Download 2-5 - MrsBudde

2-4 Deductive Reasoning
Law of Detachment
• Deductive reasoning (sometimes called logical
reasoning) is the process of reasoning logically from
given statements or facts to a conclusion.
Law of Detachment: If the hypothesis of a true
conditional is true, then the conclusion is true.
If p  q is true and p is true, then q is true.
 What can you conclude from the
given true statements?
• If a student gets an A on the Final, then the student
will pass the course. Felicia got an A on her History
Final.
• If a ray divides an angle into two congruent angles,
then the ray is an angle bisector. RS divides ARB so
that ARS  SRB.
• If two angles are adjacent, then they share a
common vertex. 1 and 2 share a common vertex.
Law of Syllogism
• Another law of deductive reasoning is the Law of Syllogism.
• The Law of Syllogism allows you to state a conclusion from
two true conditional statement when the conclusion of one
statement is the hypothesis of the other statement.
If p  q is true and q  r is true, then p  r is true.
Example: If it is July, then you are on summer vacation. If you
are on summer vacation, then you work at Smoothie King.
Conclusion: If it is July, then you work at Smoothie King.
What can you conclude from the
given information?
• If a figure is a square, then the figure is a rectangle.
If a figure is a rectangle, then the figure has four
sides.
• If you do gymnastics, then you are flexible. If you do
ballet, then you are flexible.
• If a whole number ends in 0, then it is divisible by 10.
If a whole number is divisible by 10, then it is
divisible by 5.
2-5 Reasoning in Algebra and
Geometry
Proofs
• A proof is a convincing argument that uses
deductive reasoning.
• A proof logically shows why a conjecture is
true.
• A two-column proof lists each statement on
the left and the justification (or reason) for
each statement on the right.
• Each statement MUST follow logically from the
steps before it!
Writing a Two-Column Proof
Given: m1 = m3
Prove: mAEC = mDEB
Writing a Two-Column Proof
Given: AB  CD
Prove: AC  BD
AB  CD
AB + BC = CD + BC
AB + BC = AC
CD + BC = BD
AC = BD
AC  BD
2-6 Proving Angles Congruent
Theorems
• A theorem is a conjecture or statement that you prove
true.
• Vertical Angles Theorem: Vertical angles are congruent.
• Congruent Supplements Theorem: If two angles are
supplements of the same angle (or congruent angles),
then the two angles are congruent.
• Congruent Complements Theorem: If two angles are
complements of the same angle (or congruent angles),
then the two angles are congruent.
• Theorem 2-4: All right angles are congruent.
• Theorem 2-5: If two angles are congruent and
supplementary, then each is a right angle.
Proving the Vertical Angles Theorem
Given: 1 and 3 are vertical angles
Prove: 1  3
Using the Vertical Angles Theorem
• What is the value of x?
Proof Using the Vertical Angles
Theorem
Given: 1  4
Proof: 2  3