Triangles are fascinating shapes! With just three sides, there can be many different types of triangles depending on the lengths and angles in the triangle. For example, you have probably learnt about equilateral triangles (three sides of equal length), isosceles triangles (two sides of equal lengths), and right-angled triangles (a triangle which consists of one right-angle).
In O Level Math Tuition, you will also learn about similarity and congruence. These describe the relationship between triangles which look similar: for example, one triangle that is a rotated image of another triangle, or a triangle that is a bigger version of another triangle.
What is similarity and congruence?
Congruent triangles are triangles that have exactly the same shape and size. When we say same ‘shape’, we mean that all the corresponding sides of one triangle is the same as all the corresponding sides of the other triangle. Accordingly, all the corresponding angles will also be the same. Identical triangles that are rotated or reflected are also considered congruent.
Similar triangles have the same shape but may be of a different size. This means that the corresponding angles in both triangles are the same, but one triangle is bigger than the other. Think of it as having the same triangle and zooming in or out on it!
Although most students start learning similarity and congruence in the context of triangles, the principles can be extended to other shapes as well.
Common sense tells us that in order to identify whether two triangles are congruent, we can check if all their corresponding sides and angles are equal. But in actual fact, you don’t need to know ALL the sides and angles to confirm that two triangles are congruent. Congruency tests involve checking a selected number of corresponding sides and/or angles in the triangles to see if two triangles are identical.
Why learn similarity and congruence?
Similarity and congruence sounds simple enough. Like, isn’t it obvious whether two triangles look the same or not? Apparently not.
For the exams
In the exams, diagrams are usually not drawn to scale. So, triangles which appear different might actually be similar or congruent. You will be tested on your ability to identify and find proofs for similar and congruent triangles based on mathematical evidence.
Similarity and congruence also connects closely to other math topics and principles.
Dealing with proportions
Learning about similar triangles teaches us the logic behind proportion and scale. In real life, we also use the principle of proportion to scale things to different sizes (e.g. the magnification of the microscope, or when you want to make an accurate miniature model of something).
Trigonometry
We can’t talk about triangles and not mention trigonometry! Trigonometry is the study of triangles and the ratio of the angles within a triangle. In some scenarios, you would need to apply knowledge of similar and congruent triangles to figure out the angles or lengths of certain triangles to solve the problem.
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Common challenges students face
Similarity and congruence is not a long chapter like differentiation methods or trigonometry. But it can be an obstacle to good grades if you can’t handle it like a pro! Here are some common pitfalls that students encounter:
Not knowing how to prove
This topic is all about proofs. As such, it is difficult to even practice from an assessment book, because some may not provide answers for proof questions. A good Math Tutor can coach you on knowing the critical workings and steps to include in your proofs. For example, you must always link it back to the tests of congruency and similarity and present your workings clearly.
Not having a good foundation in geometry
If you already struggle with geometry, you are likely to find this chapter a stretch for you. Take time to revisit geometry rules, such as the sum of angles in a triangle, parallel lines, perpendicular lines, labelling conventions, and so on. The more questions you do, the more intuitive these rules should be to you!
Conclusion
We hope this article gives you an idea of how – or why – to begin studying for similarity and congruence! In the event that you feel like you are unable to grasp this topic, or any other topic in math, know that you are not alone! At our tuition centre in Singapore, we help our students prepare for their school tests and exams with systematic reviews and targeted practice. Our students will never enter another test or quiz without preparation!
In small group classes, our tutors are able to provide personalised consultations and coaching, steering you towards improvement. You will be able to find out the exact areas you need to work on and the steps you need to take in order to reach your study goals! Let’s get started on your success journey today!