1. Learner Objective: Students will write simple two column proofs.
Advanced Geometry
Section 1.4 Beginning Proofs
Learner Objective: Students will write simple two column proofs.

2. 120 - 3x Is STR a right angle? Justify your response. 60 + 3x
Learner Objective: Students will write simple two-column proofs. S
Is STR a right angle?
Justify your response. x
60 + 3x E T R

3. Valid reasons include: Definitions that we have already learned.
Learner Objective: Students will write simple two-column proofs.
The type of proof we will use most often in this course is the “Two-Column” proof.
In a two column proof, the left hand column contains a list of steps in an argument that 
begins with known or given information and ends with the conclusion we wish to prove. 
For each step (statement) in the left hand column, we must supply a reason or 
justification in the right hand column. These reasons cannot just be things we make up.
Valid reasons include:
 Definitions that we have already learned.
 Previously stated and proven Theorems.
 Previously stated Postulates.
 Mathematical operations.
 Givens
 Valid Assumptions

4. If an angle is an Acute Angle,
Learner Objective: Students will write simple two-column proofs.
At this point in the course, we have covered the following definitions:  
 If an angle is an Acute Angle,  
 then its measure is greater than 0° and less than 90°.
 or
 If an angle's measure is greater than 0° and less than 90°  
 then it is an Acute Angle.
 If an angle is a Right Angle,
 then its measure is 90°.
 If an angle measures 90°,
 then it is a Right Angle.
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5. If an angle is an Obtuse Angle,
Learner Objective: Students will write simple two-column proofs.
If an angle is an Obtuse Angle,
then its measure is greater than 90° and less than 180°.
 or
If an angle's measure is greater than 90° and less than 180°, then it is an Obtuse Angle.
If an angle is a Straight Angle,
then its measure is 180°.
If an angle measures 180°,
then it is a Straight Angle.
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6. If two angles are Congruent Angles, then they have the same measure.
Learner Objective: Students will write simple two-column proofs.
If two angles are Congruent Angles,
then they have the same measure.
 or
If two angles have the same measure,
then they are Congruent Angles.
If two segments are Congruent Segments,
If two segments have the same measure,
then they are Congruent Segments.
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7. All of the above definitions can be used as reasons in a proof.
Learner Objective: Students will write simple two-column proofs.
All of the above definitions can be used as reasons in a proof.
In addition, the following can also be used:
Given
Assumed:
 Straight Lines and Straight Angles
 Collinearity of points
 Betweenness of points
 Relative positions of points
Addition
Subtraction
Multiplication
Division
Substitution

8. A theorem is a mathematical statement that can be proved.
Learner Objective: Students will write simple two-column proofs.
A theorem is a mathematical statement that can be proved.
A postulate is a mathematical statement that cannot be proved.
When a theorem is presented in this course, we will first prove the theorem. 
Once the theorem has been proved, we can then use this theorem to help prove 
homework problems.
These theorems will also be used later to prove more advanced theorems.

9. Given: is a right Angle is a right Angle Prove: OUR FIRST THEOREMS!
Learner Objective: Students will write simple two-column proofs.
OUR FIRST THEOREMS!
Theorem: If two angles are both right angles, then they are congruent.
(First we will draw a diagram that represents the condition “If two angles are right angles”. We will state a “Given” based 
upon this condition and will then state a “Prove” based upon the conclusion “then they are congruent”.)
Given: is a right Angle
is a right Angle
Prove: A B

10. Theorem: If two angles are both right angles, then they are congruent.
Learner Objective: Students will write simple two-column proofs.
Theorem: If two angles are both right angles, then they are congruent.
Given: is a right Angle
is a right Angle
Prove: A B
PROOF:

11. Theorem: If two angles are both right angles, then they are congruent.
Learner Objective: Students will write simple two-column proofs.
Theorem: If two angles are both right angles, then they are congruent.
Given: is a right Angle
is a right Angle
Prove: A B
PROOF:
starts with “given” 
information or with a valid 
assumption based upon the 
diagram

12. Theorem: If two angles are both right angles, then they are congruent.
Learner Objective: Students will write simple two-column proofs.
Theorem: If two angles are both right angles, then they are congruent.
Given: is a right Angle
is a right Angle
Prove: A B
PROOF:
Before using a “reason”, 
the condition of the 
statement must have 
already been shown to be 
true

13. Theorem: If two angles are both right angles, then they are congruent.
Learner Objective: Students will write simple two-column proofs.
ends with the “Prove” statement
Theorem: If two angles are both right angles, then they are congruent.
Given: is a right Angle
is a right Angle
Prove: A B
PROOF:
The conclusion of each 
"reason" must match the 
statement made on that line.

14. Theorem: If two angles are both right angles, then they are congruent.
Learner Objective: Students will write simple two-column proofs.
ends with the “Prove” statement
Theorem: If two angles are both right angles, then they are congruent.
Given: is a right Angle
is a right Angle
Prove: A B
PROOF:
ends with the “Prove” 
statement

15. Learner Objective: Students will write simple two-column proofs.
Theorem: If two angles are both straight angles, then they are congruent.

16. Learner Objective: Students will write simple two-column proofs.
Now that these theorems have been proved, they can now be used in future proofs.

17. Learner Objective: Students will write simple two-column proofs.

18. Learner Objective: Students will write simple two-column proofs.

19. Learner Objective: Students will write simple two-column proofs.

20. Learner Objective: Students will write simple two-column proofs.
Homework
1.4: 2-4, 7-15

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39. Learner Objective: Students will write simple two-column proofs.