# Now we are going to look at Geometry Proofs

advertisement ```Geometry
2.5/2.6 Introduction to Proofs
Name: _________________________
Last class, we worked on writing logical statements and making arguments on whether or not
those statements were true or false. Today we are going to use properties from algebra, to prove
different statements. Let’s start with a little review…
Properties of Equality from Algebra
Let a, b and c be real numbers.
Addition Property: If 𝑎 = 𝑏, then _____________________________.
Subtraction Property: If 𝑎 = 𝑏 , then __________________________.
Multiplication Property: If 𝑎 = 𝑏 then _______________________.
Division Property: If 𝑎 = 𝑏 and c  0 then ______________________.
Distributive Property: 𝑎(𝑏 + 𝑐) = ______________________
*Combining Like Terms is part of the Distributive Property.*
Substitution Property: If a = b then either a or b may be substituted for the other in any
equation (or inequality).
Properties of Equality from Geometry
Properties of Congruence from Geometry
Reflexive Property: 𝑎 = ____________
Reflexive Property: 𝑎  ______________
Symmetric Property: If 𝑎 = 𝑏, then
___________.
Symmetric Property: If 𝑎  𝑏, then
____________.
Transitive Property:
Transitive Property:
If 𝑎 = 𝑏 and 𝑏 = 𝑐 then ________.
If 𝑎  𝑏 and 𝑏  𝑐 then __________.
Let’s try some examples of Algebra Proofs.
Example 1:
Statement
8𝑥 – 7 = 12𝑥 + 9
− 7 = 4𝑥 + 9
−16 = 4𝑥
−4 = 𝑥
𝑥 = −4
Reason
Given
Example 2
Statement
Reason
3( 2𝑥 – 4) + 4𝑥 = 18
6𝑥 – 12 + 4𝑥 = 18
10𝑥 – 12 = 18
10𝑥 = 30
𝑥 = 3
Example 3:
Statement
4(5𝑥 − 9) = −2(𝑥 + 7)
Reason
Given
Now we are going to look at Geometry Proofs:
Proof – a logical argument that shows a statement is TRUE.
Two – Column Proof : numbered _____________________ and corresponding __________________
that show an argument in a logical order.
Valid Reasons for a Proof:
When we write proofs, we always write the _____________________ information first.
The last statement in a proof should always be ___________________________________.




Postulates are rules that are accepted without proof.
Rules that are proved are called theorems.
Postulates and theorems are often written in conditional form (if- then).
Unlike the converse of a definition, the converse of a postulate or theorem cannot be
assumed to be true!
Let’s see if we can try some examples.
Directions: Draw a conclusion based on the diagram or given information.
1. Given: 𝑀 𝑖𝑠 𝑡ℎ𝑒 𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 ̅̅̅̅
𝑋𝑌
𝑋
Statement: ________________________________
𝑀
𝑌
Reason: ________________________________
2. Given: ⃗⃗⃗⃗⃗⃗
𝐵𝐷 𝑏𝑖𝑠𝑒𝑐𝑡𝑠 ∡𝐴𝐵𝐶
𝐵
𝐷
𝐶
Statement: ________________________________
Reason: ________________________________
3.
1
2
Statement: ________________________________
Reason: ________________________________
4. Given: ℓ ⊥ 𝓂.
ℓ
1
𝓂
Statement: ________________________________
Reason: ________________________________
5. Given: ∡1 𝑎𝑛𝑑 ∡2 𝑎𝑟𝑒 𝑠𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦.
Statement: ________________________________
Reason: ________________________________
6. Make a conclusion about ∡𝐴𝐵𝐷 𝑎𝑛𝑑 ∡𝐷𝐵𝐶.
𝐴
𝐷
𝐵
𝐶
Statement: ________________________________
Reason: ________________________________
̅̅̅̅.
7. Give a statement about 𝐴𝐷
𝐴
𝐵
𝐷
𝐶
Statement: ________________________________
Reason: ________________________________
Complete the 2-column proof for each of the following:
1.
Given: 𝐵 𝑖𝑠 𝑡ℎ𝑒 𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 ̅̅̅̅
𝐴𝐶 .
Prove: 𝐴𝐵 + 𝐶𝐷 = 𝐵𝐷
A
C
B
STATEMENTS
1. 𝐵 𝑖𝑠 𝑡ℎ𝑒 𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 ̅̅̅̅
𝐴𝐶 .
1.
̅̅̅̅
2. ̅̅̅̅
𝐴𝐵 ≅ 𝐵𝐶
2.
3.
3. Definition of Congruence
4. 𝐵𝐶 + 𝐶𝐷 = ____________
4. Segment Addition Postulate
5.
5.
2.
𝐴𝐵 + 𝐶𝐷 = 𝐵𝐷
D
REASONS
Given: ∠1 𝑎𝑛𝑑 ∠2 𝑎𝑟𝑒 𝑠𝑡𝑟𝑎𝑖𝑔ℎ𝑡 𝑎𝑛𝑔𝑙𝑒𝑠
Prove: ∠ 1 ≅ ∠2
1
2
STATEMENTS
1. ∠1 𝑎𝑛𝑑 ∠2 𝑎𝑟𝑒 𝑠𝑡𝑟𝑎𝑖𝑔ℎ𝑡 𝑎𝑛𝑔𝑙𝑒𝑠
REASONS
1.
2. 𝑚∠1 = 180&deg; ; 𝑚∠2 = 180&deg;
2.
3.
3. Transitive Property of Equality
4. ∠ 1 ≅ ∠2
4.
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