Geometry- Unit 3

two-column proofs-This type of proof typically has the following components:

1. Given statement that lists all given information 2. Prove statement that states what is being proved 3. diagram, if applicable 4. two columns: one on the left labeled "Statements" and one on the right labeled "Reasons"

Associative Property of Addition

Changing the groupings of the addends does not affect the sum. (a+b)+c=a+(b+c)

Commutative Property of Addition

Changing the order of the addends does not change the sum. a+b = b+a

Supplementary angles add up to

180 degrees

Recall that a straight angle has a measure of

180°.

Complementary angles are two angles whose sum is

90°.

Theorem

A mathematical statement which we can prove to be true

Commutative Property of Multiplication

Changing the order of the factors does not change the product. ab=ba

Alternate Exterior Angles Theorem

If a transversal intersects two parallel lines, then alternate exterior angles are congruent.

Alternate Interior Angles Theorem

If a transversal intersects two parallel lines, then alternate interior angles are congruent.

Corresponding Angles Postulate

If a transversal intersects two parallel lines, then corresponding angles are congruent

Same-Side Interior Angles Theorem

If a transversal intersects two parallel lines, then same-side interior angles are supplementary.

Transitive Property

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

Division Property

If a=b and c does not equal 0, then a/c = b/c

Substitution Property

If a=b, then a can be substituted for b in any equation or expression

Addition Property

If a=b, then a+c=b+c.

Multiplication Property

If a=b, then axc=bxc

Subtraction Property

If a=b, then a−c=b−c.

What is the Converse of the Same-Side Interior Angles Theorem?

If two lines and a transversal form same-side interior angles that are supplementary, then the lines are parallel.

Identity Property of Multiplication

If you multiply a number by one, the product is the same as that number. a x 1 = a

p → q

Looking at the symbols above, the p represents the part of the statement following if and the q represents the part following then. The symbols can be read as "If p then q" with the arrow (→) representing then. The negation (~) is another symbol that is used, and it means the opposite. For example, ~p represents the opposite of p and is read as "not p." The truth value of a conditional is either true or false. To prove it false, a counterexample is necessary. Conditionals that have the same truth value are called equivalent statements.

Skew lines are

NOT coplanar and do not intersect and they are not parallel. They lie in different planes

flow proof

This type of proof also uses deductive reasoning, statements, and reasons but follows the form of a flowchart.

Suppose the lines were painted so that The equation states angle 1 = 55°.and The equation states angle 2 = 125°.. Use the converse statements from Step 1 to determine if lines v and w parallel. Explain your reasoning.

Using the Converse of the Same-Side Interior Angles Theorem, lines v and w are parallel since the sum of the degree measure angle 1 and angle 2 is 180 degrees, making them supplementary

biconditional

When a conditional and its converse are true, they can be combined into a biconditional. A conditional statement takes the form "if x then y." A converse reverses that to become "if y then x." If both of these are true, then it is possible to write a biconditional. In a biconditional, a hypothesis and conclusion can be connected with the words "if and only if."

A transversal is

a line that cuts across two or more lines at distinct spots

What is the name of the special angle pair for angle 1 and angle 2?

angle 1 and angle 2 are same side interior angles.

Reflexive Property

a=a

You can name a line using

any two points that lie on it.

Properties are

characteristics that are always true. You can use mathematical properties to reason and make arguments about geometry.

Angles don't have to be adjacent to be

complementary

Vertical angles are

congruent

Skew lines lie in

different planes and are neither parallel nor intersecting.

whereas skew lines lie on

different planes and do not intersect.

deductive reasoning, or logical reasoning

drawing a conclusion based on facts

Intersecting lines have

exactly one point in common.

Symmetric Property

if a=b, then b=a

A line segment contains an

infinite number of points, but it is named by its two endpoints.

Parallel lines

lie in the same plane and never intersect.

Congruent angles are angles that have the same

measure. If you were to place one angle on top of the other angle it would match exactly. The symbol congruent to is used to show that angles are congruent. When determining if angles are congruent, it is important that your answer is based only on the measures of the angles. Sometimes angles can appear to be congruent, but when you measure them they have different measures.

Lines that do not intersect may be

parallel or skew.

Lines can be

parallel, intersecting, or skew.

A line segment is

part of a line with two endpoints.

Two lines that intersect and form 90 degree angles are

perpendicular lines

Adjacent angles are angles that

share a side and a have a common vertex.

a line is a

straight path made up of a series of points that extends infinitely in opposite directions. You need two points to determine a line.

Angles don't have to be adjacent to be

supplementary

Segments or rays are parallel or skew if

the lines on which they lie are parallel or skew.

Parallel lines lie on

the same plane and do not intersect

Vertical angles are the opposite angles formed when

two lines intersect

two-column proofs

which is a type of formal proof in which the format involves separating statements and reasons into two columns. Each statement is aligned with the reason that supports it.

Lines x and y are parallel. This is written as

x||y