Download Intro to proofs 8

Introduction to Proofs
Lesson 8
Question of the Day: Match the definition with a term from the box. Place the letter in the blank
1.) _______: two angles that add to be 90
2.) _______: equidistant from the endpoints of a segment; gives two congruent pieces
3.) _______: two or more adjacent angles that form a line
4.) _______: three or more angles with the same vertex
5.) _______: bisects a segment and forms right angles with the segment
6.) _______: angles inside the parallel lines on opposite sides of the transversal
7.) _______: add up pieces of a segment to get the entire segment
8.) _______: angles opposite each other when two lines cross
9.) _______: the angles of a triangle add to 180
10.) ______: angles outside the parallel lines on opposite sides of the transversal
11.) ______: lines that intersect at right angles
12.) ______: angles inside the parallel lines on the same side of the transversal
13.) ______: base angles are congruent; two sides are congruent
14.) ______: add up the small pieces of an angle to get the whole angle
15.) ______: two angles that add to be 180
16.) ______: Angles on the same location of the parallel lines
17.) ______: two or more lines that never intersect
18.) ______: to be equal
a. line
b. angles on a line
c. midpoint
d. ray
e. vertical angles
i. vertex
j. segment addition
f. complementary angles
g. angle addition
h. angle
k. perpendicular bisector
l. angles at a point
m. congruent
o. alternate interior
p. alternate exterior
q. triangle sum
r. same side interior
s. corresponding angles
t. isosceles triangle
u. perpendicular lines
v. parallel lines
n. supplementary angles
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~Deductive Reasoning: the ability to identify the steps used to reach a conclusion based on known facts
~Geometric Proof: A step-by-step explanation that identifies the steps used to reach a conclusion about
a geometric statement.
- can be written in two columns, as a paragraph, or flow chart
-statements: geometric concepts (definitions and theorems) applied to the picture
-goes in the left column
- reasons: in the right column; what concept (definitions and theorems) was used
-ex: angles on a line, midpoint
-Every new concept (step) of the proof is a row
~Hints to writing proofs:
1.) Mark the figure with the information given. This is how you find out where to start. The
conclusion to be proved is the end of the proof.
2.) Always keep in mind what is trying to be proved and what needs to be known to get there.
3.) Don't skip any steps, even simple ones.
~Algebraic Properties of Equality:
PROPERTY
ALGEBRA EXAMPLE
GEOMETRY EXAMPLE
Reflexive: An object is equal to
itself.
Transitive: If two things are
equal to the same thing, they are
equal to each other.
Substitution: Things that are
equal may be plugged in for each
other.
Bringing the concrete (specific) to the abstract (general):
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102
Use the picture to illustrate the exterior angle of a
triangle theorem.
w
Given a triangle with line BC, prove <x +<y = <z .
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EXAMPLES:
1.) GIVEN: AB = CD Prove: AC = BD
Statements
Reasons
Given
Segment Addition
Segment Addition
Substitution Property
Transitive Property
2.) Given: m<1 + m<2 = 66°,
m<1 + m<2 + m<3 = 99°,
m<3 = m<1, m<1 = m<4
Prove: m<4 = 33°
1
2
3
Statements
m<1 + m<2 = 66°, m<1 + m<2 + m<3 = 99°,
m<3 = m<1, m<1 = m<4
4
Reasons
Substitution
m<3 = 33
m<3 = m<4
Substitution
̅̅̅̅ Prove: PQ =
3.) Given: Q is the midpoint of 𝑃𝑅
1
𝑃𝑅
2
Statements
Reasons
PQ + QR = PR
Substitution
2(PQ) = PR
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