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Section 1.12 Spatial Invariants (p. 58)
Goal: Search for geometric invariants, such as points of concurrency and collinearity of
points.
Vocabulary building:
1. angle bisector – a line segment which cuts an angle into two equal parts
2. concur – to come to an agreement; to come together; to meet
3. concurrent lines – 3 or more lines which intersect at the same point
4. collinear points – 3 or more points which are on the same line
5. perpendicular bisector – line or line segment which intersects another line segment to
form right angles and to divide the segment into two congruent parts
Mathematical Symbols:
1. perpendicular 
Main ideas:
1. Look at the pictures at the top of p. 58. (We did this experiment in an earlier lesson.)
Notice the number of sides on the original shape, and count the number of sides on the
polygon created inside the original. If the original polygon had 4 sides, the interior
polygon has 8.
2. Look at the picture at the top of p. 59. Read through the material on this page.
3. During class we worked on the “For You to Do” at the bottom of the page.
4. Continue to read p. 60. Put the “Concurrence of Perpendicular Bisectors” theorem and
the “Concurrence of Angle Bisectors” theorem into your notes. Make sure you sketch the
pictures to go with them.
5. Finally, read p. 61. Make sure you understand what “collinear points” are. In class we
worked through the “For You to Do”. We found that when the experiment was finished,
the centers of all the circles were collinear.
INVESTIGATION 2A – THE CONGRUENCE RELATIONSHIP
(p. 72)
Many time in life we are asked to compare and contrast things, to look for similarities
and differences. This section of chapter 2 will show us how we can do that
mathematically.
Section 2.1 Getting Started (p. 73)
Goal: Define congruence
Vocabulary building:
1. congruent figures – two figures are congruent if they have the same size and shape
regardless of location or orientation
ABC  ABC even though they are facing different directions.
Mathematical symbols:
1. congruent 
Main ideas:
1. Read carefully through p. 73. Pay attention to the pictures of the quadrilaterals. Notice
how they are all squares but are not all the same size and do not “sit” the same direction.
2. Make sure you understand that “congruence” is made up of two different symbols, and
equal sign and the sign for “similar” which means alike but not exactly alike. In the case
of congruence, the “not exactly alike” refers to the fact that the figures do not have to
face the same direction (orientation). The equal part refers to the fact they must be the
same size and shape.
Section 2.2 Length, Measure and Congruence (p. 75)
Goal: Interpret statements about congruent figures and write statements using the correct
notation.
Vocabulary building:
1. angle – a figure created when 2 rays intersect; named by a point on the first ray, the
vertex, and a point on the second ray; BCD or DCB both names are ok as long as the
vertex letter is in the middle (If there is only one angle, it can be named by its vertex C
B
C
D
2. equal – compares numbers
3. line segment – part of a line; is named by its 2 endpoints which are labeled with capital
letters; the name of a line segment can be written as AB or AB
A
B
4. vertex – the point where two rays or segments intersect to form an angle
Mathematical Symbols:
1. angle - 
2. measure of an angle - m
Main ideas:
1. In geometry it is important to know the difference when a shape is being described and
when a measurement is being described. “Equal” works with numbers, measurement.
“Congruent” refers to shapes.
2. Mathematical symbols help you to know whether you can work with equal (=) or
congruence (  ). AB and AB both talk about the line segment between points A and B.
AB refers to the length of the segment (ex. It might measure 2 inches), while AB names a
line segment, a geometric shape.
JK = 1.5 inches
RS = 1.5 inches
JK = RS because their lengths are the same.
JK  RS because the bar across the top talks about the shape not
the measurement.
JK  RS because their shapes are identical.
3. Read the Tony and Sasha section very carefully. Pay attention to the statements about
the line segments.
4. On p. 76 the book discusses congruence and equality for angles. Pay attention to the
paragraph and the picture. You may want to put this in your notes.
5. Pay close attention to the Check Your Understanding questions. These will help you to
know if you understand the difference between equal and congruent.
Section 2.3 Corresponding Parts (p. 79)
Goal: Understand the meaning of corresponding parts.
Vocabulary building:
1. corresponding – to be like something else in size and placement
2. quantitative measure – measurements such as area and perimeter which measure
quantity
2. similar – alike in shape but not always alike in size; ex. All circles are similar.
4. tick marks – marks (tiny line segments) on a drawing which indicate that parts are
congruent.
Mathematical Symbols:
1. tick marks
Notice the single mark for AB and the triple mark for
BC . Since the markings are different, AB is not congruent to BC . Notice also the mark
at B . Angles can’t use tick marks but can be marked to show congruence to another
angle.
Main ideas:
1. It is not always convenient to construct shapes. If we are not going to measure
something, we can indicate congruence with symbols on our drawings. (see “tick marks”
above)
2. Read through p. 79. Pay attention to the pictures.
3. At the bottom of p. 79 the book discusses congruence of triangles. Remember triangles
are made up of sides and angles. When we talk about congruent triangles, (or any
congruent shapes) we need to be careful of how we look at the figures and how we talk
about their parts.
ABC  FED
The way these are named is important. Notice A has
the same marking as F . These two angles are said to
be “corresponding” since they are marked the same and
if we flipped DEF over, it would sit exactly on top of
ABC.
When we talk about the congruence of two geometric figures, it is very important to
name them so their corresponding parts are named at the same time. In the figure of the
two triangles, when we talk about ABC  FED , we automatically know that
AB  FE because their letters are named in the same order. Also, AC  FD.
ABC correspond s to FED which tel ls us those angles are congruent.
4. corresponding parts of congruent figures are congruent (abbreviation: CPCTC) –If two
figures are congruent, than all parts (sides or angles) which correspond (match up) are
congruent. (This goes with #3 in the notes.)
Section 2.4 Triangle Congruence (p. 82)
Goal: Understand the meaning of corresponding parts and test for congruence in
triangles.
Vocabulary building:
1. converse – switching the “if’ and the “then” part of a statement; ex. If today is
Wednesday, then we have an early out. – The converse would be “If we have an early
out, then today is Wednesday.”
2. postulate – a statement which is accepted as a fact without having to prove it
Mathematical Symbols:
Main Ideas:
1. Review section 2.3 before starting this section.
2. Carefully read and think about the For Discussion questions on p. 82. These are
important to the lesson.
3. When comparing triangles, you can sometimes prove congruence by comparing 3 parts
of one triangle to 3 parts of another triangle. For example, if I measure 3 angles of
ABC and 3 angles of XYZ, and their measures are the same, are the triangles congruent?
If AB  XY and A  X and AC  XZ is that enough informatio n to prove the triangles are congruent?
Read and take some notes over “Classifying Your Information” on p. 82.
4. The top of p. 83 lists some possible ways to think about proving triangle congruence.
Unfortunately, not all of them work.
5. Triangle Congruence Postulates – These 4 postulates will prove that one triangle is
congruent to another: ASA (angle-side-angle), SAS (side-angle-side), SSS (side-sideside), and AAS (angle-angle-side).
Angle-side-angle (ASA) – When looking at the triangles, 2 angles and the side in
between them should match
A
A
S
A
A
S
Side-angle-side (SAS) – When looking at the triangles, 2 sides should match with the
angle between them
S
A
S
S
S
A
Side-side-side (SSS) – when looking at the triangles, three sides of one triangle are equal
in length to three sides of another triangle
S
S
S
S
S
S
6. Understand that there is NOT as SSA postulate. Having two sides next to each other
which are congruent and an angle which is not between them will guarantee nothing
about the third side of both triangles.
7. Look over Check Your Understanding on p. 84.
INVESTIGATION 2B – PROOF AND PARALLEL LINES (p. 88)
In this section of chapter 2, we will study parallel lines and the relationships which are
formed when a third line is put in the picture.
Section 2.5 Getting Started p. 89
Goal: Identify pairs of congruent angles when a transversal cuts parallel lines.
Vocabulary building:
1. transversal – a line that intersects 2 or more lines
2. parallel lines – lines in the same plane which never intersect
3. midline – connects the midpoints of 2 sides of a triangle
4. exterior angle – an angle formed outside of a polygon
1
5. supplementary angles – 2 angles whose sum is 180 degrees
Notes:
The big idea in this lesson is that the measure of an exterior angle is equal to the sum of
the two interior angles farthest from it.
Section 2.6 Deduction and Proof (p. 91)
Goal: Make assumptions and write proofs to understand the need for proofs in
mathematics.
Vocabulary building:
1. proof – (math def.) a sequence of steps, statements, or demonstrations that lead to a
valid conclusion
2. deductive process – use logical thinking to get an answer
3. postulate – a statement which is assumed to be true
4. assumption – something taken for granted
5. median – a segment drawn from a vertex of a triangle to the midpoint of the opposite
side
6. vertical angles – formed when 2 lines intersect; (look like the letter X)
Notes:
1. In writing a proof it is important to provide evidence, reasons why something is true.
2. Read Minds in Action on p. 92. Sasha is trying to prove to Ivan why his angles must be
congruent. This is an example of a proof:
We are told that AC = 14 in and AB = 14
in. The measure of angle CAB = 30
degrees. (These are considered “given”s.)
Sasha tells Ivan to draw a median from A to M (see vocabulary for this lesson) from C.
Now, CM  BM because of the definition of a midpoint.
AM  AM because everything is equal to itself.
CAM  BAM because of SSS.
Finally, Sasha says ACM  ABM because corresponding
parts of congruent figures are congruent.
The items in black are considered statements while the items in red are the reasons the
statements are true. This is an example of how a proof works.
3. Try this one:
A straight angle is a line. It has 180 degrees. AC is a line (straight angle). If
mBDC  72 then mBDA  180  72  108.
BQ is also a line (straight angle). Since mBDC  72 , then mCDQ  180  72  108.
Finally, by using the same strategy, we would find the measure of angle QDC = 72. This
allows us to notice that vertical angles are equal.
4. Write the vertical angle theorem in your notes. (p. 93.)
Section 2.7 Parallel Lines (p. 99)
Goal: Identify pairs of congruent angles when a transversal cuts parallel lines.
Vocabulary building:
1. parallel lines – lines in the same plane which do not intersect
2. alternate interior angles – angles on opposite sides of a transversal between lines (in
the diagram see angles 3 and 5; see angles 4 and 6)
3. corresponding angles – found on the same side of the transversal, occupy similar
positions (in the diagram see angles 1 and 5; angles 4 and 8; angles 2 and 6; and angles 3
and 7)
4. vertical angles – formed when two lines intersect
5. consecutive angles – found on the same side of the transversal between the lines (in the
diagram see angles 4 and 5; angles 3 and 6)
6. alternate exterior angles – found on opposite sides of the transversal outside the lines
(in the diagram see angles 1 and 7; angles 2 and 8)
7. corollary – a consequence which comes from a theorem
8. equidistant – equal distance from the same point
Notes:
1. It is very important to understand the vocabulary in this section. Make sure you have
studied the diagram.
2. AIP theorem – If two lines from congruent alternate interior angles with a transversal,
then the two lines are parallel
3. Read through p. 100 which is the proof of the AIP theorem.
4. Make sure you read the rest of the section. We will also find that when parallel lines
are cut by a transversal corresponding angles are equal in measure, consecutive angles are
supplementary, and alternate exterior angles are equal in measure.
Section 2.8 The Parallel Postulate (p. 105)
Goal: Identify pairs of congruent angles when a transversal cuts parallel lines.
Vocabulary building:
Notes:
Parallel postulate – if point p is not on line l, exactly one line through p will be parallel to
l
p
l
PAI theorem – if two parallel lines are cut by a transversal, alternate interior angles are
congruent.
line n is parallel to line l; angles 1 and 2 are
alternate interior angles and are congruent.
Use the picture and what you know about alternate interior, vertical, corresponding,
straight, and consecutive angles to find the missing angle measures.
m1  70
m2  110
m3  115 (straight angle w/65)
m4  65 (vertical angles)
m5  115 (vertical to angle 3 or straight angle w/65)
m6  70 (straight angle w/angle 2 or triangl e  180)
(I’ve included just some of the answers to give you an idea.)
Triangle-angle-sum theorem – The interior angles of a triangle equal 180 degrees.
Unique perpendicular theorem – If point p is not on line  , there is only one line through
p which is perpendicular to  (See the picture for the parallel postulate, think about a line
going straight down to make a right angle with line  .)
Investigation 2C – Writing Proofs
Section 2.9 Getting Started p. 115
Section 2.10 What Does a Proof Look Like? P. 117
Goal: Use a variety of ways to write and present proofs.
Vocabulary:
Notes:
See p. 117 for an example of a 2 column proof. Two column proofs must have a leftcolumn for Statements and a right-column for Reasons. The statements tell what you
know while the reasons explain how you know. All proofs begin with the “given”
because that is all you know when you start. The very last “statement” is what you are
trying to prove, your goal.
See p. 118 at the top for an example of a paragraph proof. A paragraph proof is written in
logical sentences. You must keep it detailed enough so it makes sense. You still begin
with your “given”.
In the middle of p. 118 is an example of an outline style proof. This is more informal than
the 2 column proof but still used a series of statements and reasons. Notice the symbols
for “because” and “therefore”. The reasons are put in ( ).
Section 2.11 Analyzing the Statement to Prove p. 123
Goal: Identify the hypothesis and conclusion of a given statement.
Vocabulary:
1. hypothesis – the “if” part of an “if . . . then” statement; what you assume to be true.
2. conclusion – the “then” part of an “if . . . then” statement; states what you need to
prove
3. equiangular – all angles in the figure have the same measure
Notes:
An “if . . . then” statement is made of a hypothesis and a conclusion.
Ex. If I am hungry, then I will have a snack.
The “I am hungry” is the hypothesis while the “I will have a snack” is the conclusion.
“if . . .then” statements do not have to be true.
The goal of this lesson is to be able to identify the hypothesis and the conclusion and to
tell if the statement is true or false.
Ex. If a triangle is isosceles, then the two base angles are congruent.
The hypothesis is “a triangle is isosceles”, and the conclusion is “the two base angles are
congruent.” This is a true statement since an isosceles triangle has 2 sides of equal
measure which would make two of the angles be congruent.
Ex. Vertical angles are congruent.
The words “if” and “then” are not here, however, think of the statement this way:
If angles are vertical, then they are congruent.
This is a true statement.
Section 2.12 Analysis of a Proof p. 125
Goal:Use a variety of ways to write and present proofs.
Vocabulary:
Notes:
1. “analysis” is creating the proof; deciding what to use; like a rough draft of a paper
2. “visual scan” – This is what we have been doing so far. Draw a picture, mark
congruent pieces, see what you have.
3. Look at example 1 p. 126. It explains how to analyze and use a visual scan.
4. Notice that CPCTC is used often as a reason.
5. Flow chart strategy. This is a top-down analysis which starts with what you know and
then moves down providing reasons.
6. A flow chart can easily be turned into a 2 column proof.
Section 2.13 The Reverse List p. 131
Section 2.14 Practicing Your Proof Writing Skills p. 135
Goal: Use the perpendicular bisector theorem and the isosceles triangle theorem to prove
that parts of a figure are congruent.
Vocabulary:
1. perpendicular bisector – a segment which divides another segment into two equal parts
while making a right angle at the point of intersection
2. Isosceles triangle – a triangle which has at least 2 congruent sides and 2 congruent
angles
Notes: