WARM-UP
Rewrite each of the following statements in “If-then” form as the conditional, and
converse. then write a biconditional and determine if it is . is true or false.
1. 𝑥 2 = 64; 𝑥 = −8
2. Vietnamese New Years is on January 3.
3. AB+BC=AC, B is between AC
CAHSEE prep
GEOMETRY GAME PLAN
Date
9/30/13 Monday
Section / Topic
Notes: 2.5 Proving Statements about segments
Lesson Goal
Students will be able to write proofs with reasons
about congruent segments.
Geometry California Standard 2.0
Students write geometric proofs.
Homework
P. 104-107 (#1-11, 16, 31-34)
Announcements
Math tutoring is available every Mon-Thurs in
Room 307, 3-4PM! Test next Tuesday.
PROVING STATEMENTS
ABOUT SEGMENTS
What is the difference between congruence and
equality?
Building a Proof



When writing a proof, you can only use facts that
have previously been proved (theorems), facts that
are assumed true without proof (postulates), and
definitions.
Proofs can be written in paragraph form or in a
two-column form.
We will use two-column form most often.
Two-Column Proofs: Key Elements
①
②
③
④
⑤
Given: state the “given” facts
Diagram: a figure that shows what is given
Prove: a statement of what you have to prove
Statements: (Left Column) numbered logical
statements that lead to your conclusion
Reasons: (Right Column) numbered reasons that
justify your statements (definitions, postulates,
properties of algebra/congruence, previously
proven theorems)
Properties of Equality (review)


These two properties are often interchangeable:
Substitution Prop. of Equality: If a=b, then we can
substitute (plug in) a for b, or b for a.


If x+a=c AND a=b, then x+b=c (Plug it in, plug it in!)
Transitive Prop. of Equality: If a=b & b=c, then a=c.

If Mr. Madden is the same height as Simon, and Simon is the
same height a Bradley, then Mr. Madden and Bradley are
the same height. =)
Properties of Equality (review cont.)


Mirror, mirror, on the wall…
Reflexive Prop. of Equality: a=a
 Anything
is equal to itself (Think about
your reflection in the mirror!)

Symmetric Property of Equality: If a=b, then b=a.
 Think
about something symmetrical… if you flip it, it still
looks the same. You can always flip an equation, Left
to Right. (Flip it good!)
Properties of Congruence

These 3 properties work for congruence also:

Reflexive: For any segment AB, AB ≅ AB.

Symmetric: If AB ≅ CD, then CD ≅ AB.

Transitive: If AB ≅ CD & CD ≅ EF, then AB ≅ EF.
Given:
AB = BC, C is the
midpoint of BD
Prove: AB = CD
Statement
Reason
1.
1.
AB = BC,
C is the midpoint of BD
2.
2.
BC = CD
3.
AB = CD
3.
Given
Def. Midpoint
Substitution
Given:
AB=CD
Prove: AC=BD
Statement
Reason
1. AB=CD
1.
2.
2.
AB+BC=AC
3.
BC+CD=BD
4.
5.
BC+AB=BD
AC=BD
Given
Segment Add. Post.
3. Segment Add. Post.
4.
5.
Substitution
Substitution
Given:
AC=BD
Prove: AB=CD
Statement
Reason
1. AC=BD
1.
2.
2.
AB+BC=AC
3.
BC+CD=BD
4.
5.
AB+BC = BC+CD
AB=CD
Given
Segment Add. Post.
3. Segment Add. Post.
4.
5.
Substitution
Subtr. Prop of =
Given: QR = RS
Prove: QS = 2 RS
Statement
Reason
1.
1.
2.
2.
3.
3.
4.
4.
Given: LE = RM, EG = AR
Prove: LG = MA
Statement
1. LE = RM and EG = AR
2. AR+RM=AM
Reason
1.
Given
2.
Segment
addition
3. LE+EG=LG
3. Segment addition
4. RM+AR=LG
4.Substitution
5. LG = MA
5. Substitution
Example 5: Using Segment
Relationships
• In the diagram, Q is the midpoint of PR.
Show that PQ and QR are equal to ½ PR.
• GIVEN: Q is the midpoint of PR.
• PROVE: PQ = ½ PR and QR = ½ PR.
R
Q
P
Statements:
1.
2.
3.
4.
5.
6.
7.
Q is the midpoint of PR.
PQ = QR
PQ + QR = PR
PQ + PQ = PR
2 ∙ PQ = PR
PQ = ½ PR
QR = ½ PR
Reasons:
1.
2.
Given
Definition of a midpoint
3.
Segment Addition Postulate
4.
5.
6.
7.
Substitution Property
Distributive property
Division property
Substitution
(over Lesson 2-2)
Write using two column proofs!
Slide 1 of 1
(over Lesson 2-2)
Slide 1 of 1
Example 2: Using Congruence
• Use the diagram and the given information
to complete the missing steps and reasons in
the proof.
• GIVEN: LK = 5, JK = 5, JK ≅ JL
K
• PROVE: LK ≅ JL
J
L
Statements:
Reasons:
1.
2.
3.
4.
5.
6.
1.
2.
3.
4.
5.
6.
________________
________________
LK = JK
LK ≅ JK
JK ≅ JL
________________
Given
Given
Substitution
_________________
Given
Substitution
Statements:
Reasons:
1.
2.
3.
4.
5.
6.
1.
2.
3.
4.
5.
6.
LK = 5
JK = 5
LK = JK
LK ≅ JK
JK ≅ JL
LK ≅ JL
Given
Given
Substitution
Def. Congruent seg.
Given
Substitution
Def. of congruent segments
AB = BC
DE = EF
Substitution (or Transitive)
Substitution (or Transitive)
Def. of congruent segments