Geometry (all content)
Review the triangle congruence criteria and use them to determine congruent triangles.
What's so great about triangle congruence criteria?
Two figures are congruent if and only if we can map one onto the other using rigid transformations. Since rigid transformations preserve distance and angle measure, all corresponding sides and angles are congruent. That means that one way to decide whether a pair of triangles are congruent would be to measure all of the sides and angles.
The triangle congruence criteria give us a shorter way! With as few as of the measurements, we can often show that two triangles are congruent.
We can break up any polygon into triangles. So showing that triangles are congruent is a powerful tool for working with more complex figures, too.
What are the triangle congruence criteria?
|Side-side-side (SSS)||When all three pairs of corresponding sides are congruent, the triangles are congruent.|
|Side-angle-side (SAS)||When two pairs of corresponding sides and the corresponding angles between them are congruent, the triangles are congruent.|
|Angle-side-angle (ASA)||When two pairs of corresponding angles and the corresponding sides between them are congruent, the triangles are congruent.|
|Angle-angle-side (AAS)||When two pairs of corresponding angles and one pair of corresponding sides (not between the angles) are congruent, the triangles are congruent.|
|Hypotenuse-leg (HL)||When the hypotenuses and a pair of corresponding sides of right triangles are congruent, the triangles are congruent.|
Want to learn more about the triangle congruence criteria? Check out this video.
Why isn't side-side-angle a triangle congruence criteria?
When two pairs of corresponding sides and one pair of corresponding angles (not between the sides) are congruent, the triangles might be congruent, but not always.
This criterion usually is not enough information when the corresponding angles are opposite to the shorter of the two known sides in the triangle. We especially need to steer clear when the figure might not be to scale.
Can we be sure that two triangles are not congruent?
A triangle only has sides and angles. If we know distinct side measures or distinct angle measures, then we know the two triangles cannot be congruent. Sometimes we know measures because they are in the diagram. Other times we use tools like the Pythagorean theorem or the triangle internal angle measure sum to figure out missing measures.
Sometimes there just isn't enough information to know whether the triangles are congruent or not. If we only have congruent angle measures or only know two congruent measures, then the triangles might be congruent, but we don't know for sure.
Drawing are not always to scale, so we can't assume that two triangles are or are not congruent based on how they look in the figure. That's especially important when we are trying to decide whether the side-side-angle criterion works. If the congruent angle is acute and the drawing isn't to scale, then we don't have enough information to know whether the triangles are congruent or not, no matter how they look in the drawing.
Check your understanding
Are the triangles congruent?
The triangles are not drawn to scale.
The triangles are not drawn to scale.
Want to practice more problems like this? Check out this exercise.
Want to join the conversation?
- In the "check your understanding," I got the problem wrong where it asked whether two triangles were congruent. Both triangles listed only the angles and the angles were not the same. I put no, checked it, but it said it was wrong. I thought that AAA triangles could never prove congruency. Please help!(39 votes)
- There are 3 angles to a triangle. The question only showed two of them, right? You could calculate the remaining one. For example, a 30-60-x triangle would be congruent to a y-60-90 triangle, because you could work out the value of x and y by knowing that all angles in a triangle add up to 180. Then, you would have 3 angles.(32 votes)
- if there are no sides and just angles on the triangle, does that mean there is not enough information?(24 votes)
- A triangle will always have a sum of 180° (3 angles a+b+c=180°). So a huge triangle will have the same sum of angles as a tiny one, but the area will be different.
I hope this was a little helpful!(46 votes)
- how is are we going to use when we are adults ?(21 votes)
- If you were to come at this from the perspective of the purpose of learning and school is primarily to prepare you for getting a good job later in life, then I would say that maybe you will never need Geometry. Assuming of course you got a job where geometry is not useful (like being a chef). Now, if we were to only think about what we learn, when we are young and as we grow older, as to how much money it’s going to make us, what sort of fulfillment is that? You could argue that having money to do what you want is very fulfilling, and I would say yes but to a point.
The reason why people find such enjoyment in watching the olympics is not because we just like to see people win money and awards (though that is an element) but because it is amazing and inspiring to see people who have put in the effort and discipline to become as skilled as they are (which is also why people love the movie Rocky) Humans are inspired by effort. I would also argue that humans are fulfilled by effort and discipline. When people exercise or learn the piano, usually there is this annoyance or a desire to just sit on the couch, but once those things are done we feel this sense of accomplishment, maybe even happiness. Things that require effort whether it be mental or physical makes people feel fulfilled because doing those things push your boundaries, and when we push what we thought was our limit of intelligence or skill we realize that our boundaries are much farther than we imagined.
I am not saying learning to make money is bad, but I think you need a balance between learning to make money and learning to understand the world, yourself, or to improve yourself.(10 votes)
- They tell me this AFTER I had to figure it for the quiz?(15 votes)
- I'm still a bit confused on how this hole triangle congruent thing works. I think I understand... but i'm not positive. Could someone please explain it to me in a simpler way? Thank you very much.(7 votes)
- Ok so we'll start with SSS(side side side congruency). If all the sides are the same, they would need to form the same angles, or else one length would be different. For SAS(Side Angle Side), you would have two sides with an angle in between that are congruent. If you have an angle of say 60 degrees formed, then the 3rd side must connect the two, or else it wouldn't be a triangle. So, the third would be the same as well as on the first triangle. For ASA(Angle Side Angle), say you had an isosceles triangle with base angles that are 58 degrees and then had the base side given as congruent as well. Whatever the other two sides are, they must form the angles given and connect, or else it wouldn't be a triangle. Hope this helps(12 votes)
- If a triangle is flipped around like looking in a mirror are they still congruent if they have the same lengths(7 votes)
- If the side lengths are the same the triangles will always be congruent, no matter what.(7 votes)
- when am i ever going to use this information in the real world?(7 votes)
- These concepts are very important in design. For example, when designing a roof, the spoiler of a car, or when conducting quality control for triangular products. It is much faster to compare known angles or sides than it is to measure every single side. Sometimes, it is not possible. For example, when building a roof, you want to make sure it matches the design. If you don't want to climb the whole thing after it is built and use a giant measuring tape, you can simple measure one side and two angles.(6 votes)
- if all angles are the same it is right i feel like this was what i was taught but it just said i was wrong(7 votes)
- I'm really sorry nobody answered this sooner. It's a good question.
That happens to me. I remember something so clearly, and then I find out it never happened. I was really shocked when I was doing an exercise and they said that -x squared was a negative number. I was sure it was supposed to be positive, and I was in the middle of reporting a mistake. But then I looked it up and found out I was wrong.
But I understand what they are saying about triangles. Imagine if you had a triangle, lets say the angles were 30, 60, and 90. If I have one where the short side is 20 inches, I can scale it up, make another 30, 60, 90 triangle, maybe the short side is 25 inches, and it has all the same angles, but it is bigger. That's why you have to know at least one of the sides if you know all of the angles. I hope that helps :-) Good luck!(5 votes)
- if the 3 angles are equal to the other figure's angles, it it congruent?(4 votes)
- No since the sides of the triangle could be very big and the angles might be the same. A triangle can only be congruent if there is at least one side that is the same as the other.(5 votes)