Geometry Midterm Review RBC

Irrational Number

Any real number that cannot be written as a fraction

Rational Number

A number that can be expressed as a fraction of two integers, where the denominator is not zero

Real number

Any number that we can think of, except complex numbers

Collinear points

The points that lie on the same straight line or in a single line

Line

A straight set of points that extend in opposite directions

Plane

The geometrical concept of a flat surface with no edges or thickness

Point

An exact location on a plane

Postulate

A statement accepted to be true without proof

Segment

A part of a line or curve between two points

Ray

A part of the line having one fixed point and the other point does not have an end

Opposite Ray

Two rays that have the same endpoint and extend in opposite directions

Angle

a figure which is formed by two rays or lines that shares a common endpoint

Supplementary angle

Those angles that sum up to 180 degrees

Complementary angle

Either of two angles whose sum is 90

Parallel

Two or more lines that are always the same distance apart and never intersect, even if they are extended infinitely in both directions

Perpendicular

A straight line that makes the right angle (90 degrees) with the other line

Angle Bisector

A line that splits an angle into two equal angles

Construction

Drawing shapes, angles or lines exactly with compasses and rulers

Perpendicular Bisector

A line that divides a given line segment exactly into two halves forming 90 degrees angle at the intersection point

Midpoint

A point that divides the line segment into two equal halves

Distance

The numerical measurement of the length of a line segment, a line with a distinct starting and stopping point

Conjecture

A mathematical statement which appears to be true, but has not been formally proven

Counterexample

An example that disproves a statement, proposition, or theorem by satisfying the conditions but contradicting the conclusion

Inductive Reasoning

A specific truth which is known to be true, and then applying this truth to more general concepts

Biconditional

True statements that combine the hypothesis and the conclusion with the key words 'if and only if

Conditional

A statement that can be written in the form “If P then Q,” where P and Q are sentences

Contrapositive

"if not-B then not-A " is the contrapositive of "if A then B "

Converse

q → p

Hypothesis

A proposition that is consistent with known data, but has been neither verified nor shown to be false

Inverse

The operation that undoes what was done by the previous operation

Truth table

Provides a method for mapping out the possible truth values in an expression and to determine their outcomes

Truth value

A value indicating the relation of a proposition to truth

Deductive Reasoning

The process of reasoning to reach a logical conclusion from one or more statements

Law of Detachment

If P leads to Q and P is true, then Q must be true

Law of Syllogism

A valid argument form of deductive reasoning that follows a set pattern. It is similar to the transitive property of equality, which reads: if a = b and b = c then, a = c

Linear Pair

Two adjacent angles that add up to 180°

Paragraph Proof

A series of logical statements proving a mathematical concept true written in paragraph form

Proof

The logical way in which mathematicians demonstrate that a statement is true

Theorem

A statement that can be proved to be true based on known and proven facts

Two-column proof

A method used to present a logical argument using a table with two columns

Postulate 1-2: Segment Addition Postulate

If points A, B, and C are on the same line with B between A and C, then AB + BC = AC

Postulate 1-3: Protractor Postulate

Given ray BA and a point C not on ray BA, a unique real number from 0 to 180 can be assigned to ray BC

Postulate 1-4: Angle Addition Postulate

If point D is in the interior of angle ABC, then the measure of angle ABD + the measure of angle DBC = the measure of angle ABC

Reflexive Property of Congruence

AB=AB

Symmetric Property of Congruence

AB=CD then CD=AB

Transitive Property of Congruence

AB=CD and CD=EF then AB=EF

1.1 Vertical Angles Theorem

Vertical angles are congruent

1.2 Congruent Supplements Theorem

If two angles are complementary to congruent angles (or to the same angle) then they are congruent

1.3 Congruent Complements Theorem

If two angles are complementary to congruent angles (or to the same angle) then they are congruent

Theorem 1.4 Right Angles

All right angles are congruent

Theorem 1.5

If two angles are congruent and supplementary, then each is a right angle

1.6 Linear Pairs Theorem

The sum of the measures of a linear pair is 180

Adjacent angles

Two angles that are side by side and share a common vertex and a common side

Vertical Angles

Located across from one another in the corners of the "X" formed by two straight lines

Transversal

A line that passes through two lines in the same plane at two distinct points

Same-Side Interior Angles Postulate

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

Alternate Interior Angles Theorem

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

Corresponding Angles Theorem

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

Alternate Exterior Angles Theorem

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

Converse of the Corresponding Angles Theorem

If two lines and a transversal form corresponding angles that are congruent, then the lines are parallel

Converse of the Alternate Interior Angles Theorem

If two lines and a transversal form alternate interior angles that are congruent, then the lines are parallel

Converse of the Same-Side Interior Angles Postulate

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

Converse of the Alternate Exterior Angles Theorem

If two lines and a transversal form alternate exterior angles that are congruent, then the lines are parallel

Theorem 2-8

If two lines are parallel to the same line, then they are parallel to each other

Theorem 2-9

If two lines in the same plane are perpendicular to the same line, then they are parallel to each other

Theorem 2-10

Through a point not on a line, there is one and only one line parallel to the given line

Triangle Angle-Sum Theorem

All angles in a triangle add up to be 180

Triangle Exterior Angle Theorem

The measure of each exterior angle of a triangle equals the sum of each of the measures of its two remote interior angles

Theorem 2-13

Two non-vertical lines are parallel if and only if their slopes are equal. Any two vertical lines are parallel.

Theorem 2-14

Two non-vertical lines are perpendicular if and only if the product of their slopes are -1. A vertical line and a horizontal line are perpendicular to each other.

Rigid motion

Transformation that preserves length and angle measure

Reflection

An object is flipped to create a mirror or congruent image

Image

The new position of a point, a line, a line segment, or a figure after a transformation

Line of reflection

An image will reflect through a line

Preimage

The image of a graph or shape before it is taken through a transformation

Transformation

Changing a shape's position or which way the shape points

Angle of Rotation

The measure of the amount that a figure is rotated about a fixed point

Center of rotation

The point at which a picture turns

Rotation

When an object is turned clockwise or counterclockwise around a given point

Glide Reflection

Translating a figure by sliding it along a line, then reflecting the figure over an axis of reflection

Point symmetry

When, given a central point on a shape or object, every point on the opposite sides is the same distance from the central point

Reflectional symmetry

When a line is drawn to divide a shape in halves so that each half is a reflection of the other

Rotational symmetry

A shape can be spun around a single point and look the same as it did before it was spun

Theorem 3-1

A tranlation is a composition of reflections across two parallel lines

Theorem 3-2

Any rotation is a composition of reflections across two lines that intersect at the center of rotation. The angle of rotation is twice the angle formed by lines of reflection

Theorem 3-3

The composition of two or more rigid motions is a rigid motion.

Theorem 3-4

Any rigid motion is either a translation, reflection, rotation, or glide reflection

Corollary to Theorem 3-4

Any rigid motion can be expressed as a composition of reflections

Congruent angles

Two or more angles that have the same measure

Congruent segments

Line segments that are equal in length

Congruence transformation

A moved figure that retains the same size, shape, angles, and side lengths of the original image

Congruent

To have the same shape and size

Equilateral triangle

A triangle that has all its sides equal in length

Isosceles triangle

A triangle with two equal sides

Corresponding angles

Pairs of angles formed on the same side of the transversal and in the same relative position

Hypotenuse

The longest side of a right-angled triangle compared to the length of the base and perpendicular

Pythagorean Theorem

The square of the length of the hypotenuse of a right triangle equals the sum of the squares of the lengths of the other two sides

Isosceles Triangle Theorem and the Converse

If two sides are congruent, then the angles opposite those sides are congruent.

If two angles of a triangle are congruent, then the sides opposite those angles are cogruent

Theorem 4-2

If a line or line segment bisects the vertex angle of an isosceles triangle, then it is also the perpendicular bisector of the opposite side.

Side-Angle-Side (SAS) Congruence Criterion

If two sides and the included angle of one triangle are congruent to two sides and the angle of another triangle, then the two triangles are congruent