1.2.1c3 13.notebook
September 05, 2013
1.2.1c3 13.notebook
September 05, 2013
Unit 1: Reasoning and Proof
Lesson 2: Geometric Reasoning and Proof
Essential Question: How do you prove theorems about lines, angles, and triangles?
1.2.1c3 13.notebook
September 05, 2013
EQ for 1.2.1: How are linear pairs of angles related?
1. In the diagram at the right, lines k and n intersect at the
point shown, forming angles numbered 1, 2, 3, 4.
a. If measure of angle 1 = 72, what can you say about measure
of angle 2, 3 and 4? What assumptions are you using to obtain your
answers?
b. If measure of angle 2 is 130, what can you say about measure
of angles 1, 3 and 4?
c. In general, what relationships between pairs of angles do you think are true? Make a list
of them.
d. Will the general relationships you listed for Part c hold for any pair of intersecting
lines?
e. Write an if-then statement about linear pairs of angles that you think is always correct.
You may want to begin as follows. If two angles are a linear pair, then... .
f. Write an if-then statement about vertical angles that you think is always correct. You
may want to begin as follows. If two lines intersect, then ... .
1.2.1c3 13.notebook
September 05, 2013
EQ: HOw are vertical angles related and why is that the case?
In mathematics, statements of basic facts that are accepted as true without proof are
called postulates. (or axioms). These assumed facts will be helpful in supporting your
reasoning in the remainder of this unit and in future units. Begin by assuming the
following postulate concerning linear pairs of angles.
Linear Pair Postulate If two angles are a linear pair, then the sum of their measures is
1800.
2. Study the attempt at the right by one group of students at Washington High School to
prove the conjecture they made in Part f of Problem 1. Based on the labeling of the
diagram, they set out to prove the following.
If lines n and k intersect at the point shown, then m<1 = m<3.
They reasoned as follows:
(1) Since lines n and k intersect,
<1 and <2 are a linear pair.
so, m<1 + m<2 = 180 degrees.
(2) Since lines n and k intersect,
<2 and <3 are linear pair.
so, m<2 + m<3 = 180 degrees.
(3) If m<1 + M<2 = 180 and
m<2 + m<3 = 180, then
m<1 + m<2 = m<2 + m<3
(4) If m<1 + m<2 = m<2 + m<3,
then m<1 = m<3
(1) This is correct. Since <1 and <2 are a linear pair, then
by the Linear Pair Postulate, the sum of their measures is
180 degrees.
(2) This is correct. Since <2 and <3 are a linear pair, then
by the Linear Pair Postulate, the sum of their measures is
180 degrees.
(3) This is correct. If both m<1 + m<2 and m<2 + m<3 equal
1800 then the two sums must be equal to each other.
(4) This is correct because of the Subtraction Property of
Equality allows us to subtract m<2 from each side of the
equation. m<1 + m<2 = m<2 + m<3
-m<2 -m<2
Left with m<1 = m<3
b. Now write an argument to show the following: If lines n and k intersect at the point
shown, then m<2 = m<4. Give reasons justifying each of your statements.
(1) Since lines n and k intersect, <2 and <3 are a linear pair so,
m<2 + m<3 = 180.
(2) Since lines n and k intersect, < 3 and <4 are a linear pair so,
m<3 + m<4 = 180.
(3) If m<2 + m<3 = 180 and m<3 + m<4 = 180, then
m<2 + m<3 = m<3 + m<4.
(4) If m<2 + m<3 = m<3 + m<4, then m<2 = m<4.
Can use <2 & < 3; <3 & <4
or
<1 & <4; <1 & <2
1.2.1c3 13.notebook
September 05, 2013
In mathematics, a statement that has been proved using deductive reasoning from
definitions, accepted facts, and relations is called a theorem.
The statement proved in Problem 2 is sometimes referred to as the Vertical Angles
Theorem, vertical angles have equal measure.
3. Recall that two intersecting lines (line segments or rays) are perpendicular (
only if they form a right angle.
) if and
a. Rewrite this definition as two if-then statements.
b. Claim: Two perpendicular lines form four right angles. Is this claim true or false?
Explain your reasoning.
c. Study the following strategy that Juanita used to
prove the claim in Part b. First, she drew and labeled
the diagram at the right.
Then she developed a plan for proof based on her diagram.
2 1
3 4
1.2.1c3 13.notebook
Linear Pair Postulate
September 05, 2013
If two angles are a linear pair,
then the sum of their measure is 1800
4
3
2
1
Ex: <1 & <2