A Lesson on Basic Geometry Proof

1. Geometry Lesson
Bob Roach
ED538
Fall 2010

2. Geometry Grade 9-11
California Geometry Standards Grades 8
through 12
2.0 Students write geometric proofs,
including proofs by contradiction.

3. 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.

4. Warm Up Exercise Answer (1)
All rectangles have congruent diagonals.
ABCD is a rectangle. We conclude that
ABCD has congruent diagonals.
a) All rectangles have
congruent diagonals
b) ABCD is a rectangle
c) ABCD has
congruent diagonals
Statements Reasons
a) Given
b) Given
c) Law of
Detachment

5. Warm Up Exercise Answer (2)
All squares have four congruent sides. GHIJ
has four congruent sides.
a) All squares have
four congruent sides
b) GHIJ has four
congruent sides.
c) No conclusion
Statements ReasonsReasons
a) Given
b) Given
c) p → q ; q

6. Warm Up Exercise Answer (3)
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
Statements Reasons
a) Given
b) Given
c) Given
d) Syllogism using conditional a and b
a) p → q
q → r
___________
p → r
e) Law of Detachment c and d

8. 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

9. 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

10. 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)

11. Proof of Theorem 2-1 (Symmetric
Part)
Statements Reasons
a) Given
b) Definition of ≅ line segments
c) Symmetric property(=)
d) Definition of ≅ line segments
Given: PQ ≅ RS
Prove: RS ≅ PQ
P Q
SR
a) PQ ≅ RS
d) RS ≅ PQ
b) PQ = RS
c) RS = PQ

12. Example of Line Segment Proof
Statements Reasons
a) Given
b) Definition of ≅ line segments
c) Addition property(=)
d) Segment addition postulate
e) Substitution property (=)
f) Definition of ≅ line segments
Given: PQ ≅ XY
QR ≅ YZ
Prove: PR ≅ XZ
X Z
a) PQ ≅ XY
QR = YZb) PQ = XY
Y
P RQ
QR ≅ YZ
c) PQ + QR = XY + YZ
d) PR = PQ + QR
c) PQ + QR = XY + YZ
XZ = XY + YR
f) PR ≅ XZ
e PR = XZ

13. Tears of Joy - Govi