Unit 1
Proof, Parallel and Perpendicular Lines
SpringBoard Geometry Pages 1-100
(add in comma after the course and write the unit and dash
before pages)
Overview
In this unit, students study formal definitions of basic figures, the axiomatic
system of geometry and the basics of logical reasoning. They are then introduced
to mathematical proof by applying formal definitions and logical reasoning to
develop proofs about basic figures. Finally, students learn how to write
equations of parallel and perpendicular lines.
Standards:
Embedded Assessment 1:
Standards in this Unit:
Geometric Figures and Basic Reasoning-The Art
and Math of Folding Paper (page 37-38)
MAFS.912.G-CO.1.1 Know precise definitions of angle,
circle, perpendicular line, parallel line, and line segment,
based on the undefined notions of point, line, distance
along a line, and distance around a circular arc.
MAFS.912.G-GPE.2.6 Find the point on a directed line
segment between two given points that partitions the
segment in a given ratio.
MAFS.912.G-CO.3.9 Prove theorems about lines and
angles; use theorems about lines and angles to solve
problems. Theorems include: vertical angles are
congruent; when a transversal crosses parallel lines,
alternate interior angles are congruent and
corresponding angles are congruent; points on a
perpendicular bisector of a line segment are exactly
those equidistant from the segment’s endpoints.
Make use of geometric figures (Lesson 1-1, 1-2)(describe
step by step process to create geometric figure)
Complete two column Algebraic proofs (Lesson 2-1, 22)(write a prove statement when given a statement)
Axiomatic system of geometry(write an if then statement,
given a conditional statement write the converse, inverse,
and contrapositive, write a biconditional statement)(Ex:
That means that from a small, basic set of agreed-upon
assumptions and premises, an entire structure of logic is
devised.) (Lesson 3-1, 3-2, 3-3)
MAFS.912.G-GPE.2.5 Prove the slope criteria for parallel
and perpendicular lines and use them to solve geometric
problems (e.g., find the equation of a line parallel or
perpendicular to a given line that passes through a given
point).
1
Embedded Assessment 2:
Distance, Midpoint, and Angle Measurement- A
Walk in the Park (61-62) (include “page”)
(no bold)Use Segment, Midpoint, and Angle Bisectors
(include more detail about…used for finding
distance)(Lesson 4-1, 4-2)
Use Distance and Midpoint formulas on a coordinate
plane (Lesson 5-1, 5-2)
Embedded Assessment 3:
Angles, Parallel Lines, and Perpendicular LinesGraph of Steel (99-100) (include “page”)
(no bold)Create Geometric proofs about line segments
and angles (Lesson 6-1, 6-2)
Using and Proving Parallel and perpendicular lines with
proofs (Lesson 7-1, 7-2, 7-3)
Writing Equations with the slops of parallel and
perpendicular lines (Lesson 8-1, 8-2)
(Refer to finding measure of angles, justify reasoning of
equations)
Vocabulary (Academic/Math):
compare and contrast, justify, argument, interchange, negate, format, confirm, inductive reasoning,
conjecture, deductive reasoning, proof, theorem, axiomatic system, undefined terms, two-column
proof, conditional statement, hypothesis, conclusion, counterexample, converse, inverse,
contrapositive, truth value, bi-conditional statement, postulates, midpoint, congruent, bisect,
bisector of an angle, parallel, transversal, same-side interior angles, alternate interior angles,
corresponding angles, perpendicular, perpendicular bisector(no bold)
2
Support for Lesson 1-1
Learning Targets for lesson are found on page 3.
Main Ideas for success in 1-1
 Identify, describe, and name points, lines, line segments, rays, and planes using correct notation.
 Identify and name angles.
 Vocabulary used in this lesson includes: point, line, plane, line segment, ray, and angle
 Standards for Mathematical Practice to be demonstrated are: Critique the reasoning of others, and Reason abstractly
Practice Support for Lesson 1-1

 This example is a model to help solve Practice problem 16.
EXAMPLE:
Identify all possible names of this geometric figure.
Since the figure has points of J, K, M, and L the figure should be called plane JKLM or you can use the figure letter R and
call it plane R.

 This example is a model to help solve Practice problem 17.
EXAMPLE:
Identify all rays in the figure.
To identify a ray you have to look for the endpoint that the ray starts at and then extends through a point. You then use
the endpoint as the first letter and the point it extends through as the second point. So there are three rays ,
,
and
.

 This example is a model to help solve Practice problem 18.
EXAMPLE:
Explain why in the figure above
is not a correct name for
.
B is not correct because it describes the entire angle ABD, not the smaller portion of ABC.

1
Practice Support for Lesson 1-1 Continued:
2
Additional Practice:
Answers to additional practice:
1. a. angle; T, STQ, QTS
b. line segment; AC,CA
2. C
3
Support for Lesson 1-2
Learning Targets for lesson are found on page 7.
Main Ideas for success in 1-2.
 Describe angles and angle pairs.
 Identify and name parts of circles.
 Vocabulary used in this lesson includes: “compare and contrast” complementary angles, supplementary angles, acute
angles, right angles, obtuse angles, straight angles, chord, diameter, radius, and justify
 Standards for Mathematical Practice to be demonstrated are: Make sense of problems, Model with mathematics,
Construct viable arguments, Reason quantitatively
Practice Support for Lesson 1-2

 This example is a model to help solve Practice problem 11.
EXAMPLE:
a. List all possible correct names for segment
a. Since
.
has the endpoint on the sides of the circle we can call it a chord,
but it also goes through the center of the circle so we can call it a diameter.
b. all possible correct names for segment
b. Since
.
has its endpoints at the center and on the edge of the circle we call it a radius.
c. List all possible correct names for segment
c. Since
.
has endpoints on the circle but does not go through the center
we call it a chord.
4
Practice Support for Lesson 1-2 Continued:
 This example is a model to help solve Practice problem 12
EXAMPLE:
Identify the pairs of Supplementary angles in the figure above.
Since supplementary angles have to be 180 degrees we look for linear pairs which equals 180.
PKO, PKM; PKL, LKN; OKL, LKM; PKM, NKM are all linear pairs.
 This example is a model to help solve Practice problem 13
EXAMPLE:
Name the angles that appear to be right in the figure above.
PKL, and
NKL because they both equal 90 degrees.
 This example is a model to help solve Practice problem 14.
EXAMPLE:
Can two acute angles be Supplementary?
No, for an angle to be acute it must be less than 90 degrees, so therefore two angles less than 90 cannot add to more
than 178 degrees.
 This example is a model to help solve Practice problem 15
EXAMPLE:
Is it possible to draw two non-adjacent angles that share vertex A, and are complementary?
Yes it is possible. As shown below two angles can be non-adjacent (meaning not directly next to each other and touching)
and still have a sum of 90 degrees. If angle 1 was 45 degrees, and angle 2 was 45 degrees then the two would add to 90
and still be non-adjacent.
5
Additional Practice 1-2:
1.
2.
3.
additional practice answers:
1.
2.
3.
6
Support for Lesson 2-1
Learning Targets for lesson are found on page 13.
Main Ideas for success in 2-1:
 Make conjectures by applying inductive reasoning.
 Recognize the limits of inductive reasoning.
 Vocabulary used in this lesson includes: Inductive Reasoning, conjecture,
 Standards for Mathematical Practice to be demonstrated are: Reason quantitatively, Critique the reasoning of others,
and Make use of structure,
Practice Support for Lesson 2-1
 This example is a model to help solve Practice problems 11.
EXAMPLE:
Use inductive reasoning to find the next 3 items in the sequence.
a. A, C, E, G,I, …
. Since we see A then C we skipped B, also we see C then E so we skipped D. therefore J, L, N would be next.
b. 2, 4, 6, 10, 16…
b. Since we see 2 then 4 followed by 6 we might think the sequence is even numbers, but we skip 8 and go to 10 followed
by 16. If we look at it as a sum we see 2+4=6, and 4+6=10, and 6+10=16. Therefore 26, 36, 62 would be the next three
items in the sequence.
 This example is a model to help solve Practice problems 12.
EXAMPLE:
Write the first five terms of two different sequences that have 2 as the second term.
1, 2, 3, 4, 5 (you would just need to count up from ones.)
0, 2, 2, 4, 6, 10 (you would start at 0 + 2, and add the two adjacent terms together to get the next.)
 This example is a model to help solve Practice problems 13.
EXAMPLE
Generate a sequence with 4 terms using this description: The first term in the sequence is 2, and each other term is three
more than twice the previous term.
(ie. 2 x + 3 if x is the previous term)
2, 7, 17, 37
7
Practice Support for Lesson 2-1 Continued:
 This example is a model to help solve Practice problems 14.
EXAMPLE:
The diagram shows the first three figures in a pattern. Each figure is made of small triangles. How many small triangles will
th
be in the 4 figure of the pattern? Support your answer.
We notice that the first is 1 and the second is 4 then 9. From there we try to demine the pattern
the number in the sequence ie:
and
, so from this patern we take
is the pattern the x is
=16. So 16 triangles will exist in
th
the 4 figure of this sequence.
 This example is a model to help solve Practice problems 15.
EXAMPLE:
The first two terms of a number pattern are 3 and 5. Joe conjectures that the next term will be 7. Mike conjectures that
the next term will be 8. Whose conjecture is reasonable? Explain.
Both students can be correct 3+5=8 and with a series of odd numbers we get 3,5,7.
8
Additional Practice 2-1:
1.
2.
3.
additional practice answers:
1.
2.
3.
9
Support for Lesson 2-2
Learning Targets for lesson are found on page 18.
Main Ideas for success in 2-2:
 Use deductive reasoning to prove that a conjecture is true.
 Develop geometric and algebraic arguments based on deductive reasoning.
 Vocabulary used in this lesson includes: argument, deductive reasoning, proof, and theorem
 Standards for Mathematical Practice to be demonstrated are: Express regularity in repeated reasoning, Model with mathematics,
Reason abstractly, Construct viable arguments, Critique the reasoning of others, and Reason quantitatively
Practice Support for Lesson 2-2
 This example is a model to help solve Practice problems 15.
EXAMPLE:
Use expressions for even and odd integers to confirm the conjecture that the sum of an odd integer and an odd integer is an even
integer.
m + m+2 = 3m+2 and if m is odd all products of (3m+2) will equal an even integer.
 This example is a model to help solve Practice problems 16.
EXAMPLE:
Prove this conjecture geometrically: Any even integer can be expressed as the sum of an even integer and an even integer.
A figure representing an even integer can be broken apart into a rectangle representing an even integer and a square representing an
even integer. The even integer can be expressed as the sum of the even integer and the even integer 2.
 This example is a model to help solve Practice problems 17.
EXAMPLE
Use deductive reasoning to prove that the solution of the equation x − 4 = − 2 is x = 2. Be sure to justify each step in your proof.
X – 4 = -2 Given
X – 4 + 4 = -2 + 4 Addition Property of Equality
X + 0 = 2 Addition
X = 2 Identity Property
10
Practice Support for Lesson 2-2 Continued:
 This example is a model to help solve Practice problems 18.
EXAMPLE:
Based solely on the pattern in the table, Kevin states that the number of edges of a polyhedron is equal to its number of
Vertices plus the Faces minus 2. Is Andre’s statement a conjecture or a theorem? Explain.
Shape
Vertices
Faces
Edges
Square Pyramid
5
5
8
Square Prism
8
6
12
Tetrahedron
4
4
6
A conjecture. Kevin makes the statement based on three types of polygons. He has not proved that the statement is true
for every polygon. A theorem is an item that first needs to proven to be used as a true statement.
 This example is a model to help solve Practice problems 19.
EXAMPLE:
DNA found at a crime scene is consistent with those of an escaped prisoner. Based on this evidence, an investigator
concludes that the suspect was at the crime scene. Is this an example of inductive or deductive reasoning? Explain.
Inductive reasoning, the evidence shows that the hair could have come from the prisoner, but it does not prove that the
suspect was actually at the crime scene. It could have been planted there by a second person. So, the reasoning is
inductive rather than deductive. Deductive reasoning is when your case or idea is based on a proof that is built on rules of
logic
11
Additional Practice 2-2:
1. Use expressions for even integers to show that the product of two even integers is an even integer.
2. Make use of structure. Use deductive reasoning to prove
that x = 5 is not in the solution set of the inequality 2x+1≤ 7.
Be sure to justify each step in your proof.
3. During the first month of school, students
recorded each day on which they had a quiz in
math class. A student stated that there is a math
quiz every Tuesday morning. Is the student’s
statement a conjecture or a theorem? Explain.
4.
Answers to additional practice:
1. Use 2p and 2q to represent two even
integers. Then (2p)(2q)=2(2pq)
We know that the expression 2pq
represents an integer because when you
find the product of two or more integers,
the result is also an integer. So the
expression “2(2pq)” is an even integer
because it is 2 times an integer.
2. Sample answer: 2x+1 ≤ 7
2x ≤ 6 Subtraction Property of Inequality
x ≤ 3 Division Property of Inequality
5 is not less than or equal to 3 because 5 is
to the right of 3 on the number line. So x =
5 is not in the solution set of x ≤ 3.
3. The student’s statement is a conjecture
because it is a generalization based on a
pattern of data. The statement is not a
theorem because it has not been proved
using deductive reasoning.
4. a. Deductive reasoning. The student’s
conclusion is based on a proof, so the
reasoning is deductive rather than
inductive.
12
Support for Lesson 3-1
Learning Targets for lesson are found on page 25
Main Ideas for success in:
 Distinguish between undefined and defined terms
 Use properties to complete and write algebraic two-column proofs
 Use undefined terms to create definitions of defined terms
 Vocabulary used in this lesson includes: axiomatic system, undefined terms, ray, collinear points, coplanar points,
angle, vertex, complementary angles, supplementary angles, two-column proofs.
 Standards for Mathematical Practice to be demonstrated are: express regularity in repeated reasoning, construct
viable arguments, look for and make use of structure.
Practice Support for Lesson 3-1
 This example is a model to help solve Practice problem 8.
EXAMPLE:
Identify the property that justifies the statement: If
, then
.
The property that justifies the statement would be the Subtraction Property of Equality because 10 has been
subtracted to both sides of the equation.
-10
-10
 This example is a model to help solve Practice problems 9 and 11.
EXAMPLE
Complete the prove statement and write a two-column proof for the equation:
Given:
Prove:
Given:
Prove:
Statements
Reasons
1.
1. Given equation
2.
2. Distributive Property
3.
3. Subtraction Property of Equality
4.
4. Addition Property of Equality
5.
5. Division Property of Equality
Complete algebraic solutions:
1
Practice Support for Lesson 3-1 Continued:
 This example is a model to help solve Practice problem 10.
EXAMPLE:
Explain why ray is considered a defined term in geometry.
Ray is a defined term because it can be defined in terms of the undefined terms line and point. A ray is a part of
a line bounded by one endpoint and extending infinitely in one direction.
 This example is a model to help solve Practice problem 12.
EXAMPLE:
Suppose you are given that
Substitution Property?
and
. What can you prove by using these statements and the
1.
2.
3.
4.
The Substitution Property says that if
, then can be substituted for in any equation or inequality.
Since we know
we can use the Substitution Property to substitute 10 for in the first equation.
This would give the equation:
Using the Subtraction Property of Equality we can subtract 2 from both sides to give the equation:
5.
Using the Division Property of Equality we can divide both sides by 4 to find
6.
We are able to prove
.
using the given statements and the Substitution Property.
2
Additional Practice 3-1:
1.
2.
3.
Answers to additional
practice:
1.
a.
b.
Given
Multiplication
Property of
Equality
Addition Property
of Equality
Division Property
of Equality
c.
d.
2.
3.
C
C
3
Support for Lesson 3-2
Learning Targets for lesson are found on page 29.
Main Ideas for success in:
 Identify the hypothesis and conclusion of a conditional statement.
 Give counterexamples for false conditional statements.
 Restate conditional statements in if-then form.
 Vocabulary used in this lesson includes: conditional statement, hypothesis, conclusion, and counterexample.
 Standards for Mathematical Practice to be demonstrated are: make use of structure, reason abstractly, critique the
reasoning of others, and construct viable arguments.
Practice Support for Lesson 3-2
 This example is a model to help solve Practice problem 8.
EXAMPLE:
o
Write the statement in if-then form: All 90 angles are right angles.
1.
2.
3.
4.
First, identify the hypothesis and the conclusion. When you rewrite the statement in if-then form, you may
need to reword the hypothesis or conclusion.
a. A conditional statement is a statement of logic that combines two statements or facts and can be
written in if-then form. The part of the statement that follows “if” is the hypothesis, and the part
that follows “then” is the conclusion.
o
Hypothesis: The measure of an angle is 90 .
Conclusion: It is a right angle.
Next, use the hypothesis and conclusion to write the conditional statement as an if-then statement. After
the “if” you write the hypothesis and after the “then” you write the conclusion.
o
a. If the measure of an angle is 90 , then it is a right angle.
 This example is a model to help solve Practice problem 9.
EXAMPLE:
Which of the following is a counterexample of this statement?
"All mammals have legs.”
A.
B.
C.
D.
Whales are mammals that do not have legs
Cats are mammals that have legs
People only have two legs
Chairs are not mammals and have legs
A counterexample is an example that can be found for which the hypothesis is true, but the conclusion is false.
So the hypothesis in this statement would be “if an animal is a mammal” and we want this to remain true, but our
conclusion, “the animal will have legs” needs to be false. The counterexample would be “A” because the hypothesis of
the statement remains true, a whale is a mammal, and the conclusion is false, a whale does not have legs.
4
Practice Support for Lesson 3-2 Continued:
 This example is a model to help solve Practice problem 10.
EXAMPLE
Identify the hypothesis and the conclusion of the statement:
If today is Friday, then tomorrow is Saturday.
The letter p and q are often used to represent the hypothesis and conclusion, respectively, in a conditional
statement. The basic form of an if-then statement would then be, “if p, then q”. So the hypothesis is
represented by “p” and follows the word if. The conclusion is represented by “q” and follows the word then.
If today is Friday, then tomorrow is Saturday.
p (hypothesis)
q (conclusion)
Hypothesis: today is Friday
Conclusion: tomorrow is Saturday
Tip: The words if and then are
not included when giving the
hypothesis and conclusion.
 This example is a model to help solve Practice problems 11 and 12.
EXAMPLE:
Christina says that
correct? Explain.
If
, then
is a counterexample that shows the following conditional statement is false. Is Christina
.
Christina is correct because the hypothesis of the conditional is true, but the conclusion is false.
Since
, this counterexample shows that the conditional statement is false. A counterexample must
show that the conclusion can be false when the hypothesis is true. In Christina’s example, the hypothesis is true
and the conclusion is false. Once you have found one counterexample for a condition statement the conditional
statement is false.
5
Additional Practice 3-2:
1.
2.
Write each statement in if-then form.
a. The only time I wake up early is when I set my alarm clock.
b. I eat breakfast at a restaurant only if it is a weekend.
c. An obtuse angle has a measure between 90° and 180°.
State or describe a counterexample for each conditional statement.
a. If
then
.
b. If three points A, B, and C are collinear, then B is between A and C.
3.
Answers to additional practice:
1.
a.
b.
c.
If I wake up early, then I set
my alarm clock.
If I eat breakfast at a
restaurant, then it is a
weekend.
If an angle is obtuse, then its
measure is between 90° and
180°.
2.
a.
b.
3.
A counterexample is
.
Two counterexamples are a
line with A between B and C
and a line with C between A
and B.
D
6
Support for Lesson 3-3
Learning Targets for lesson are found on page 32.
Main Ideas for success in:
 Write and determine the truth value of the converse, inverse, and contrapositive of a conditional statement.
 Write and interpret biconditional statements.
 Identify logically equivalent statements.
 Vocabulary used in this lesson includes: converse, inverse, contrapositive, interchange, negate, truth values, and
biconditional statement.
 Standards for Mathematical Practice to be demonstrated are: make use of structure, critique the reasoning of others,
and reason abstractly.
Practice Support for Lesson 3-3
 This example is a model to help solve Practice problems 11-13.
EXAMPLE:
Write the converse, inverse, and contrapositive of the following conditional statement.
If it thunders then it is raining.
1.
2.
3.
4.
Identify the hypothesis (p) and conclusion (q)
a. Hypothesis (p) – it thunders
b. Conclusion (q) – it is raining
Converse: If q, then p. Interchange the hypothesis and conclusion
a. If it is raining, then it thunders.
Inverse: If not p, then not q. Negate the hypothesis and negate the conclusion.
a. If it is not thundering then it is not raining.
Contrapositive: If not q, then not p. Interchange and negate both the hypothesis and conclusion. Or you could
think about it as negating the converse.
a. If it is not raining then it is not thundering.
 This example is a model to help solve Practice problem 14.
EXAMPLE:
Write the definition of coplanar lines as a biconditional statement.
A biconditional statement is a conditional statement and its converse. It typically includes the words “if and only if”
as in a definition. For a biconditional to be a true statement, it must be true both “forward and backward.” Writing a
biconditional statement is equivalent to writing a conditional statement and its converse.
Conditional: If three lines are coplanar, then they lie in the same plane.
Converse: If three lines lie in the same plane, then they are coplanar.
Biconditional: Three lines are coplanar if and only if they lie in the same plane.
7
Practice Support for Lesson 3-3 Continued:
 This example is a model to help solve Practice problem 15.
EXAMPLE
Give an example of a false statement that has a true converse.
You will want a statement that has a false conditional, but the converse is true when you interchange the hypothesis
and conclusion.
Statement: If a polygon has four sides, then the figure is a square. (false)
Converse: If a polygon is a square, then it has four sides. (true)
 This example is a model to help solve Practice problem 16.
EXAMPLE:
What conclusions can be drawn from the given statements?
Given: (1) If you exercise regularly, then you have a healthy body.
(2) You do not have a healthy body.
Conclusions: If you do not have a healthy body then you do not exercise regularly.
If you exercise regularly, then you have a healthy body. The first statement is the conditional statement
and the second statement is the beginning of the contrapositive. Since the conditional and the
contrapositive are logically equivalent then the negation of the hypothesis must be true if the negation
of the conclusion is true.
8
Additional Practice 3-3:
1.
2.
3.
Answers to additional practice:
1.
a.
Inverse: If it is not raining, then I
do not stay indoors;
Contrapositive: If I do not stay
indoors, then it is not raining.
b. Inverse: If I do not have a
hammer, then I do not hammer
in the morning; Contrapositive: If
I do not hammer in the morning,
then I do not have a hammer.
2. If people have the same ZIP code,
then they live in the same
neighborhood. If people live in the
same neighborhood, then they have
the same ZIP code.
3. D
9
Support for Lesson 4-1
Learning Targets for lesson are found on page 39.
Main Ideas for success in lesson # here:
 Apply the Segment Addition Postulate to find lengths of segments.
 Use the definition of midpoint to find lengths of segments.
 Use a ruler to measure the length of a line segment.
 Vocabulary used in this lesson includes: axiom, postulate, distance along a line, ruler postulate, segment addition
postulate, midpoint, congruent ( ), and bisect.
 Standards for Mathematical Practice to be demonstrated are: attend to precision, reason abstractly, reason
quantitatively, and use appropriate tools strategically.
Practice Support for Lesson 4-1
 This example is a model to help solve Practice problem 19.
EXAMPLE:
Given: Point O is between points D and G,
,
, and
. Find the value of .
1.
First, draw a picture to help visualize the information given. Draw a segment where D and G are the
endpoints and O is between them.
2.
Next, add in the information given about each part of the segment.
3.
The Segment Addition Postulate tells us that if point O is between points D and G then
we will use this to write an equation and solve for .
Input the values of each segment into the equation.
. So
Combine like terms
Subtract 18 from both sides
Divide both sides by 6
4.
The value of
is 7.
1
Practice Support for Lesson 4-1 Continued:
 This example is a model to help solve Practice problems 20 and 23.
EXAMPLE
If M is the midpoint of
,
and
, find the value of
and the length of
.
1.
First, draw a picture to help visualize the information given. Draw a segment where P and Q are the
endpoints and M is in the middle of them.
2.
Next, add in the information given about each part of the segment.
3.
Because M is the midpoint of
4.
Now use the Segment Addition Postulate to write an equation where
. Use this equation to
solve for .
Input the values of each segment into the equation.
, you know that
and
. Therefore
Combine like terms
Subtract
from both sides
Subtract 16 from both sides
Divide 4 on both sides
The value of
5.
Once we have the value of , we need to find the length of
.
. Substitute 10 for
is 10.
in the expression for
has a length of 58 units.
2
Practice Support for Lesson 4-1 Continued:
 This example is a model to help solve Practice problem 21.
EXAMPLE:
Point D is between points A and B. The distance between points A and D is of AB. What is the coordinate of point D?
1.
First we need to find the distance between A and B. Point A is at
and point B is at .
2.
Since the distance between A and D is of AB, we need to find of AB.
 This tells us that point D is 9 units away from A.
3.
To find the coordinate of D we need to add 9 units to the coordinate of A.
 the coordinate of D is 3
 This example is a model to help solve Practice problem 22.
EXAMPLE
Explain how to use a ruler to measure the length of a line segment.
To measure a certain line segment, place the ruler with its edge along the line segment such that the zero mark
of the ruler coincides with an endpoint. Now we read the mark on the ruler which is against the other endpoint.
That mark would be length of the line segment. If the line segment does not start at the “zero” on the ruler, to
find the length of the segment we either count the number of units between the ends of the line segment or
take the absolute value of the difference between the endpoints.
 This example is a model to help solve Practice problem 24.
EXAMPLE:
Compare and contrast a postulate and a theorem.
A postulate is understood as true without proof, while a theorem must be proven. A theorem is a proposition that
can be deduced from postulates. Usually postulates provide the starting point for the proof of a theorem. We
make a series of logical arguments using postulates to prove a theorem.
3
Additional Practice 4-1:
1. Suppose point T is between points R and V on a line. If RT = 6.3 units and RV = 13.1 units, then what is TV?
A. 2.5 units
B. 6.8 units
C. 7.8 units
D. 19.4 units
2.
3.
Suppose P is between M and N.
a. If
,
b. If
,
Use the centimeter ruler shown.
4.
a. What is the length of
b. What number on the ruler represents the midpoint of
Points P, M, and T are on a line and PT - PM = MT. Which point is between the other two? Explain your answer.
, and
, and
, what is the value of ?
, what is the value of ?
Answers to additional practice:
1.
2.
B
a.
b.
5
3
3.
a. 9cm
b. 19.5
4. M is between P and T. Starting
with PT - PM = MT, add PM to
each side to get PT = MT = PM or
PT = PM + MT. That equation
satisfies the situation that M is a
point between P and T.
4
Support for Lesson 4-2
Learning Targets for lesson are found on page 45.
Main Ideas for success in lesson #here:
 Apply the Angle Addition Postulate to find angle measures.
 Use the definition of angle bisector to find angle measures.
 Use a protractor to measure the degree of an angle.
 Vocabulary used in this lesson includes: protractor postulate, angle addition postulate, bisector of an angle, adjacent
angles, congruent angles, vertical angles, perpendicular ( ), complementary, and supplementary.
 Standards for Mathematical Practice to be demonstrated are: use appropriate tools strategically, express regularity in
repeated reasoning, construct viable arguments, and critique the reasoning of others Capitalize the first letter
Practice Support for Lesson 4-2
 This example is a model to help solve Practice problem 12.
EXAMPLE:
Point K is in the interior of m∠ABC, m∠ABC =
1.
, m∠ABK = 42°, and m∠KBC =
. What is m∠ABC?
First, draw an angle to help visualize the information given. Draw an angle whose vertex is B and the rays
creating the angle are
and
as shown below. Point K needs to be in the interior of ∠ABC as shown below.
2.
Next, add in the information given about each part of the angle.
3.
The Angle Addition Postulate tells us that
∠
∠
∠
∠
∠
Angle Addition Postulate
Substitute the values of each
angle into the equation
Combine Like Terms
Subtract
from both sides
∠
.
4. Once you have the value
of , substitute it into
the equation of ∠ABC.
∠
∠
∠
∠
Subtract 8 from both sides
Divide both sides by 5
The measure of angle ABC
is 110.
5
 This example is a model to help solve Practice problems.
Practice Support for Lesson 4-2 Continued:
 This example is a model to help solve Practice problems 13 and 15.
EXAMPLE
bisects ∠
1.
. If
∠
and
∠
∠
, then what is
?
First, draw an angle to help visualize the information
given. Draw an angle whose vertex is Q and the rays
creating the angle are
and
as shown below.
bisects the angle which means it goes through the
middle of the angle, creating two congruent angles.
2.
Next, add in the information given about each part of
the angle.
3.
Because
bisects ∠
, you know that
∠
∠
and ∠
∠
. Therefore
∠
Add this to the diagram. Include
that the congruent symbol means =.
4.
Use the Angle Addition Postulate
to write an equation where
∠
∠
∠
.
Substitute the values of each
angle and solve for y.
∠
∠
∠
Angle Addition Postulate
Substitute the values of each
angle into the equation
Combine Like Terms
Subtract
from both sides
Add 2 to both sides
Divide both sides by 2
5.
Substitute the value of y into the equation for ∠
∠
6
6.
Practice Support for Lesson 4-2 Continued:
 This example is a model to help solve Practice problem 14.
EXAMPLE
∠C and ∠D are supplementary. If ∠
and ∠
, what is the measure of each angle?
o
1. If ∠C and ∠D are supplementary this means that the sum of their angle measures is 180 . Write an equation
where ∠C + ∠D = 180 and solve for .
∠
∠
Supplementary angles are angles whose measure sums to 180
Substitute the values of each angle into the equation
Combine like terms
Subtract 8 from both sides
Divide both sides by 4
The value of
2.
is 43.
Substitute the value of into the equations for ∠C and ∠D.
∠
∠
∠
∠
∠
∠
∠
3.
Check to make sure your answers give supplementary angles; they add to 180.
 This example is a model to help solve Practice problems 16.
EXAMPLE:
Kevin knows that point B is in the interior of ∠
and he knows that ∠
this information? Explain. Make a sketch that supports your answer.
1. Draw an angle with point F being the vertex and
2. Add
the rays creating the angle are
3.
and
.
∠
. What can Kevin conclude from
so that B is in the interior of the angle
and ∠
∠
Since B is in the interior of ∠
and ∠
∠
, we can conclude that
is the angle bisector of
∠
because the bisector of an angle is a ray that divides the angle into two congruent adjacent
angles, ∠
∠
.
7
Additional Practice 4-2:
1.
Make sense of problems. Suppose that
2.
Suppose
3.
∠D and ∠E are complementary. If
bisects ∠CAR. If
∠
bisects ∠MPN. What conclusion can you make?
and
∠
∠
and
, what is
∠
∠
?
, what is ?
Answers to additional
practice:
1.
2.
3.
m∠MPQ = m∠QPN
35°
11
8
Support for Lesson 5-1
Learning Targets for lesson are found on page 51.
Main Ideas for success in lesson# here:
 Use the Pythagorean Theorem to derive the Distance Formula.
 Use the Distance Formula to find the distance between two points on the coordinate plane.
 Vocabulary used in this lesson includes: derive, Pythagorean Theorem, hypotenuse, and Venn diagram.
 Standards for Mathematical Practice to be demonstrated are: model with mathematics, attend to precision, express
regularity in repeated reasoning, reason abstractly, and reason quantitatively Capitalize the first letter
Practice Support for Lesson 5-1
 This example is a model to help solve Practice problems 21 and 22.
EXAMPLE:
Find the distance between the points with the given coordinates.
and
1.
2.
3.
The distance d between the points (x1, y1) and (x2, y2) is given by
Label the points as follows
and
Therefore,
and
.
Hint: is read as “x sub one” and it refers to the first x-value.
Substitute the values into the Distance Formula.
Distance Formula
,
4.
and
.
,
Substitute into formula
Simplify the equation.
Use order of operations to simplify the equation.
Simplify inside the parenthesis first.
Square
and
squaring is
before you add explain what
Add 16 and 49 before you take the square root
explain square root
**justify why the answer is reasonable for your
example, modeling 23 separately, you may need an
additional pg
The distance between
and
is
which is about 8.06 units.
 This example is a model to help solve Practice problems 23 and 24.
EXAMPLE:
Besides the Distance Formula, what other method could you use to find the distance between two points in the
coordinate plane?
The structure of the Distance Formula and the Pythagorean Theorem are nearly identical. When using the
Pythagorean Theorem it involves the relationship between the lengths of the legs and the hypotenuse of a
right triangle.
9
Practice Support for Lesson 5-1 Continued:
 This example is a model to help solve Practice problem 25.
EXAMPLE
Triangle ABC has vertices A(1, 1), B(4, 1) and C(1, 5). What is the perimeter of the triangle?
1.
2.
First, we need to find the length of each side of the triangle. Use the Distance Formula to find these lengths.
Use the Distance Formula to find the length of AB.
Distance Formula
Substitute into formula:
,
and
Use order of operations to simplify the equation.
Simplify inside the parenthesis first.
Square and
square #’s
before you add refer to ex#_ for how to
Add 9 and 0 before you take the square root
3.
Use the Distance Formula to find the length of AC.
Take the square root of 9. The length of AB is 3 units.
Refer to ex#_for finding sq. roots
Distance Formula
Substitute into formula:
,
and
Use order of operations to simplify the equation.
Simplify inside the parenthesis first.
Square
and
before you add refer to ex#_...
Add 0 and 16 before you take the square root
Take the square root of 16. The length of AC is 4 units.
4.
Use the Distance Formula to find the length of CB.
Distance Formula
Substitute into formula
,
and
Use order of operations to simplify the equation.
Simplify inside the parenthesis first.
Square
and
before you add
Add 9 and 16 before you take the square root Refer to
ex#_for finding sq. roots
5.
Take the square root of 25. The length of CB is 5 units.
The perimeter of a triangle is the sum of the lengths of its 3 sides. In this case we will add
.
Perimeter =
units
10
Practice Support for Lesson 5-1 Continued:
 This example is a model to help solve Practice problem 26.
EXAMPLE
Use the Distance Formula to show that
.
1.
If
, this means that the length of AB is the same
as the length of CD. To find the length of the segments we
use the Distance Formula.
2.
Use the Distance Formula to find the length of AB.
Distance Formula
Substitute into formula:
,
and
Use order of operations to simplify the equation.
Simplify inside the parenthesis first.
Square
and
before you add Refer to ex# for …
Add 25 and 1 before you take the square root
Take the square root of 26. Refer to ex#_for
finding sq. roots
The length of AB is about 5.1 units.
3.
Use the Distance Formula to find the length of CD.
Distance Formula
Substitute into formula:
,
and
Use order of operations to simplify the equation.
Simplify inside the parenthesis first.
Square
and
before you add
Add 1 and 25 before you take the square root
4.
So, by the definition of congruent segments,
Take the square root of 26.
The length of AB is about 5.1 units.
because AB=CD.
11
Additional Practice 5-1:
1.
Which expression represents the distance between points
and
?
2.
3.
The coordinates of the vertices of triangle are
,
and
. Find the perimeter of the triangle.
The coordinates of the vertices of a triangle are
,
and
.
a. Find AB.
b. Find BC.
c. Find AC.
d. Bases on the lengths of the sides, what kind of triangle is
?
Answers to additional
practice:
1.
D
2.
3.
12
Support for Lesson 5-2
Learning Targets for lesson are found on page 56.
Main Ideas for success in lesson# here:
 Use inductive reasoning to determine the Midpoint Formula.
 Use the Midpoint Formula to find the coordinates of the midpoint of a segment on the coordinate plane.
 Vocabulary used in this lesson includes: midpoint.
 Standards for Mathematical Practice to be demonstrated are: use appropriate tools strategically, make use of structure,
reason quantitatively, and make sense of problems capitalize the first letter
Practice Support for Lesson 5-2
 This example is a model to help solve Practice problems 9, 10, 11 and 12.
EXAMPLE:
Find the coordinates of the midpoint of the segment with the given endpoints.
and
The midpoint of a segment is the point that divides the segment into two congruent segments. In order to find
the midpoint of a line segment you use the Midpoint Formula:
. You are finding the average of
the x-coordinates and the average of the y-coordinates.
1.
2.
Label the points as follows
and
Therefore,
and
.
Hint: is read as “x sub one” and it refers to the first x-value.
Substitute the values into the Midpoint Formula and simplify.
Midpoint Formula
and
,
Substitute into the formula.
Simplify the numerator.
Simplify the fractions, if possible.
3.
The midpoint is located at
13
Practice Support for Lesson 5-2 Continued:
 This example is a model to help solve Practice problem 13.
EXAMPLE
Find and explain the errors that were made in the following calculation of the coordinates of a midpoint. Then fix the
errors and determine the correct answer.
Find the coordinates of the midpoint M of the segment with endpoints X(-4, 2) and Y(0, 8).
The midpoint can be found by averaging the x-coordinates of the two different points and averaging the y-coordinates
of the two different points. The mistakes made was instead of adding the two x-coordinates and adding the two ycoordinates, the second coordinate was subtracted from the first. The midpoint can be found correctly by changing
the subtraction sign to an addition sign and simplifying.
The correct coordinates of the midpoint are
.
 This example is a model to help solve Practice problem 14.
EXAMPLE:
AB is graphed on a coordinate plane. Explain how you would determine the coordinates of the point on the segment
that is of the distance from A to B.
One way to find the coordinates would be to find the midpoint M of AB. Below is an example of a line segment and
its midpoint graphed on a coordinate plane.
Next, you could find the midpoint MB since that would be of the distance from A to B. So the coordinates of the
point on the segment that is ¾ the distance from A to B would be located halfway between the midpoint of AB and B.
Point D is between points A and B. The distance between points A and D is of AB. What is the coordinate of point D?
1.
First we need to find the distance between A and B. Point A is at -6 and point B is at 6.
14
Additional Practice 5-2:
1.
2.
has endpoints
and
. Find the coordinates of the midpoint of
.
Which expression represents the midpoint of the line segment with endpoints
and
3.
For the coordinates (5, 8) and (9, 14), one is an endpoint of a line segment and the other is the midpoint. How
many possibilities are there for the other endpoint? Find each one. Explain your method.
?
Answers to additional practice:
1.
2.
(4.5, 3.5)
C
3.
15
Support for Lesson 6-1
Learning Targets for lesson are found on page 63.
Main Ideas for success in 6-1:
 Students use definitions, properties, and theorems to justify a statement.
 Write two-column proofs to prove theorems about lines and angles.
 Vocabulary used in this lesson includes: properties, postulates, and midpoint of a line segment
 Standards for Mathematical Practice to be demonstrated are: Reason abstractly, Construct viable arguments, and
Critique the reasoning of others
Practice Support for Lesson 6-1
Lines CF, DH, and EA intersect at point B. Use this figure for Items 4–8. Write the definition, postulate, or property that
justifies each statement.
 This example is a model to help solve Practice problems 4.
EXAMPLE:
If ∠6 is supplementary to ∠ABF, then m∠6 + m∠ABF = 180°. (meaning that two angles have a value that adds up to 180
degrees.)
Definition of supplementary angles (meaning that two angles have a value that adds up to 180 degrees.)
 This example is a model to help solve Practice problems 5.
EXAMPLE:
If ∠5 ≅ ∠4, then
bisects ∠ABG.
Definition of angle bisector (meaning a ray or line that has split an angle into two equal parts.)
 This example is a model to help solve Practice problems 6.
EXAMPLE
DB + BH = DH
Segment Addition Postulate (meaning that you add two segments together to get a larger segment)
 This example is a model to help solve Practice problems 7.
EXAMPLE:
If ∠CBH is a right angle, then
.
Definition of perpendicular lines (meaning two lines that intersect at a 90 degree or right angle)
1
Practice Support for Lesson Continued:
 This example is a model to help solve Practice problems 8.
EXAMPLE
If m∠5 = m∠4, then m∠5 + m∠6 = m∠4 + m∠6.
Addition Property of Equality (meaning you can replace an equal angle in an addition problem and get the same total
angle value.)
 This example is a model to help solve Practice problems 9.
EXAMPLE:
Write a statement related to the figure above that can be justified by the Angle Addition Postulate. (meaning the
addition of two angles to get a larger angle value.)
m∠CBH = m∠CBA + m∠ABH
Because the two angles combined complete the angle CBH.
2
Additional Practice 6-1:
1. Use the diagram shown.
Write a statement that can be justified by each of the following:
a. definition of angle bisector
b. Angle Addition Postulate
2. Use the diagram shown.





3
Support for Lesson 6-2
Learning Targets for lesson are found on page 66.
Main Ideas for success in 6-2:
 Complete two-column proofs to prove theorems about segments.
 Complete two-column proofs to prove theorems about angles.
 Vocabulary used in this lesson includes: Vertical angles theorem, and Two Column Proofs
 Standards for Mathematical Practice to be demonstrated are: Reason abstractly, Construct viable arguments, and
Attend to precision.
Practice Support for Lesson 6-2
 This example is a model to help solve Practice problems 4.
EXAMPLE:
Given:
BAC,
CAD,
DAE,
EAF, Ray AC bisects
BAD, Ray AE bisects
DAF.
Prove:
m
BAC, + m
Statement
BAD
Ray AE bisects
DAF
m
DAE = m
EAF
m
BAC = m
CAD
m
BAC + m
EAF = m
m
CAE
Reason
Ray AC bisects
CAD + m
EAF = m
DAE
CAD + m
1. Given
2. Bisected angles are equal in
measure. (Angle bisector)
3. Angle Addition
(referred to in 6-1)
DAE = m
4. Angle Addition
EAF = m
5. Substitution
CAE
m BAC + m
CAE
4
Practice Support for Lesson 6-2 Continued:
 This example is a model to help solve Practice problems 5. (Keep in mind that the proof is about the use of the
segment addition postulate not the actual value of the segments.)
EXAMPLE:
Given:
A-B-C-D on
AD
Prove:
AB + BC + CD = AD
Reason
Statement
Given:
A-B-C-D on
AD
1. Given
AB + BD = AD
2. Segment Addition
BC + CD = BD
3. Segment Addition
AB + BC + CD = AD
4. Substitution

 This example is a model to help solve Practice problems 6.
EXAMPLE

5
Practice Support for Lesson 6-2 Continued:
 This example is a model to help solve Practice problems 7.
EXAMPLE:
Statements
Reasons
1.
1. Given
2.
2. Each angle has one unique angle bisector.
3.
3. An angle bisector is a ray whose endpoint is the vertex of
the angle and which divides the angle into two congruent
angles.
4.
4. Reflexive Property. A quantity is congruent to itself.
5.
5. SAS - If two sides and the included angle of one triangle are
congruent to the corresponding parts of another triangle,
the triangles are congruent.
6.
6. CPCTC - Corresponding parts of congruent triangles are
congruent.
 This example is a model to help solve Practice problems 8.
EXAMPLE
How do you know that the triangle is item 7 is a Isosceles triangle?
Since Isosceles triangles are by definition triangles that have two sides and two angle’s congruent, and the proof asked us
to prove two angles congruent we know that the triangle must be isosceles.
6
Additional Practice:
1.
2.
Additional Practice Answers:
1. D
2. a. Definition of angle Bisector
3.
b. Given
c. Substitution
d. ∠ is supplementary to ∠ .
e. Definition of supplementary
angles.
3. a. m∠1 =37; m∠PTR =53
b. Angle Addition Postulate
c. m∠2 =16
d. Subtraction Property of Equality
7
Support for Lesson 7-1
Learning Targets for lesson are found on page 73.
Main Ideas for success in 7-1:
 Make conjectures about the angles formed by a pair of parallel lines and a transversal.
 Students need to prove theorems about these angles.
 Vocabulary used in this lesson includes: parallel, transversal, corresponding angles, same-side interior angles,
alternate interior angles, confirms, and means parallel.
 Standards for Mathematical Practice to be demonstrated are: Reason quantitatively, Reason abstractly, Construct
viable arguments, Express regularity in repeated reasoning, and Use appropriate tools strategically
Practice Support for Lesson 7-1
In the diagram, a || b (meaning that line a is parallel to line b, also meaning that the lines run side by side and do not
touch.)Use the diagram for Items 15–20. Determine whether each statement in Items 15–18 is true or false. Justify your
response with the appropriate postulate or theorem.


 This example is a model to help solve Practice problem 15.
EXAMPLE:
7 is supplementary to 6 .
True; Same-Side Interior Angles Theorem (meaning that the angles are located on the inside of the two lines and are on
the same side of the transversal c).

 This example is a model to help solve Practice problem 16.
EXAMPLE:
1
True, Corresponding Angles Theorem (meaning that the angles are in the same general location in there pod of four
angles grouping)
 This example is a model to help solve Practice problem 17.
EXAMPLE:
2 supplementary
False, Vertical angles like 2 and 6 are congruent.
1
Practice Support for Lesson Continued:
 This example is a model to help solve Practice problems 18.
EXAMPLE:
1
True, alternate interior angles theorem
 This example is a model to help solve Practice problems 19.
EXAMPLE:
If m 1 = 2x − 20, and m 4 = 4x + 5, what is m 1? What is m 4?
m 1 + m 4=180
same side exterior angles are supplementary (as stated in ex 15)
(2x - 20) + (4x + 5)=180
Substitution
6x - 15=180
Combine like terms
6x – 15 + 15=180 + 15
Addition property of equality
6x=195
Add
=
Division property of equality
X=32.5
m 1=2x - 20 so 2(32.5) - 20= 45 and
m 4=4x + 5 so 4(32.5) + 5=135
finally m 1=45 and m 4=135
 This example is a model to help solve Practice problems 20.
EXAMPLE:
Based on your answer to example 19, what are the measures of the other numbered angles in the diagram? Explain your
reasoning.
m 3 = m 5 = m 7 = 45°; m 2 = m 6 = m 8 = 135°.
Sample explanation: 1
5
3, 6
4, 7
3 and 4
1, and 8
2 by the Corresponding Angles Postulate.
2 by the Vertical Angles Theorem.
2
Additional Practice 7-1:
1.
2.
3.
Answers to additional practice:
1.
2.
3.
3
Support for Lesson 7-2
Learning Targets for lesson are found on page 79.
Main Ideas for success in 7-2:
 Student need to develop theorems to show that lines are parallel.
 Determine whether lines are parallel.
 Vocabulary used in this lesson includes: Converse of a theorem or postulate.
 Standards for Mathematical Practice to be demonstrated are: Make use of structure, Use appropriate tools
strategically, Reason abstractly, Attend to precision, and Model with mathematics
Practice Support for Lesson 7-2
For Items 12–14, use the diagram to answer each question. Then justify your answer.
 This example is a model to help solve Practice problem 12.
EXAMPLE:
Given that m 12= 152° and m 6 = 152°, is m || p?
Yes. I know that 12
6. I also know that 6
8 because these angles are vertical Angles that are created by the
intersection of two lines or line segments. So, by the Transitive Property (meaning if a = b, and b = c, then a = c), 12
8. 12 and 8 are congruent since they are corresponding angles, (meaning that the angles are in the same general
location in there pod of four angles grouping) so m || p by the Converse of the Corresponding Angles Postulate.
 This example is a model to help solve Practice problem 13.
EXAMPLE:
Given that m 9 = 52° and m 5 = 56°, is m || n?
No. Sample justification: 9 and 5 are alternate interior angles formed by lines m and n and a transversal. Because
these angles are not congruent, line m is not parallel to line n.
 This example is a model to help solve Practice problem 14
EXAMPLE
Given that m 6 = 124° and m 3 = 52°, is n || p?
No. Sample justification:
6 and 8 are vertical angles, so they are congruent and m 8 = 124°.
3 and 9 are vertical angles, so they are congruent and m 9 = 52°.
8 and 9 are same-side interior angles formed by lines n and p and a transversal Because these angles are not
supplementary (124° + 52° ≠ 180°), line n is not parallel to line p.
4
Practice Support for Lesson 7-2 Continued:
 This example is a model to help solve Practice problem 15.
EXAMPLE:
Two lines are cut by a transversal such that a pair of corresponding angles are right angles. Are the two lines parallel?
Explain.
Yes. Sample explanation: All right angles are congruent, so the corresponding angles are congruent. Thus, the lines are
parallel by the Converse of the Alternate Interior Angles Theorem

 This example is a model to help solve Practice problem 16.
EXAMPLE:
Describe how a construction worker can determine whether two power lines painted on the ground are parallel. Assume
that the worker has a protractor, string, and two stakes.
The worker can place the stakes outside the lines and tie the string to them so that the string crosses the power lines like
a transversal. Then the worker can measure 2 corresponding angles formed by the power lines and the string. If the
corresponding angles are congruent, then the power lines are parallel.
5
Additional Practice 7-2:
1.
2.
3.
Answers to additional practice:
1.
2. 8
3.
6
Support for Lesson 7-3
Learning Targets for lesson are found on page 84.
Main Ideas for success in 7-3:
Students need to develop theorems to show that lines are perpendicular.
 Determine whether lines are perpendicular.
 Vocabulary used in this lesson includes: perpendicular, perpendicular lines, Perpendicular Postulate, Parallel Postulate,
Perpendicular Transverse Theorem, and Perpendicular bisector.
 Standards for Mathematical Practice to be demonstrated are: Make use of structure, Reason abstractly, Critique the
reasoning of others, and Use appropriate tools strategically.
Practice Support for Lesson 7-3
In the diagram, ℓ || m, m 1 = 90°, and 5 is a right angle. Use the diagram for Items 9-11.


 This example is a model to help solve Practice problem 9.
EXAMPLE:
Explain how you know that m ⊥ p.
4 measures 90°, so it is a right angle. Because lines m and p intersect to form a right angle, they are perpendicular.
 This example is a model to help solve Practice problem 10.
EXAMPLE:
Show that m 10 = 90°
I know that ℓ || m and m ⊥ p, and ℓ ⊥ p, so line ℓ must also be perpendicular to line p. Perpendicular lines intersect to
form right angles, so 10 must be a right angle and measure 90°.
 This example is a model to help solve Practice problem 11.
EXAMPLE:
Show that ℓ || n
2 and 6 are both right angles, so they are congruent. 2 and 6 are congruent corresponding angles formed by lines
ℓ and n and a transversal, so ℓ || n by the Converse of the Corresponding Angles Postulate
7
Practice Support for Lesson 7-3 Continued:

 This example is a model to help solve Practice problem 12.
EXAMPLE:
The perpendicular bisector of
intersects
at point B. If
=15 what is
Since
=15 and it is the point of intersection of the perpendicular bisector (meaning intersecting a line at a right or 90
degree angle and then splitting that line in half,) the value is doubled and
is 30.
 This example is a model to help solve Practice problem 13.
EXAMPLE:
An angle formed by the intersection of two lines is Acute. Could the lines be perpendicular? Explain
No. An acute angle measures less than 90°, so it is not a right angle. Because one of the angles formed by the intersection
of the lines is not a right angle, the lines are not perpendicular.
8
Additional Practice 7-3:
1.
2.
3.
Additional Practice answers:
1. C
2.
3.
9
Support for Lesson 8-1
Learning Targets for lesson are found on page 89.
Main Ideas for success in 8-1:
 Make conjectures about the slopes of parallel and perpendicular lines.
 Use slope to determine whether lines are parallel or perpendicular.
 Vocabulary used in this lesson includes: Slope, Parallel, Perpendicular, Conjecture, and opposite reciprocals.
 Standards for Mathematical Practice to be demonstrated are: Reason quantitatively, Reason abstractly, Express
regularity in repeated reasoning, and Model with mathematics
Practice Support for Lesson 8-1

 This example is a model to help solve Practice problem 10
EXAMPLE:
Is
? (meaning is line TQ parallel to line SR?)
Yes, the slope of
is - which reduces to -2, and
has a slope of - which also reduces to -2. Since the slopes are
equal they are parallel.
 This example is a model to help solve Practice problem 11
EXAMPLE:
Is
? (Meaning is line QR meeting line SR at a right angle?)
Yes, The slope (the rise and run of the line) of
is and the slope of
is -2. Since the slopes are opposite reciprocals
(meaning is the fraction flipped and the power reversed) they are perpendicular (meaning that the lines meet at a right or
90 degree angle)
 This example is a model to help solve Practice problem 12
EXAMPLE
Is
?
No, The slope of
is - and the slope of
is -2. Since they are not opposite reciprocals (meaning is the fraction flipped
and the power reversed) the lines are not parallel.
1
Practice Support for Lesson 8-1 Continued:

 This example is a model to help solve Practice problems 13.
EXAMPLE:
Line m passes through the origin (0,0) and the point (3, 4). What is the slope of a line parallel to line m?
If line m passes through the origin (0,0), and through (3,4) the slope would be . Because:
Following the slope
formula learned in Algebra 1. Therefore the slope of the perpendicular line would have to be an opposite reciprocal.
That slope would be - .
 This example is a model to help solve Practice problems 14.
EXAMPLE
RST is a right angle, if
has a slope of 5 what is the slope of
?
If the slope of one leg of the angle is 5 then the other leg would have to be perpendicular to create a right angle. SO
the slope would have to be opposite reciprocals (meaning is the fraction flipped and the power reversed.) so the
other leg (ST) would have a slope of
.
2
Additional Practice 8-1:
1.
2.
3.
Answers to additional practice:
1.
2. C
3.
3
Support for Lesson 8-2
Learning Targets for lesson are found on page 92.
Main Ideas for success in 8-2:
 Write the equation of a line that is parallel to a given line.
 Write the equation of a line that is perpendicular to a given line.
 Vocabulary used in this lesson includes: Slope intercept form, Y-intercept of a line, and point slope form
 Standards for Mathematical Practice to be demonstrated are: Reason abstractly and quantitatively, construct viable
arguments, Model with mathematics, and Make sense of problems.
Practice Support for Lesson 8-2
 This example is a model to help solve Practice problem 16.
EXAMPLE:
What is the equation of a line parallel to y = −2x + 5 that passes through point (5,1)?
Since we are looking for a parallel line (meaning that the lines run side by side and do not touch) the slope must be an
equal slope. The given slope is -2, so the parallel slope is also -2. The next item is to find the y-intercept (the point that
the line crosses the y axis) Use the point slope form to write the correct equation in slope intercept form.
y - 1 = -2(x – 5)
Point slope form
y – 1 = -2x + 10
Distributive Property
y – 1 + 1 = -2x +10 +1
Addition property of equality
y = -2x + 11
Simplify
So the final equation that is parallel and goes through (5,1) is y = -2x + 11.
 This example is a model to help solve Practice problem 17.
EXAMPLE:
What is the equation of a line perpendicular to y=4x−3 that passes through point (-2,1)?
Since we are looking for a perpendicular line the slope must be an opposite reciprocal slope. The given slope is 4, so the
perpendicular slope is - . The next item is to find the y-intercept. Use the point slope form to write the correct equation
in slope intercept form.
Point slope form
y - 1 = - (x + 2)
Substitute
y–1=- x-
Distributive Property
y – 1 + 1 = - x - +1
Addition property of equality
y=- x+
Simplify
So the final equation that is parallel and goes through (-2,1) is y = - x + .
4
Practice Support for Lesson 8-2 Continued:
 This example is a model to help solve Practice problem 18. (use the diagram below for problem 18)
EXAMPLE
What is the equation of the line parallel to the given line that passes through point (2,2)
First you must determine the slope of the given line. The given line has a slope of 3. Now that we have the slope we can
find the equation in the same way we did in the previous questions. Using that we can find the equation to be y = 3x – 4.
 This example is a model to help solve Practice problem 19.
EXAMPLE:
What is the equation of the line perpendicular to the given line that passes through point (2,2)?
Again the first thing is to detriment the slope of the give line, and the slope is 3 still. Now the perpendicular slope
would be the - . Now that we have the slope we can find the equation in the same way we did in the previous
questions. Using that we can find the equation to be y = - x + .
 This example is a model to help solve Practice problem 20.
EXAMPLE:
How could you check that your answer to Item 19 is reasonable?
You could graph the line and check that it passes through the point (2, 2) and that it appears to be perpendicular to line
given.
 This example is a model to help solve Practice problem 21.
EXAMPLE:
∆DBA has vertices at the points A(−1, 4), B(0, 0), and D(4, 1). What kind of triangle is ∆DBA? Explain your answer.
The triangle has a right triangle. The reason is because the slope of
is -4 and the slope of
is - . These two slopes
are opposite reciprocals.
5
Additional Practice 8-2:
1.
2.
3.
Answers to additional practice
1.
2.
3. D
6