Investigation 1.2.1
Name: ______________________________________________
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1. In the diagram to the right, lines k and n intersect at the point
shown, forming angles numbered 1, 2, 3, 4.
a) If 𝑚∠1 = 72°, what can you say about 𝑚∠2? About 𝑚∠3? About 𝑚∠4? What assumptions are you
using to obtain your answers?
b) If 𝑚∠2 = 130°, what can you say about 𝑚∠1? About 𝑚∠3? About 𝑚∠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 Par c hold for any pair of intersecting lines? Test your
conjectures using specific examples.
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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….
In the remainder of this lesson, you will continue to use inductive reasoning to discover possible relations among lines
and angles, but you will also use deductive reasoning to prove your conjectures are always true. To reason deductively,
you must first have some basic facts from which to reasons. 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 180°
2. Study the attempt below 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.
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If lines n and k intersect at the point shown, the 𝒎∠𝟏 = 𝒎∠𝟑.
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They reasoned as follows
(1) Since lines n and k intersect, ∠𝟏 and ∠𝟐 are a
linear pair. So, 𝒎∠𝟏 + 𝒎∠𝟐 = 𝟏𝟖𝟎°
(2) Since lines n and k intersect, ∠𝟐 𝒂𝒏𝒅 ∠𝟑 are a
linear pair. So, 𝒎∠𝟐 + 𝒎∠𝟑 = 𝟏𝟖𝟎°
(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.
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a) Explain why each of the statements in the
students’ reasoning is or is not correct.
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
In mathematics, a statement that has been proved using deductive reasoning from definitions, accepted facts,
and called a theorem. The statement proved in Problem 2 is sometimes referred to as Vertical Angles
Theorem; vertical angles have equal measure.
In Course 1, you may have proved the Pythagorean Theorem. After a theorem has been proved, it may be used
to prove other conjectures. As your geometric work in Course 3 progresses, you will want to know which
theorems have been proved.
3. Recall that two intersecting lines (line segments or rays) are perpendicular (⊥) if and only if they
form a right angle.
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.
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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
I know that if l ⊥ m, they form a right angle, say ∠𝟏. A right angle has
measure 90°. Use the fact that ∠𝟏 and ∠𝟑 are vertical angles to show
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𝒎∠𝟑 = 𝟗𝟎°. Use the fact that ∠𝟏 and ∠𝟐 are a linear pair to show 𝒎∠𝟐 = 𝟗𝟎°.
Then use the fact that ∠𝟐 and ∠𝟒 are vertical angles to show 𝒎∠𝟒 = 𝟗𝟎°.

She then wrote her proof in a two-column statement-reason form.
Statements
1. l ⊥ m
2. l and m form a right angle. Call it ∠1.
3. 𝑚∠1 = 90°
4. ∠1 and ∠3 are vertical angles.
5. 𝑚∠3 = 𝑚∠1 = 90°
6. ∠1 𝑎𝑛𝑑 ∠2 are a linear pair.
7. 𝑚∠1 + 𝑚∠2 = 180°
8. 𝑚∠2 = 180° − 𝑚∠1 = 90°
9. ∠2 𝑎𝑛𝑑 ∠4 are vertical angles.
10. 𝑚∠2 = 𝑚∠4
11. 𝑚∠4 = 90°
12. ∠1, ∠2, ∠3, 𝑎𝑛𝑑 ∠4 are right angles.
Reasons
1. Given
2. Definition of perpendicular lines
3. Definition of right angle
4. Definition of Vertical Angles
5. Vertical Angles Theorem
6. Definition of Linear Pair
7.
8.
9.
10.
11.
12.
i.
How does the diagram that Juanita drew show the information given in the claim?
ii.
Check the correctness of Juanita’s reasoning and supply reasons for each of statements 7 – 12.
iii.
Describe a plan of proof of the above claim that does not involve use of the Vertical Angles Theorem.
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