Language Objective: SWBAT answer critical questions about how polygon classifications are related to eachother in
writing.
1) Are all pairs of sides in a rectangle parallel? Why?
2) The definition of an isosceles triangle says that it must have “at least two congruent sides.” Given this, is an
equilateral triangle an isosceles triangle? Explain why or why not using full sentences.
3) Are there any quadrilaterals that can be classified as two different quadrilaterals? Explain and provide
examples.
25
26