## Geometric Proofs On Lines and Angles

#### Aligned To Common Core Standard:

**High School Congruence** - HSG-CO.C.9

What are Geometric Proofs?
We all have heard about proof. Generally speaking, proof is something that you need to establish a fact or determine something as true.
But how do we prove something in geometry? How do we prove that the two angles are congruent or not? This is where geometric proofs play their role. So, let’s begin with defining geometric proofs and discussing their types later on.
**GEOMETRIC PROOFS** -
A geometric proof is an approach of determining whether the statement is false or true by making use of logic, reasoning, facts and deductions to conclude an argument.
A well-developed proof has its every statement supported by:
**Theorems:** statements that can be proven true by using reasoning or support of previously established facts
**Postulates:** statements that are assumed as truths without the need of proofs
**Axioms:** statements that are considered as established and self-evident truths
**TYPES OF GEOMETRIC PROOFS** -
Geometric proofs are classified into:
**Direct Proof** -
In direct proofs, conclusions are derived from facts by using theorems and axioms and without making further assumptions.
**Indirect Proof** -
In indirect proofs, the statement to be proven is assumed as false. If the assumption results in an impossibility, then the supposed statement has to be proven true.
**Paragraph Proof** -
Paragraph proofs are logical arguments written in the form of a paragraph, supporting every step with evidences and details to provide a definite conclusion.
**Two-column Proof** -
Two-column proof comprises two columns with statements listed in one column while the reasons and logics for each statement stated in the second column. Their content is similar to paragraph proof but their form is different.

### Printable Worksheets And Lessons

- Angle Proof Step-by-step
Lesson - It's a great idea to review the meaning of supplemental,
complementary, and opposite angles before looking at this section.

- Guided Lesson -
The first one is a tricky one. It actually gets easier as it goes
along.

- Guided Lesson Explanation
- This is setup up as an abbreviated explanation. We expect you
to understand your basic definitions of angles.

- Practice Worksheet -
After you complete this, you should feel very accomplished with
this skill and begin to understand the power of knowing, just one
angle.

- Matching Worksheet
- Match the angles to what they are asking you for.

- Coordinate Geometry Proofs Worksheet
Five Pack - With just a dab of information, you need to prove
midpoints, angles, and geometric shapes exist.

- Direct Euclidean Proofs Worksheet
Five Pack - We are looking for abbreviated proofs here.

- Proofs in the Coordinate Geometry
Worksheet Five Pack - The coordinate plane does make this a
bit easier than the other sets.

#### Homework Sheets

The first one focuses on angles, the second on lines and angles. The third one puts it all together.

- Homework 1 - We can see that
- Homework 2 - Vertical angles are equal is the lead here.
- Homework 3 - Knowing that two lines are parallel, you can learn a lot.

#### Practice Worksheets

We start to really concentrate on numerical angles here for the first time.

- Practice 1 - Given line ABD, m
- Practice 2 - Find the value of x in each case.
- Practice 3 - Find the missing angles.

#### Math Skill Quizzes

This is applied geometric at it's best! These problems have endless real world connections.

### Simple Strategies for Solving Geometric Proofs

When you first start working with these types of problems you can easily get overwhelmed. The best thing to start by creating a plan for yourself. We would encourage you to start by talking it out or writing a short outline of how you should proceed with the problem. Once you have a brief outline, go over the plan to make sure that you did not leave anything out. The dots should connect from one to another. It is then helpful to plug numbers into those values to make sure that you are on the right path. Things that can help you along the way is to spot reference angles and triangles. Look congruent triangles because they can help you prove two sides are the same through a number of different theorems. Finding parallel lines is often golden. This will allow you to prove matching angles and spot balancing angles. You can find most of this information by referring to the information that is given to you. Once you have all this in place, write your proof. Once complete, reverse engineer your proof to make sure that it works. Everything should flow equally in both directions as you progress through your proof. Following these simple steps often will be the key to your success.