Geometry is the study of shapes and angles and can be challenging for many students. Many of the concepts are totally new and this can lead to anxiety about the subject. There are a lot of postulates/theorems, definitions, and symbols to learn before geometry begins to make sense. Through combining good study habits and a few study pointers, you will have success in learning geometry.
EditSteps
EditGetting the Grade
- Attend every class. Class is a time to learn new things and solidify the information that you may have learned in the previous class. If you don’t go to class, it is much more difficult to stay current with the material.
- Ask questions in class. Your teacher is there to make sure you have a solid grasp on the material. If you have a question, don’t hesitate to ask it. Some of the other students in the class likely have the same question.
- Prepare for class by reading the lesson you are going to cover ahead of time and know the formulas, theorems, and postulates by heart.
- Pay attention to your teacher while you are in class. You can talk to your classmates at break or after school.
- Draw diagrams. Geometry is the math of shapes and angles.[1] To understand geometry, it is easier to visualize the problem and then draw a diagram. If you're asked about some angles, draw them. Relationships like vertical angles are much easier to see in a diagram, if one isn't provided, draw it yourself.
- Form a study group. Study groups are a good way to learn the material and clarify concepts you don’t understand. Having a group that meets on a regular schedule will also force you to stay on top of the material and try your best to comprehend it. Studying with classmates is useful when you come across more difficult topics. You can work through them together to figure them out.
- One of your study mates may understand something that you don’t and help you out with it. You might also be able to help them understand something and learn it better by teaching them.
- Do all of the assigned homework. Homework is assigned because it helps you learn all of the concepts in the material. Doing the homework teaches you what you really understand and what topics you might need to put more time into.
- If you come across a topic in your homework that you are struggling with, focus on that topic until you understand it. Ask you classmates or your teacher to help you out.
- Teach the material. When you have a firm understanding of a topic or concept, you should be able to teach it to someone else. If you can’t explain it to them so that they also understand, you likely don’t get it as well as you thought either. Teaching material to others is also a good way to enhance your own memory or recall of the topic.[2]
- Try teaching your sibling or parent some geometry.
- Take the lead in a study group to explain something you know really well.
- Do lots of practice problems. Geometry is as much a skill as a branch of knowledge. Simply studying the rules of geometry will not be enough to get an A, you need to practice solving problems. This means doing your homework and working extra problems for any trouble areas.
- Make sure to do as many practice problems as you can from other sources. Similar problems may be worded in a different way that might make more sense to you.
- The more problems you solve, the easier it will be to solve them in the future.
- Seek extra help. Sometimes going to class and talking to your teacher just isn’t enough. You might need to find a tutor who has more time to focus specifically on what you are struggling with. Working with someone one-on-one can be very useful in understanding difficult material.
- Ask your teacher if there are tutors available through the school.
- Attend any extra tutoring sessions held by your teacher and ask your questions.
EditLearning the Basics
- Know Euclid’s five postulates of geometry. Geometry is founded upon the basis of five postulates put together by the ancient mathematician, Euclid.[3] Knowing and understanding these five statements will help you understand many of the concepts in geometry.
- 1. A straight line segment can be drawn joining any two points.
- 2. Any straight line segment can be continued in either direction indefinitely in a straight line.
- 3. A circle can be drawn around any line segment with one end of the line segment serving as the center point and the length of the line segment serving as the radius of the circle.
- 4. All right angles are congruent (equal).
- 5. Given a single line and a single point, only one line can be drawn directly through the point that will be parallel to the first line.
- Understand the Pythagorean Theorem. The Pythagorean Theorem states that a2 + b2 = c2.[4] It is the formula that allows you to calculate the length of the side of a right triangle if you know the lengths of the other two sides. A right triangle is a triangle with a 90° angle. In the theorem, a and b are the opposite and adjacent (straight) sides of the triangle, while c is the hypotenuse (angled line) of the triangle.
- For example: Find the length of the hypotenuse of a right triangle with side a = 2 and b =3.
- a2 + b2 = c2
- 22 + 32 = c2
- 4 + 9 = c2
- 13 = c2
- c = √13
- c = 3.6
- Know the difference between similar and congruent shapes. Similar shapes are those that have identical corresponding angles and corresponding sides that are proportionally smaller or larger than each other. In other words, the polygon will have the same angles, but different side lengths. Congruent shapes are identical, they are the same shape and size.[5]
- Corresponding angles are identical angles in two shapes. In a right triangle, the 90 degree angles in both triangles are corresponding. The shapes do not have to be the same size for their angles to be corresponding.
- Learn about complementary and supplementary angles. Complementary angles are those angles which add together to make 90 degrees, supplementary angles add to 180 degrees. Remember that vertical angles are always congruent, as are alternate interior and alternate exterior angles. Right angles are 90 degrees, while straight angles are 180.
- Vertical angles are the two angles formed by two intersecting lines that are directly opposite each other.[6]
- Alternate interior angles are formed when two lines intersect a third line. They are on opposite sides of the line they both intersect, but on the inside of each individual line.[7]
- Alternate exterior angles are also formed when two lines intersect a third line. They are on opposite sides of the line they both intersect, but on the outside of each individual line.[8]
- Remember SOHCAHTOA. SOHCAHTOA is a mnemonic device used to remember the formulas for sine, cosine, and tangent in a right triangle. When you want to find the sine, cosine, or tangent of an angle, you use the following formulas: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, and Tangent = Opposite/Adjacent.[9]
- For example: Find the sine, cosine, and tangent of the 39° angle of a right triangle with side AB = 3, BC = 5 and AC = 4.
- sin(39°) = opposite/hypotenuse = 3/5 = 0.6
- cos(39°) = adjacent/hypotenuse = 4/5 = 0.8
- tan(39°) = opposite/adjacent = 3/4 = 0.75
EditWriting a 2-Column Proof
- Draw a diagram after reading the problem. Sometimes the problem will be provided without an image and you will have to diagram it yourself to visualize the proof. Once you have a rough sketch that matches the givens in a problem, you might need to re-draw the diagram so that you can read everything clearly and the angles are approximately correct.
- Make sure to label everything very clearly based on the information provided.
- The clearer your diagram, the easier it will be to think through the proof.
- Make some observations about your diagram. Label right angles and equal lengths. If lines are parallel to each other, mark that down as well. If the problem does not explicitly state two lines are equal, can you prove that they are? Make sure you can prove all of your assumptions.
- Write down the relationships between various lines and angles that you can conclude based on your diagram and assumptions.
- Write down the givens in the problem. In any geometric proof, there is some information that is given by the problem. Writing them down first can help you think through the process needed for the proof.
- Work the proof backwards. When you are proving something in geometry, you are given some statements about the shapes and angles, then asked to prove why these statements are true. Sometimes the easiest way to do this is to start with the end of the problem.[10]
- How does the problem come to that conclusion?
- Are there a few obvious steps that must be proved to make this work?
- Make a 2-column grid labeled with statements and reasons.[11] In order to make a solid proof, you have to make a statement and then give the geometric reason that proves the truth of that statement. Underneath the statement column, you will write a statement such as angle ABC = angle DEF. Under the reason you will write the proof for this. If it is given, simply write given, otherwise, write the theorem that proves it.
- Determine which theorems apply to your proof. There are many individual theorems in geometry that can be used for your proof. There are many properties of triangles, intersecting and parallel lines, and circles that are the basis for these theorems. Determine what geometric shapes you are working with and find the ones that apply to your proof. Reference previous proofs to see if there are similarities. There are too many theorems to list, but here are a few of the most important ones for triangles:[12]
- CPCTC: corresponding parts of the congruent triangle are congruent
- SSS: side-side-side: if three sides of one triangle are congruent to three sides of a second triangle, then the triangles are congruent
- SAS: side-angle-side: if two triangles have a congruent side-angle-side, then the two triangles are congruent
- ASA: angle-side-angle: if two triangles have a congruent angle-side-angle, then the two triangles are congruent
- AAA: angle-angle-angle: triangles with congruent angles are similar, but not necessarily congruent
- Make sure your steps flow in a logical fashion. Write down a quick sketch of your proof outline. Write down the reasons for each step. Add the given statements where they belong, not just all at once in the beginning. Re-order the steps if necessary.
- The more proofs you do, the easier it will be to order the steps properly.
- Write down the conclusion as the last line. The final step should complete your proof, but it still needs a reason to justify it. When you have finished the proof, look it over and make sure there are no gaps in your reasoning. Once you have determined that the proof is sound, write QED at the bottom right corner to signify it is complete.
EditTips
- STUDY EVERY DAY. Review today's notes, yesterday's notes and always review what you have learned before so you don't forget any postulates/theorems, definitions or symbols/notations.
- Look at other websites and videos for things you don't understand.
- Keep flash cards with formulas on them to help you remember them and review them frequently.
- Get phone numbers and emails of several people in your geometry class so they can help you while you're studying at home.
- Take a class in the summer beforehand so you don't have to work hard during the school year.
- Meditate. This helps.
EditWarnings
- Do not procrastinate
- Do not cram.
EditThings You'll Need
- A Straight-Edge
- A Compass
- A Scientific Calculator
- Graph-paper
- A Protractor
- Pencils (all work should be done with pencil)
- A Highlighter
- Colored Pencils
EditRelated wikiHows
- Learn Trigonometry
- Do a Double Linear Interpolation
- Write a Congruent Triangles Geometry Proof
- Survive Geometry Class
EditSources and Citations
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