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# Geometry lesson

## on Oct 26, 2010

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A Lesson on Basic Geometry Proof

A Lesson on Basic Geometry Proof

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## Geometry lessonPresentation Transcript

• Geometry Lesson
Bob Roach
ED538
Fall 2010
California Geometry Standards Grades 8 through 12
2.0 Students write geometric proofs, including proofs by contradiction.
• Warm Up Exercise
On a piece of paper to be handed in, copy each of the three statement sets then draw (write) a conclusion if possible:
1) All rectangles have congruent diagonals. ABCD is a rectangle.
2) All squares have four congruent sides. GHIJ has four congruent sides.
3) If l is perpendicular to m, then ABC is a right angle. If ABC is a right angle then BCD is complementary to BDA. l is perpendicular to m.
• Warm Up Exercise Answer (1)
All rectangles have congruent diagonals. ABCD is a rectangle. We conclude that ABCD has congruent diagonals.
Statements
Reasons
Given
Given
Law of Detachment
All rectangles have congruent diagonals
ABCD is a rectangle
ABCD has congruent diagonals
• Warm Up Exercise Answer (2)
All squares have four congruent sides. GHIJ has four congruent sides.
Statements
Reasons
Reasons
Given
Given
p -> q ; q
a) All squares have four congruent sides
b) GHIJ has four congruent sides.
c) No conclusion
• Warm Up Exercise Answer (3)
Statements
Reasons
Given
Given
Given
Syllogism using conditional a and b
p -> q
q -> r
___________
p -> r
e) Law of Detachment c and d
a) l ⟂ m -> ABC is a right angle
b) ABC is a right angle -> BCD is complementary to BDA
c) l ⟂ m
d) l ⟂m-> BCD is complementary to BDA
e) BCD is complementary to BDA
• Homework Questions?
• Five Essential Parts of a Proof
State the theorem to be proved
List the given information
If possible draw a diagram to illustrate
State what it is to be proved.
Develop the proof using deductive reasoning
• What is a Theorem?
Statements that are proved by using:
Definitions
Postulates
Undefined terms (line, point, plane,...)
Deductive reasoning
are called Theorems
We then use proved theorems to prove other theorems
• Verifying Segment Relationships
Theorem 2-1: Congruence of segments is reflexive, symmetric, and transitive
AB ≅ AB (Reflexive)
If AB ≅ CD then CD ≅ AB (Symmetric)
If AB ≅ CD and CD ≅ EF then AB ≅ EF (Transitive)
Proofs: Use properties of real numbers with the definition of segment congruence (have same length)
• Proof of Theorem 2-1 (Symmetric Part)
P
Q
Given: PQ ≅ RS
Prove: RS ≅ PQ
S
R
Statements
Reasons
Given
Definition of ≅ line segments
Symmetric property(=)
Definition of ≅ line segments
a) PQ ≅ RS
b) PQ = RS
c) RS = PQ
d) RS ≅ PQ
• Example of Line Segment Proof
Given: PQ ≅ XY
QR ≅ YZ
Prove: PR ≅ XZ
Z
X
Y
R
P
Q
Statements
Reasons
a) PQ ≅ XY
QR ≅ YZ
Given
Definition of ≅ line segments