If you are shown two triangles and are told to prove that they are congruent, do you recall which properties you can use to show triangle congruence? Think about this for a minute then click on the Show Me Button below to check your thinking.
It is not enough to just tell someone that two triangles are congruent by, let's say, SSS. You must write a proof showing all the steps that have led you to this conclusion. Watch the following video to see how this is done.
As you watch this video, use the study guide to follow along if you'd like. Click the button below to download the study guide.
There are five postulates, sometimes called properties or theorems, used to prove triangle congruence. In this video, I would like to show you how you might use these postulates to complete a proof showing two triangles are congruent – the exact same shape and size. In Geometry class, you were asked to cut a rectangle along one of its diagonals, and you saw that the two triangles that were made looked like they were the same shape and size. A conjecture followed: If a rectangle is cut along its diagonal, then the resulting two triangles are congruent. This conjecture is written in an appropriate if P then Q form for a conditional, so we should be able to prove if it is true. Let’s start with the given information.
It would be good to identify the hypothesis and conclusion first. Then, since we are working with a geometric shape, side lengths and angle measures that result from the given information should be listed next. Pause the video now, and write down an appropriate hypothesis and conclusion from the conditional statement. Check your answer in a moment by resuming the video… A good hypothesis might be, “an object, or shape, is a rectangle”. And the conclusion is that “the triangles formed by one of the diagonals are congruent”. So, since our original object is a rectangle, we know that side AB is congruent to side CD and side BC is congruent to side DA. Also, from the fact the original shape is a rectangle, we know the measure of angle B is ninety degrees and the measure of angle D is ninety degrees.
With this given information, which postulate for triangle congruence can be used? Can there be more than one? Think about it for a minute and write down which postulate you think might be used and why. Resume playback of the video in a moment to check your answer… There are several postulates that can be used to prove triangle congruence in this example. We might use the HL postulate since the hypotenuse and a leg of ΔABC and ΔCDA are congruent. We could use the SSS postulate since three pairs of sides are congruent. We might use the SAS postulate since two pairs of sides and their included angle are congruent. We could even use the other two postulates if we consider the fact that rectangles have two pairs of parallel sides, but our example proof below will not consider this fact.
Once you establish what postulate you would like to use, you should write the proof and verify nothing has been left out. The proof you write in your notes could be formal like the example I show, or it can be an informal paragraph proof that lists the arguments in a series of sentences in a paragraph. I’d like to use the HL Postulate in this example. I will start listing all statements and the reasons for each. Also, I will mark the diagram with the information as I list it.
First, our given information shows that shape ABCD is a rectangle… Our second statement might say that angle B and angle D are right angles, by the definition of a rectangle... Then we could say that triangle ABC and triangle CDA are right triangles, by the definition of right triangles… In our fourth statement we might again use the definition of rectangles to show that side AB is congruent to side CD… Then we could go on to say that side AC when a part of triangle ABC is congruent to side CA when a part of triangle CDA… Since we have two right triangles whose hypotenuses are congruent and one of each triangle’s legs are congruent, then the two triangles are congruent by the HL postulate. Our proof is now complete since our final statement since it supports the conclusion of the conditional statement above.
Can you create a similar proof by using one of the other postulates mentioned? Give it a try on your own. You can check your work in a moment by resuming the video. Good Luck!
