Video Transcript
The triangle in the given figure is constructed in the following way. 𝑀 is the midpoint of line segment 𝐴𝐵, line segment 𝑀𝑃 is the line parallel to line segment 𝐵𝐶, and line 𝑀𝑄 is the line parallel to line segment 𝐴𝐶.
We’re then given six questions regarding angles, lengths, and the congruency of triangles. Before we go through these questions, let’s have a look at this diagram. First of all, we’re told that 𝑀 is the midpoint of this line segment 𝐴𝐵. We have our first pair of parallel lines 𝑀𝑃 and 𝐵𝐶. And our second pair of parallel lines are 𝑀𝑄 and 𝐴𝐶.
So let’s take a look at our first question, which asks, what do we know about the measures of angles 𝐴𝑀𝑃 and 𝐴𝐵𝐶?
We can mark these two angles onto the diagram. We can remember that 𝑀𝑃 and 𝐵𝐶 are parallel and 𝐴𝐵 is a transversal of these two lines. This, therefore, means that angle 𝐴𝑀𝑃 and angle 𝐴𝐵𝐶 are corresponding, meaning that they will be the same size. So we can answer the first question that these two angles must be equal.
The second question asks, what do we know about the measures of angles 𝐴𝑃𝑀 and 𝑀𝑄𝐵?
Let’s have a look at these angles on the diagram. Notice that we have a pair of parallel lines 𝐴𝐶 and 𝑀𝑄 and the transversal 𝑀𝑃, which means that angle 𝐴𝑃𝑀 will be alternate and, therefore, equal to the angle 𝑃𝑀𝑄. When we then consider our other pair of parallel lines 𝑀𝑃 and 𝐵𝐶 and a transversal 𝑀𝑄, then we know that angle 𝑃𝑀𝑄 will also be alternate with the angle 𝑀𝑄𝐵. We can then give our answer for the second question that angles 𝐴𝑃𝑀 and 𝑀𝑄𝐵 are equal.
The third question asks, what do we know about the lengths of line segment 𝐴𝑀 and line segment 𝐵𝑀?
We can look at these two lengths on the diagram. And we can recall that we were told that 𝑀 is the midpoint of line segment 𝐴𝐵. This means that line segment 𝐴𝑀 and line segment 𝐵𝑀 must be the same length. We can then give our answer to this question that they are equal.
The fourth question asks, are triangles 𝐴𝑀𝑃 and 𝑀𝐵𝑄 congruent? If yes, state by which congruence criteria.
Let’s look at these two triangles on the diagram. We have triangle 𝐴𝑀𝑃 on the top and triangle 𝑀𝐵𝑄 underneath. We’re asked if these two triangles are congruent, which means the same shape and size. In order to prove congruency in triangles, there’s a number of different criteria which involve pairs of corresponding angles and pairs of corresponding sides.
Let’s review what we found out in the first three questions. We saw our first pair of congruent angles. Angle 𝐴𝑀𝑃 is equal to angle 𝐴𝐵𝐶. The second pair of equal angles are angle 𝐴𝑃𝑀 and angle 𝑀𝑄𝐵. And we also showed that there’s a pair of equal lengths. The line segment 𝐴𝑀 and the line segment 𝐵𝑀 are equal in length. We’ve therefore demonstrated that there’s two pairs of corresponding angles equal and a pair of corresponding sides equal. Our answer for the fourth question would therefore be yes, they’re congruent by the angle-angle-side or AAS criteria. Note that we couldn’t use the ASA criteria as the side isn’t included between the two angles.
The next question asks, therefore, what can be said of the lengths of line segments 𝐴𝑃, 𝑀𝑄, 𝑀𝑃, and 𝐵𝑄?
We can note where these line segments are on the diagram. And it might be useful here to apply what we know about the congruency of our two triangles. If we begin by looking at line segment 𝐴𝑃, as this has a congruent triangle, then we know that, in triangle 𝑀𝐵𝑄, there will be a corresponding length. It will in fact be the line segment 𝑀𝑄. As corresponding lengths will be equal, we’ve found a relationship between two of the line segments that we were asked for.
Let’s have a look at the third line segment 𝑀𝑃. The corresponding length in triangle 𝑀𝐵𝑄 will be the length 𝐵𝑄. This was the other length we were asked for in the question. So we can give our answer that 𝐴𝑃 equals 𝑀𝑄 and 𝑀𝑃 equals 𝐵𝑄.
The final question here asks, since 𝑀𝑃𝐶𝑄 is a parallelogram, what can be said of the points 𝑃 and 𝑄?
Let’s have a look at this parallelogram on the diagram. When we’re answering questions like this, we’ll need to apply what we know about parallelograms. But we’ll also need to remember the information that we’ve worked out in the previous parts of the question, for example, the fact that we have two congruent triangles in the diagram.
We can remember that in a parallelogram, opposite sides are parallel and congruent. We already know that the line 𝑀𝑃 and the line 𝑄𝐶 are parallel. But because we’re told they’re part of a parallelogram, we now know that they’re of the same length. Are there any other lengths that will be the same length as the line segment 𝑀𝑃?
We can remember that 𝐴𝑀𝑃 is a congruent triangle with 𝑀𝐵𝑄. So the corresponding length will be 𝐵𝑄 in the triangle 𝑀𝐵𝑄. So now we know that we have three congruent lengths, how does this help us answer the question about points 𝑃 and 𝑄?
Well, if we look at the point 𝑄, we can see that we’ve got two congruent line segments on either side. As 𝐵𝑄 is equal to 𝑄𝐶, this means that 𝑄 must be a midpoint of the line segment 𝐵𝐶. Let’s see if we can prove that 𝑃 would also be a midpoint. Starting with the parallelogram, we know that we’ve got another pair of parallel and congruent sides, the line segment 𝑃𝐶 and the line segment 𝑀𝑄.
Using congruent triangles, we know that this length of 𝑀𝑄 in triangle 𝑀𝐵𝑄 will be congruent with this length of 𝐴𝑃 in triangle 𝐴𝑀𝑃. So once again, if we look at this point 𝑃, we can see that it lies in between two congruent line segments. We can, therefore, give our answer to this final question part. 𝑃 and 𝑄 are midpoints because 𝐴𝑃 equals 𝑃𝐶 and 𝐵𝑄 equals 𝑄𝐶.
