Geometry Journal 2

Nicolle Busto 9-1

9-1

Conditional Statement• It is a statement that establishes a necessary

condition for a thing to happen.

Examples: 1. If it rains then the grass will be wet. 2. Converse: If the grass is wet then it rained.3. Inverse: If it doesn’t rain then the grass will not be wet. 4. Contrapositive: If the grass is not wet then it didn’t rain.

Counter example• It’s an example that

proves that a statement is false.

Example:1. If the grass is wet then it rained. -The grass could be wet, if you watered the grass.

2. If a number is divisible by 5 then its unit digit is 5.-20

3. If a number is odd then the number is prime.-9

Definition A description of a mathematical concept that can be written as a bi-conditional statement.

A line is perpendicular to a plane iff their intersection is exactly a 90° angle.

Perpendicular lines: Two lines are perpendicular iff they intersect at 90°.

Bi-Conditional statement

It is a statement where both conditions are needed.

They are used to write definitions

They are important because they can be used to establish if a definition is true or false.

Example 1: A rectangle is a square iff their four sides are equal.

Example 2: An angle is acute iff its measure is less than 90°.

Example 3: A triangle is equilateral iff its three sides are congruent.

Deductive reasoning

Is the process of using logic to draw conclusions.

Symbolic notation: Uses letters to represent statements and symbols to represent the connections between them. It works by reaching conclusions from a starting statement.

Examples: p = today is a school day. Q = I am going to the school today. - P Q2. P = today the ice

cream store is open. Q = I am going to the store today.

- P Q3. P = If two numbers are odd. Q = their sum is even.

Laws of Logic

1. Law of Detachment: If P Q is a true statement and P is true then Q is true.

·Example 1: If an animal is a dog it has a tail. - The pet of Busto family is a dog the pet of Busto family has a tail. ·Example 2: If a number is even it is divisible by two.- 120 is an even number 120 is divisible by two.·Example 3: It rains today the streets will be wet.- Tomorrow will rain the streets will be wet tomorrow.

2. Law of Syllogism: If P Q is true and Q R is true, then P R is true.

·Example 1: If an animal is a fish it lives in water, if an animal lives in water it is wet all the time. - A shark is a fish a shark is wet all the time.·Example 2: If an animal is a mammal it has warm blood, if an animal has warm blood it can control it’s inner temperature.- A dog is a mammal it can control it’s inner temperature. ·Example 3: If a number is divisible by 12 it is divisible by 6, if a number is divisible by 6 it is even.- 36 is divisible by 12 36 is even.

Algebraic Proof

• You use the properties and work with the initial statement until you reach the desired conclusion.

Statements Reasons

1. 3x + 7 = 40 - 7 - 7

Given

3x / 11 = 33 / 11 Subtraction property of equality

X = 3 Division property of equality

2. 5x – 8 = 42 + 8 +8

Given

5x = 50 Addition property of equality

5x / 5 = 50 / 5X = 10 Division property of equality

3. -3x + 4 = 34 -4 -4

Given

-3x / -3 = -30 / -3 Subtraction property of equality

X = - 10 Division property of equality

Properties of equalityAddition property If a = b, then a + c = b + c

Subtraction property If a = b, then a – c = b – c

Multiplication property If a = b, then ac = bc

Division property If a – b then c /= o, then a/c = b/c

Reflexive property a = a

Symmetric property If a = b, then b =a

Transitive property If a = b and b = c, then a = c

Substitution property If a = b, then b can be substituted for a in any expression.

Properties of congruenceReflexive property --- ---

EF ~= EFSymmetric property If <1 ~= <2, then <1 ~= <2

Transitive property If figure A ~= figure B and figure B ~= figure C, then figure A ~= figure C

Segments and Angles properties

You can work with the measures of angles or measures of sides using the laws of algebra.

Property Segments Angles

Reflexive PQ = PQ M<1 = M<1

Symmetric If AB = CD, then CD = AB If m<A = m<B, then m<B = m<A

Transitive If GH = JK and JK = LM, then GH = LM

If m<1 = m<2 and m<2 = m<3, then m<1 = m<3

Property Segments Angles

Reflexive PQ ~= PQ M<1 ~= M<1

Symmetric If AB ~= CD, then CD ~= AB

If m<A ~= m<B, then m<B ~= m<A

Transitive If GH ~= JK and JK ~= LM, then GH ~= LM

If m<1 ~= m<2 and m<2 ~= m<3, then m<1 ~~= m<3

P Q R S T U

1. PQ = PQ2. PQ = RS, then RS = PQ3. If PQ = RS and RS = TU, then PQ =

RS

Proof process1. Write the conjecture to be proven.

2. Draw a diagram to represent the hypothesis of the conjecture.

3. State the given information and mark it on the diagram.

4. State the conclusion of the conjecture in terms of the diagram.

5. Plan your argument and prove the conjecture.

Two proof- column

• You list your steps on the proof in the left column. You write the matching reason for each step in the right column.

• They give you the given, and what you need to proof. Sometimes they give you the plan.

Statement Reason

<1 and <2 are right angles Given

M<1 = 90, m<2 = 90 Def or rt. Angles

M<1 = M<2 Trans prop. Of =

<1 ~= <2 Def. of ~= angles

Given: <1 and <2 are right anglesProve: <1 ~= <2Plan: Use the definition of a right angle to write the measure of each angle. Then use the Transitive Property and the definition of congruent angles.

Statement Reason

AB ~= XY, BC ~= YZ Given

AB = XY, BC = YZ Def. coongruent segments

AB + BC = XY + YZ Addition porp of =

AB + BC = ACXY + YZ = XZ

Segment Addition Postulate

AC = XZ Subs. Prop of =

AC ~= XZ Def. congruent segments

Given: AB ~= XY BC ~= YZProve: AC ~= XZ

Statement Reason

PQ ~= RS Given

PQ = RS Def. of congruent segments

RS = PQ Tsymmetric prop. =

RS ~= PQ Def. of congruent segments

Given: PQ ~= RSProve: RS ~= PQ

Linear pair postulate

• Linear pair: a pair of adjacent angles whose no common sides are opposite rays.

• LPP: If two angles form a linear pair, then they are supplementary angles.

EXAMPLES:1. 3.

2.

TheoremsTheorem Hypothesis Conclusion

Congruent Supplements: If two angles are supplementary to the same angle (or two congruent angles), then the two angles are congruent.

<1 and <2 are supplementary. <2 and <3 are supplementary.

<1 ~= <3

Linear pair: if two angles form a linear pair than they are supplementary.

<a and <b form a linear pair.

<a and < b are supplementary.

Right angle congruence: all right angles are congruent,

<a and <b are right angles.

<a ~= <b

Congruent complements: if two angles are complementary to the same (or to two congruent angles), then the two angles are congruent.

<1 and <2 are complementary. <2 and <3 are complementary.

<1 ~= <2

Vertical angles theorem • Vertical angles are

congruent.1.

2.

3.

Common Segment Theorem

• Give collinear points A, B, C, and D arranged as shown, if AB ˜= CD, then AC ˜= BD.

• AB ˜= BD.• AC ˜= BD.

EXAMPLES: 1.