Geometry is one of the foundational building blocks of mathematics. Your first geometry class might even be the first real-world application of maths that you do at school. Along with arithmetic, it is one of the oldest branches of mathematics, with the Ancient Greeks studying geometry alongside philosophy in their quest to understand the Universe. In fact, we still use a lot of the concepts from Ancient Greek geometry today, such as Pythagoras’ Theorem.
Geometry, simply put, is the branch of mathematics that measures and seeks to understand the distance, shape, size, and relative position of figures. We often apply geometry to shapes like triangles and squares as a means of applying this knowledge, but geometry is also frequently used in engineering and architecture.
Fortunately for modern students, humans have been studying geometry for thousands of years. There are a few hard and fast rules and approaches that you can familiarise yourself with in order to master basic geometry.
Here, we’ve assembled our top favourite tips for studying geometry:
1. Know Your Complementary and Supplementary Angles
All shapes, referred to in geometry as “figures”, follow a set of rules governing the size of their internal and external angles. Complementary angles are those that form a sum of 90 degrees. In other words, if two angles comprise a right angle, they are complementary. The angles in right triangles that are not the right angle will be complementary.
Supplementary angles add up to 180 degrees. Angles on either side of a straight line, for example.
Knowing these two facts is a great base for building your knowledge of other angles, and will allow you to solve problems involving them.
2. Congruent Angles
Congruent angles are those that have equal measure. Euclid, an ancient Greek mathematician, was the first to describe congruence in triangles. In fact, his ancient textbook, The Elements, is the major source of geometric knowledge as we know it in the modern day.
Two congruent angles are often found on opposite sides of an “x” shape. The concept of congruence can also be applied to triangles, where triangles are considered congruent if they have all three sides equal (side side side, or SSS), two sides and the angle between them equal (side angle side, or SAS), or two angles and a side equal (angle side angle, or ASA). Oftentimes, quadrilateral (four-sided) shapes are made out of two congruent triangles.
3. Parallel Lines
Parallel lines, as initially described by Euclid, are lines that will never meet. They run alongside each other, theoretically forever.
In application, the properties of parallel lines and the lines intersecting them can be applied to other two dimensional objects, such as quadrilaterals. Intersecting lines across two parallel lines will form four interior angles and four exterior angles. From this, you can apply your knowledge of congruent, and complementary angles to work out the measure of these angles. Some of these will be identical angles formed by the line segments.
4. Understand Geometry Terms
Just like when you study algebra and you need to learn some new terminology to get to grips with the new concepts, geometry has its own vocabulary that you will want to familiarise yourself with.
A few key geometry terms include:
- Parallel and perpendicular lines
- Three different types of triangles (scalene triangle, equilateral triangle, isosceles triangle)
Make sure you understand these terms. Perhaps forming a study group or asking for homework help will ensure that you have a working knowledge of geometry terminology and how it applies to the work you’re doing.
5. Draw Diagrams
While in other areas of maths you stick to formulas and equations, geometry is concerned with measurements. For measurements to truly make sense, you’ll need to get familiar with drawing diagrams.
This is actually why most maths exercise books come with grid lines! It’s to make drawing diagrams and organising your work easier. When you’re drawing diagrams, be sure to use the right equipment, and be consistent with your measurements. You don’t need to always draw precise angles, but ensuring that your diagrams are neat, legible, and easy to follow will help you follow your own work, and help the person marking your work understand what you’re doing.
6. Practice problems
Like anything, you’ll need to practice your geometry to really get better at it. A theoretical understanding of the concepts is a great starting point, but you’ll want to apply this knowledge over and over in various settings to make sure it’s truly sticking.
Australia’s Board of Studies supplies past exam papers for HSC students looking to practice ahead of the big day. You can do practice exams, or find practice geometry resources, to sharpen your skills.
When doing practice papers, be sure to mark the test when you’re done. Once you’ve marked the exam, go over any questions you got wrong. See if you can identify where you went wrong, and make sure to practice that particular area and check your theory before taking the next practice test.
7. Get Familiar with Basics
Euclid’s five postulates are as follows. Pretty much all geometry and its applied areas of study follow these five basic rules.
- A straight line can be drawn joining any two points.
- You can continue any line segment, in either direction, indefinitely, as long as it continues to be a straight line.
- You can draw a circle around any line segment with one end of the line segment serving as the centre point, and the length of the line segment serving as the radius of the circle.
- All right angles are congruent, that is, equal.
- If you have one line and one point not on the line, only one line can be drawn directly through the point that will be parallel to the first line.
8. The Rules of Triangles
Triangles probably seemed mundane when you were in kindergarten, but when it comes to geometry they do some pretty wonderful things. We use triangles to build pretty much all other shapes, with the notable exception of the circle. Squares, for example, are really just two triangles joined along the longest side.
Triangles are governed by a few key rules, and the different types of triangles are similarly governed by subsets of these rules.
These rules include:
- The sum of interior angles in a triangle is always 180 degrees
- An isosceles triangle has two identical sides, which end in two equal angles
- Equilateral triangles are always comprised of three equal sides and three congruent angles
You can extrapolate from these rules as well. For example, if a quadrilateral is more or less the combination of two triangles, the sum of internal angles of a quadrilateral is always going to be 2 x 180 degrees, which is 360 degrees. Trigonometry is also useful for navigation.
9. Have The Right Tools
This relates to drawing your diagrams. Make sure you have the appropriate tools to complete your geometry and draw the diagrams. That means you need a straight ruler, a compass, and a protractor. These will help you draw clear, concise diagrams, and help you to measure angles, lines, and other geometric figures.
10. Memorise Pythagoras
Pythagoras was another Ancient Greek philosopher, who lived around the 6th century BC. Pythagoras’ Theorem has been proven many times in many different ways, perhaps more than any other mathematical theorem. The Pythagorean Theorem can be extrapolated from and applied to many other areas, and generalised beyond even Euclidean geometry.
The Pythagorean Theorem states namely that the area of the square whose side is the hypotenuse (that is, opposite the ninety degree angle) is equal to the sum of the areas of the squares on the other two sides. In other words:
a2 + b2 = c2
Simply put, you need to memorise this. Commit this basic formula to your memory. Some maths exams will give you a sheet with all the necessary formulas, but not all of them.
Geometry does not have to be difficult. In fact, by committing yourself to understand and working with the basic concepts, you can apply your knowledge and extrapolate it further. You will need to use your geometric tools to draw diagrams, practice problems, and memorise a few key concepts, including Pythagoras. Other than that, your usual healthy study habits will come in very handy here as we apply mathematics to the real world!
You can make use of the study features offered by Zookal Study, including cheap textbooks, maths homework help, and help with algebra, to further improve your understanding of geometry. Often, geometry and algebra intersect, such as in the case of the various formulas for finding the surface area or volume of a given figure.