This is Common Core State Standards support video in mathematics. The standard is HSG.CO.B.8. This standard reads: Explain how the criteria for triangle congruence (Angle-Side-Angle, Side-Angle-Side, Side-Side-Side) follow from the definitions of congruence in terms rigid motions. Now, Angle-Side-Angle, Side-Angle-Side, and Side-Side-Side, there’s some difference of opinion as to whether these are postulates or theorems, but we won’t worry about that right now. That isn’t part of this standard, but I’ll probably stick with just calling them postulates.
Let’s look at some other standards that are connected to this one. There is a standard back in eighth grade, 8.GA.2, that is related to this same topic and it states: Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them. So it’s very much related to this one. At the high school level right before this standard, there’s standard HSG.CO.B6, which states: Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. Corresponding to that is HSG.CO.B.7: Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. Note that the terminology is very, very similar. In fact, the activities that we can do for standard HSG.CO.B.8 could probably be used to also show and explain these other standards also.
Let’s make sure that we’re on the same page as far as the definition of congruence, and in fact, standard HSG.CO.B.7 pretty much states what the definition is. Congruence would be defined as two triangles being congruent if and only if the lengths of the corresponding sides are equal and the measures of the corresponding angles are equal. So we have to meet all those conditions. Let’s make sure that we all have the same understanding in terms of rigid motions. Rigid motions include translations, which are just slides, then reflections and rotations. Note that the location or the orientation can change, but the size is always going to remain constant.
Well, let’s start off with Angle-Side-Angle. If we have these two triangles congruent, triangle ABC being congruent to triangle XYZ, then, based on this postulate, Angle-Side-Angle, we would start off with two different angles, but they are congruent to each other, angle BAC and angle YXZ. And then on ray AB, I would mark off a certain size segment. Then on ray XY, I would mark off the same size segment. So these two segments would have the same measure, and then from endpoint B, we would construct an angle that’s a certain size; and then from endpoint Y, we would construct the same size angle. So based on Angle-Side-Angle, these two triangles should be congruent.
So let’s look at this context. We have a triangle ABC and a triangle XYZ, and we need to show using this postulate and using rigid motions that they are, in fact, congruent. Let’s start off with our two congruent angles, angle BAC and angle YXZ. What we need to do is to slide one triangle over to where those two angles would match up. There’s different ways to do this, but let’s go ahead, and we’ll slide this triangle YXZ over to where vertex X and vertex A on the two respective angles correspond. So let’s slide it over. Now we’re going to have to do a rotation in order for these two angles to match up. So let’s rotate it clockwise. There we have it. Let’s make sure that we have our labeling. So we have angle BAC and angle YXZ that are congruent, and now we have them overlapping. And as it turns out, we’ve already got the triangles overlapping.
So we’ve shown these two angles are congruent. You have one overlapping the other, and since the triangles are already overlapping, we can tell that segment AB and segment XY are congruent. And then angles ABC and XYZ overlap and they match up exactly, so they’re congruent. So by using rigid motions and just by using the first pair of angles, then the pair of sides and then the next pair of angles, we’ve shown these two triangles are, in fact, congruent. So we’ve met these conditions. We showed that this first pair of angles, BAC and YXZ, are congruent. Then we moved it over to where segment AB and XY overlap, and then, in turn, angle ABC and XYZ overlapping and again showing that these are congruent and we have them overlapping each other.
So taking a slightly different perspective, we have these two triangles. We’ve already done rigid motions and shown that they were congruent, and the rigid motions that we used were translations; we slid it over and then we rotated to get the first pair of angles to match up, and that was enough for the rigid motions. It ended up that the triangles were already overlapping. Now taking this a step further, we met the conditions Angle-Side-Angle. If we look at this in terms of the rays for the angles that are formed here at vertex B and vertex A, notice that ray BC has to intersect ray AC in such a way where we have this certain size segment formed. The same thing is going to happen over here with triangle XYZ, and so those two segments, in fact, have to be congruent. Then in turn, segments AC and XZ have got to be congruent.
Now let’s move to Side-Angle-Side, so we have these two triangles, ABC and triangle DEF. So based on a Side-Angle-Side, we start off with an equal pair of sides, and then from endpoint A, we construct a certain size angle and from endpoint D on the other situation over here, we construct the same size angle. And then along ray AC, we measure off a certain size segment: and then we do the same thing on ray DF. Now if we were to connect B and C to form this triangle ABC, and over in the other context we connect points E and F, then we have triangle DEF formed, and based on our postulate Side-Angle-Side, that should be enough for these two triangles to be congruent.
Well, let’s do something similar to what we’ve already done, and we need to do some rigid motions to, in fact, show that if we meet those conditions, Side-Angle-Side, that the two triangles are, in fact, congruent. So we have our equal pair of sides, AB and DE, and we need to slide one over to where we get two of the endpoints to match. Again, there’s different ways to have done this, but let’s go ahead and move one over to where we have endpoints A and D overlapping. So let’s do that. All right, so A and D overlap, but our segments AB and DE are still in different orientations. So we need to do a rotation here, so we rotate it counterclockwise. So now we do have segments AB and DE overlapping, and we know that they’re congruent.
Next, Side-Angle-Side; we need to look at the angles BAC and EDF. In order to get the angles to line up, match over each other, then we need to reflect angle BAC across segment AB. So let’s do that, let’s reflect it across here. Okay, so now we do have the angles overlapping, and they are congruent. Then the third part of Side-Angle-Side, we have sides AC and DF and they’re already overlapping. We know they’re congruent, and then when we zoom out and look at the overall triangles, one does overlap the other. So we know that the two triangles are congruent. Again, we’ve met all conditions, Side-Angle-Side. We used a translation to get one endpoint of one segment to match over the other one. Then we had to do a rotation to get the segments to coincide, but then we had to do a reflection. We had to do another step in order to get the pair of angles to coincide.
Let’s look at our third context, Side-Side-Side, and let’s say we have these two triangles, ABC and triangle PLM. So based on Side-Side-Side, we would have segments AB and PL congruent. We would have segments BC and LM congruent, and segments AC and PM congruent. So we start off on the left over here; we have three segments. On the right we have three segments, and we do have them paired up to where we have three different pairs that are congruent to each other. So on the left-hand side, if we put those together to form a triangle, and then over here on the right, we put those three segments together to make a triangle, based on Side-Side-Side. Those conditions would be sufficient to show that those two triangles are, in fact, congruent.
Now let’s follow some of the same procedures that we did a while ago, and we need to use our rigid motions to get one triangle to fit over the other to prove that they are congruent. So let’s start off with our first pair of segments, AB and PL. Again, different ways to go about this, but let’s go ahead and slide one triangle over to where endpoints P and A coincide or overlap. So let’s slide it over. Okay, so now we have endpoints A and P overlapping, but we don’t have the segments overlapping. So we need to do a rotation. So let’s rotate it clockwise. Okay, so now we’re at the point where we’ve shown that we’ve got the first pair of sides overlapping, AB and PL.
Now we need to look at our second segments, in this case BC and LM. Not overlapping, so we’re going to have to do something to get that to happen. So we need to reflect segment BC across segment AB, or some people might want to call that segment PL. It’s the same segment. At this point they’re overlapping, so let’s go ahead reflect it across that segment, and here’s the result. So we’ve taken care of two parts of Side-Side-Side so far, and now we already have the third pair of segments overlapping each other. So we know that AC and PM are congruent, and so now we’ve shown that the two triangles are, in fact, congruent by knowing that all three matching pairs of sides are congruent to each other. The rigid motions that were used—we used a translation to slide one endpoint over to fit exactly over the matching endpoint of the respective segment. Then we had to use a rotation to get that first pair of segments to match, and then we had to do a reflection to get the second pair of segments to match.
In looking at the bigger picture here, all of the cases are going to begin with a translation to overlap a vertex of one triangle with the corresponding vertex of a second triangle, and most pairs of triangles will not have the same orientation. That’s going to be very rare, especially at this level. If that’s the case, then some type of a rotation will be necessary in order for the targeted pair angles or the targeted pair of segments to overlap. And once you have that first pair of angles or that first pair of segments overlapping, you more than likely will need some type of reflection for a second pair of corresponding angles or a second pair of segments to overlap. So you might need to do all three types of transformations.
If we look at the standards for mathematical practice for this standard; make sense of problems and persevere in solving them, reason abstractly and quantitatively, construct viable arguments and critique the reasoning of others, and model with mathematics—if we look at these first four, we can pretty much establish that we did use the second, third, and fourth ones. And then if we look at the last four of the standards for mathematical practice, we did use appropriate tools strategically, we attended to precision, and we did look for and express regularity in repeated reasoning. So we filled in some of the details for standard HSG.CO.B.8. Again, it’s just a matter of taking those postulates and doing your rigid motions to show that as long as you met those conditions, the two pairs of triangles that you’re dealing with will, in fact, be congruent.