Video Transcript
The triangle in the given figure is constructed in the following way. π is the midpoint of line segment π΄π΅, line segment ππ is the line parallel to line segment π΅πΆ, and line ππ is the line parallel to line segment π΄πΆ.
Weβre then given six questions regarding angles, lengths, and the congruency of triangles. Before we go through these questions, letβs have a look at this diagram. First of all, weβre told that π is the midpoint of this line segment π΄π΅. We have our first pair of parallel lines ππ and π΅πΆ. And our second pair of parallel lines are ππ and π΄πΆ.
So letβs take a look at our first question, which asks, what do we know about the measures of angles π΄ππ and π΄π΅πΆ?
We can mark these two angles onto the diagram. We can remember that ππ and π΅πΆ are parallel and π΄π΅ is a transversal of these two lines. This, therefore, means that angle π΄ππ and angle π΄π΅πΆ 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 π΄ππ and πππ΅?
Letβs have a look at these angles on the diagram. Notice that we have a pair of parallel lines π΄πΆ and ππ and the transversal ππ, which means that angle π΄ππ will be alternate and, therefore, equal to the angle πππ. When we then consider our other pair of parallel lines ππ and π΅πΆ and a transversal ππ, then we know that angle πππ will also be alternate with the angle πππ΅. We can then give our answer for the second question that angles π΄ππ and πππ΅ are equal.
The third question asks, what do we know about the lengths of line segment π΄π and line segment π΅π?
We can look at these two lengths on the diagram. And we can recall that we were told that π is the midpoint of line segment π΄π΅. This means that line segment π΄π and line segment π΅π must be the same length. We can then give our answer to this question that they are equal.
The fourth question asks, are triangles π΄ππ and ππ΅π congruent? If yes, state by which congruence criteria.
Letβs look at these two triangles on the diagram. We have triangle π΄ππ on the top and triangle ππ΅π underneath. Weβre asked if these two triangles are congruent, which means the same shape and size. In order to prove congruency in triangles, thereβs a number of different criteria which involve pairs of corresponding angles and pairs of corresponding sides.
Letβs review what we found out in the first three questions. We saw our first pair of congruent angles. Angle π΄ππ is equal to angle π΄π΅πΆ. The second pair of equal angles are angle π΄ππ and angle πππ΅. And we also showed that thereβs a pair of equal lengths. The line segment π΄π and the line segment π΅π are equal in length. Weβve therefore demonstrated that thereβs two pairs of corresponding angles equal and a pair of corresponding sides equal. Our answer for the fourth question would therefore be yes, theyβre congruent by the angle-angle-side or AAS criteria. Note that we couldnβt use the ASA criteria as the side isnβt included between the two angles.
The next question asks, therefore, what can be said of the lengths of line segments π΄π, ππ, ππ, and π΅π?
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 π΄π, as this has a congruent triangle, then we know that, in triangle ππ΅π, there will be a corresponding length. It will in fact be the line segment ππ. As corresponding lengths will be equal, weβve found a relationship between two of the line segments that we were asked for.
Letβs have a look at the third line segment ππ. The corresponding length in triangle ππ΅π will be the length π΅π. This was the other length we were asked for in the question. So we can give our answer that π΄π equals ππ and ππ equals π΅π.
The final question here asks, since πππΆπ is a parallelogram, what can be said of the points π and π?
Letβs have a look at this parallelogram on the diagram. When weβre answering questions like this, weβll need to apply what we know about parallelograms. But weβll also need to remember the information that weβve 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 ππ and the line ππΆ are parallel. But because weβre told theyβre part of a parallelogram, we now know that theyβre of the same length. Are there any other lengths that will be the same length as the line segment ππ?
We can remember that π΄ππ is a congruent triangle with ππ΅π. So the corresponding length will be π΅π in the triangle ππ΅π. So now we know that we have three congruent lengths, how does this help us answer the question about points π and π?
Well, if we look at the point π, we can see that weβve got two congruent line segments on either side. As π΅π is equal to ππΆ, this means that π must be a midpoint of the line segment π΅πΆ. Letβs see if we can prove that π would also be a midpoint. Starting with the parallelogram, we know that weβve got another pair of parallel and congruent sides, the line segment ππΆ and the line segment ππ.
Using congruent triangles, we know that this length of ππ in triangle ππ΅π will be congruent with this length of π΄π in triangle π΄ππ. So once again, if we look at this point π, we can see that it lies in between two congruent line segments. We can, therefore, give our answer to this final question part. π and π are midpoints because π΄π equals ππΆ and π΅π equals ππΆ.