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Two Column Proofs - Problem 4
Given two overlapping triangles that share a side, it is possible to show that two segments of each triangle are congruent. First, mark which angles are congruent, as given in the problem. Then, after redrawing the triangles separately, the correspondence of angles is much clearer, such as the congruence of other angles or sides. Recall that a segment is congruent to itself. Then, from properties of triangles such as ASA or AAS, the two triangles are congruent. By CPCTC, corresponding sides are congruent.
In the proof, statements are facts and reasons are the theorems or properties that cause the statement to be true (such as SAS or CPCTC).
When we’re talking about two column proofs, another common problem is this, overlapping triangles. If you look at this closely you see that there is a whole bunch of triangles in here. We’ve got this big triangle ABD, we’ve got another big triangle ADC, now we’ve got these two smaller triangles here and we’ve got this kind of triangle AED that doesn’t really correspond to anything. So we know we’re not going to use that smaller triangle because our goal here is to show that two triangles are congruent.
So let’s look at what we’re given and what we’re being asked to show. We’re being asked to show that AB and CD are congruent, so it’s this segment right here and this segment right here. So that doesn’t really help us to narrow it down between the two triangles. Let’s mark the first two angles congruent. BDA which means D is your vertex, so we have BDA. So I’m going to mark angle BDA as congruent to angle CAD, so we have CAD, so I’m going to mark those two angles as congruent.
The second pair of angles, BAC so we have BAC, and I’m going to use a different number of markings, congruent to BDC, so we have BDC. So at this point I’m going to guess that we’re not going to have enough information to say that these two smaller triangles are congruent. So we’re going to have to use the larger triangles, so I’m going to redraw those two triangles over here. A great step when you’re trying to solve these problems.
So I’m going to redraw triangle ABD and I’m going to redraw this other triangle here but I’m going to flip it so we’re going to have A is going to correspond to angle D, so I’m going to write D there. Angle B is going to angle C and our last vertex is angle A, so let’s mark in what we know. Well we see that this angle A is a combination of these two congruency marks and angle D over here is a combination of 2 which means angle D and A must be congruent here which is going to be the same thing for these two other angles. These two angles must also be congruent.
But angle-angle is not enough to say that two triangles are congruent, so we need one more piece of information. And if you look at this we have line segment AD and line segment AD, that’s going to be congruent to itself. So we see now that we can use our angle-side-angle short cut to say that these two triangles are congruent which will then allow us to show AB and CD are congruent.
So let’s go over and start by showing angle-side-angle in our statement and reason. So our first statement will be two angles are congruent and those are the ones that are given. So I’m going to say just say angle BDA is congruent to angle CAD and our reason, it’s part of the given. We said angle-side-angle is our short-cut, so let’s show the two sides.
Side AD this base right here is going to be congruent to itself. Whenever you say is congruent to itself that’s reflexive property. And our third statement we’ve got our two angles, we have our sides we need to show the other pair of angles and that’s going to be angle BAC congruent to angle BDC and our reason there is given. So we now have enough information to say that two triangles must be congruent, which is our fourth statement. And I’m going to say that triangle ABD, so we’re using this larger triangle is congruent to triangle, and I can make it better congruence mark than that, is congruent to, well what’s going to correspond to angle A?
To do that we’re going to go back to our drawings and I’m going to say that in this triangle, A corresponds to D. So when I go back over here and say that A and D must correspond, but what correspond to B? So let’s go back to our drawing and I see that angle B and angle C correspond. So B corresponds to C and our last 2 D corresponds to A, so we have triangle ABD corresponding to triangle DCA and our reason we said was angle-side-angle.
But we’re not done because we haven’t shown that these two sides are congruent so that will be our fifth and final statement; that line segment AB is congruent to line segment CD and our reason is that if these two triangles are congruent, matching parts must also be congruent. And we said that CPCTC, Corresponding Parts of Congruent Triangle are Congruent.
So the two key steps here. The first key step was marking your diagram, separating the triangles and then seeing that AD has to be congruent to itself by the reflexive property.