Study Guide

Congruent Triangles

Congruent Triangles Introduction

When she was younger, your little sister used to be a pain to babysit. Whenever she wasn't crying or screaming, she was spitting up. Plus, her diaper was nothing short of a war zone. Things got more interesting (and way more dangerous) when she learned how to escape from her crib and tried fitting silverware into electrical outlets. When she started talking, you got solid proof that the word "poopyhead" could be used over 127 times in under ten minutes. And yes, you counted.

Now that she's about ready to enter kindergarten, she's a lot easier to handle. She calls you by your real name, poops in the toilet, and only cries when boo-boos are involved. You could almost say that you enjoy babysitting her now. Almost. (You keep it on the down low, though. Hanging out with your little sister is hardly an invitation to sit with the cool kids at lunch.)

One evening, you two decide to play her new matching memory game, but it seems to be going nowhere. Every time you flip over two cards that you think match, your sister objects and says it doesn't count. Why? She tells you it's not just a matching game. It's a congruence game. It's not enough to flip over any old pair of triangles because they might be very, very different. For you to win, the two triangles need to be exactly the same shape. They need to be congruent.

When did your sister become so strict about games? Or start paying attention to the rules? Or learn the word "congruence," for that matter? Though, to be fair, she does have a point. It pays to learn what makes two shapes congruent. Maybe not a six-figure salary, but we can promise you it'll come in handy.

For instance, if you're a construction worker building a roof or an architect designing a building or a student taking a geometry class, you'll need to know how to recognize and create congruent shapes. Triangles are among the simplest shapes in the two-dimensional world, so why not start there?

And if it means beating your sister at that silly memory game, even better.

Congruent Triangles Resources


Constructions in Geometry
What do we mean when we talk about "construction" in geometry? No need to bust out your hard hat or anything. All we mean when we talk about construction is creating a particular line or shape using only a straightedge (like a ruler or the spine of your 20-year-old textbook) and a compass (not the kind that points north). This website can help you construct whatever geometric figure you might need. Use caution though, since it's still a construction site.

Pythagoras: A Biography
Pythagoras is like a celebrity in the world of geometry. Just like meeting Paul McCartney or Sting, you might want to get a little background on his life before you're formally introduced. You'll get to know him really soon, so this biography will give you more than enough information on what he's all about.

Encyclopedia of Triangle Centers
How many centers does a triangle have? It's not like a circle that only has one center. In fact, you'd probably be better off asking how many centers a triangle doesn't have.

Hardest Easy Triangle Problems
Think you have what it takes to solve the hardest easy problems in geometry? If you can master these few problems, you know you're in good shape. (Is the hardest easy problem the same as the easiest hard problem? We'll debate while you work it out.)


Translation and Reflection Song
Do you remember the lyrics to every catchy pop song you've ever heard, but still have trouble memorizing the congruence transformations? Boy, have we got the solution for you.

Step-by-Step Triangle Proofs
Having trouble with the SSS, SAS, ASA, and AAS congruence rules? Not sure how to organize your proofs about triangles? Got nothing better to do? Ultra-clear explanations and easy-to-understand proofs are comin' your way fast.

SAS Whoa!
Are you ready to headbang to some triangle congruence? You'll dig the baseline, feel the vocals, and love the concepts. Once that melody is stuck in your head, you'll be ready to rock

Games and Tools

Classifying Congruence Transformations
Think you know about congruence transformations? Test your knowledge of translations, rotations, and reflections and be sure you can shift, turn, and flip shapes and visualize congruence.

Which Triangles Are Congruent?
Pull out those S's and A's, because you'll need 'em to prove all these triangles are congruent. Make sure you know the triangle congruence rules down pat, and these questions will be a breeze.