### Video Transcript

Determine the lengths of π΄π and π΅π.

Within this diagram, we can see that we have two triangles: triangles π΄π΅πΆ and π΄ππ, each with a mixture of sides and angles marked. We can see that the angles in the two triangles are the same; they both have an angle of 104 degrees and an angle of 39 degrees, which means the third angle must also be the same as the angles sum in a triangle is always 180 degrees.

We now consider whether these two triangles are congruent to each other. As if they are, this will help us with calculating the length of π΄π and π΅π. Letβs start by listing what we know. The two angles of 104 degrees are the same, so angle π΅πΆπ΄ is equal to angle ππ΄π. The A in brackets is used to indicate a statement about equal angles. Next, letβs look at the two sides that have been marked as 37 centimetres: sides πΆπ΄ and π΄π. So we have the statement πΆπ΄ is equal to π΄π and the S in brackets indicates that this is a statement about a side.

Now letβs look at the angles of 39 degrees: angles πΆπ΄π and π΄ππ. We have the angle πΆπ΄π is equal to angle π΄ππ. And again the A in brackets indicates that this is a statement about angles. So looking at our three statements, we can see that we do have enough information to conclude that these two triangles are congruent to each other. The ASA tells us that this is the angle-side-angle congruence criteria that we can use. So we conclude that triangle π΅πΆπ΄ is congruent to triangle ππ΄π.

Now how does this help with us answering the question, which was to determine the length of π΄π and π΅π. Well, letβs look which pairs of sides correspond with each other between the two triangles. Side π΄π corresponds with side πΆπ΅; therefore, π΄π must be equal to 38 centimetres. The third side of each of the triangles must correspond with each other, so that side π΄π΅ and ππ.

We know that ππ is equal to 59 centimetres, and therefore π΄π΅ is also equal to 59 centimetres. However, the question didnβt ask us to find π΄π΅; it asked us to find π΄π, which weβve done and π΅π. In order to calculate π΅π, we can take the whole length of π΄π΅ and then subtract the length of π΄π, both of which we know. So π΅π is equal to 59 minus 38, which is 21. So we have our answer to the problem. π΄π is equal to 38 centimetres; π΅π is equal to 21 centimetres.