Parallel Lines

Parallel Lines are lines that do not intersect at any point. Five ways to prove that lines are parallel are:

1. Converse of Corresponding Angles Postulate: If two lines are cut by a transversal, meaning that the two corresponding angles are congruent.
2. Converse of Alternate Interior Angles Theorem: If two lines are cut by a transversal, meaning that a pair of alternate interior angles are congruent.
3. Converse of Alternate Exterior Angles Theorem: If two lines are cut by a transversal, meaning that a pair of alternate exterior angles are congruent.
4. Converse of Corresponding Same-side Interior Angles Theorem: If two lines are cut by a transversal, meaning that a pair of same-side interior angles are supplementary.
5. Parallel Postulate

Transformations

There are three major transformations that have to do with shapes. They are: Translation, Reflection, and Rotation.

• A translation is when two of the same shape is drawn, but one of them is moved to another area.
• A reflection is a mirror of one object.
• A rotation is a turn of an object.

Now, there are two other terms that are needed to know when talking about transformations. They are: pre-image and image.
• Pre-image is the shape before the transformation.
• Image is after the transformation.

Lines within a Coordinate Plane

In a coordinate plane, there perpendicular lines. A perpendicular line is when one line makes a 90 degree angle with with another line.

A coordinate plane consists of four quadrants, a x-axis, a y-axis, and the origin (0,0) as shown in the picture below.

When given a pair of coordinate points, it will be set up a (x,y), (x,y). In order to graph these points to create a line, an equation must be used.

• Slope-intercept form: y=mx+b
• Point-slope form: y-y1= m(x-x1)

Either equation will work fine when the slope and a pair of coordinate points are found. First, once two pairs of points are present, the slope can be found with the slope formula.

• Slope formula: m=y2-y1
•                                                                         x2-x1

Once the slope is found, a line can now be drawn when everything is put into one pf the top two equations.

Triangles Congruent?

What does it mean to say that triangles are congruent?

Well to say that a triangles are congruent, means that they have the same side-length and shape. Congruent angles, are angles that have the same measure. Below, is an example of angle and side-length congruence of a triangle. Sides A,B, and C are all congruent because they have the same side-length. Also angles m<1, m<2, and m<3 are all congruent because they have the same angle measurments.

Defining Triangles

A triangle is a three sided shape that is classified by either the angle measures or the side-lengths. Triangles classified under angle measures consist of four different types: Acute, Equiangular, Right, and Obtuse. To know what a triangle is classified as a specific angle measurement triangle, look at each angle. If the triangle has three angles that are less than 90 degrees, then that means it is an acute triangle. An equiangular triangle is when all three angles in the triangle are congruent and acute. If it is a right triangle, that means that there must be one angle that is equal to 90 degrees. Lastly, to tell if a triangle is obtuse, there must be one angle that is larger than 90 degrees.

Next, triangles can be classified as in side-lengths. There are three different types of side-length triangles: Equilateral, Isosceles, and Scalene. As shown in the picture below, each side-length triangle represents a different number of congruent sides. Equilateral Triangles have three congruent sides. An Isosceles Triangle has two congruent sides. Lastly, a Scalene Triangle has no congruent sides. 

Inductive and Deductive Reasoning

Inductive Reasoning is the reasoning that a statement is true due to specific cases and reasons to be true. One way to use inductive reasoning is to create a conclusion from some-sort of reasoning. A term to know when talking about inductive reasoning is a Conjecture. A conjecture is when a statement is belived to possibly be true due to it relying on inductive reasoning.

•       An example of an Inductive Reasoning problem is...

Deductive Reasoning is when logic is used to make conclusions. This is when a proof comes in to describe and show how to prove statements. The are two laws that have to do with deductive reasoning. They are: Law of Detachment, and Law of Syllogism.

Effective Arguments and Proofs

To write a proof, there must be two columns. One column states the Statements, the other column states the Reasons. When writing a proof, the Statement column says the order in which steps it takes to go from the given, to the proven. In the Reason column, this is where theorems, properties, and other steps are written to tell the meaning of the statements in the other column.There are two different types of proofs: Algebraic Proof, and Geometric Proof.

To the side, there is an example of a geometric two-column proof.

Geometry Around the World

Geometry is used in multiple ways to improve how we work everyday. Many jobs include needing a knowledge about algebra and geometry. Some of these jobs include being an architect or an interior design worker. Both jobs need the use of knowledge of geometry.

 This photo shows how a knowledge of geometry needed to be known when deciding how to build this structure, because as shown, there are multiple shapes involved in these buildings.