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In the new linear GCSE maths paper, you will be required to solve various mathematical problems involving congruent triangles. The specific questions you will be expected to answer will vary depending upon which examination board with which you are registered, but as a rule you will be required to:. Listed below are a series of summaries and worked examples to help you solidify your knowledge about congruent triangles. During your GCSE maths exam, you will be required to prove whether two triangles are congruent.

You may also be required to solve various mathematical problems using the principles of congruent shapes.

Key Concepts

Two or more triangles are congruent if one of the following rules applies:. If the three sides of the first triangle are equal to the three sides of the second triangle, then the triangles are congruent to one another. You can mark these SSS triangles in the following manner:. If two sides of the first triangle are equal to two sides of the second triangle, and the included angle is equal, then the two triangles are congruent.

You can mark these SAS triangles in the following manner:. If two angles of the first triangle are equal to two angles of the second triangle, and one similarly located side is equal, then the two triangles are congruent. You can mark these AAS triangles in the following manner:. If the hypotenuse and one other side of the first right-angled triangle are equal to the hypotenuse and corresponding side of the second right-angled triangle, then the two triangles are congruent. You can mark these RHS triangles in the following manner:. All of these rules can be used to prove whether triangles are congruent, as well as to calculate various measurements relating to two triangles:.

Example a - Given that these two triangles are congruent, calculate the perimeter of triangle XYZ.

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Solution a - As well as using the rules of congruence, this question also requires you to recognise and recall the properties of various triangles. If these two triangles are congruent, all of their angles will be the same. To do so, we would essentially be repeating the process we used to find our missing variables from above, but keeping the variables intact.

We have three different triangles and three missing angles, so their equations would look like:. We know that all of those equations will find us one of the three unmarked angles. Note: there is a third—and even faster—way to solve this problem that involves quadrilaterals. Check out our guide to SAT polygons for more info!

Though there are many different types of triangle problems on the SAT, they tend to stand out from the crowd. Because of this, it is difficult to break down one exact path for problem-solving triangle questions. Using your formulas is the absolute most crucial step for any triangle problem. Remember that dealing with a multi-shape triangle problem is like working with dominoes.

Each successive piece of information makes way for finding the next piece of information.


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You will always have enough data to go on—just focus on finding one shape and one piece of information at a time, and the dominoes will fall into place. Draw your own diagrams if you are given none. Draw on top of your diagrams when you are given pictures. Write in your givens and all the measurements you find along the way to your missing variable or variables , and mark congruent lines and angles. Finally, give yourself a break from dealing with variables and complex algebra if you need to.

Congruency of Right Triangles (LA & LL Theorems)

It can be far too easy to make a mistake when dealing with variables alone, so if you have the time to spare, go for plugging in your own numbers! You have more information and strategies at your disposal than you think. Just keep them on hand and organized in your head and you'll be set. Now let's test your triangle knowledge on more, SAT math problems. In the figure above, a regular polygon with 8 sides has been divided into 8 congruent isosceles triangles by line segments drawn from the center of the polygon to its vertices. What is the value of x? Since the 8 triangles are congruent, the measures of each of the 8 angles are the same.

Since the triangles are isosceles, the measures of the two angles are the same. Now, though it may not look it, we are told that E is the midpoint of line AB. Triangles will show up, without fail, at least a few times on every single SAT usually in about 1 to 3 problems.

Geometry – Mathplanet

The good news is that you will be given multiple formulas to aid you through these types of questions, but the drawback is that the test is timed, and so you should only waste time going to your formula box if you are all out of options. Know your definitions, try to memorize your formulas, and do your best to keep a clear head as you go through your test. And, as always, practice, practice, practice! Now that you've done your paces on your triangles, it's time to make sure you are prepared for all the math topics you'll see on the SAT. All of our math guides will take you through strategies and practice problems for all the topics covered on the math section, from integers , to ratios , circles to polygons and more!

Feeling anxious about test day?

Make sure you know exactly what to do and bring to ease your mind and settle your nerves before it's time to take your SAT. Running out of time on the SAT math section? Look no further than our guide to help you beat the clock and maximize your SAT math score.

Angling to get a perfect score? Check out our guide to getting a perfect , written by a perfect-scorer.

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Geometry For Dummies Cheat Sheet

Check out our best-in-class online SAT prep classes. We guarantee your money back if you don't improve your SAT score by points or more. Our classes are entirely online, and they're taught by SAT experts. If you liked this article, you'll love our classes. Along with expert-led classes, you'll get personalized homework with thousands of practice problems organized by individual skills so you learn most effectively. We'll also give you a step-by-step, custom program to follow so you'll never be confused about what to study next. Courtney scored in the 99th percentile on the SAT in high school and went on to graduate from Stanford University with a degree in Cultural and Social Anthropology.

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