### Video Transcript

The triangles π΄π΅πΆ and πΆπ·πΈ are similar. Determine which of the following is equivalent to π΄π΅ over πΈπ·. Circle your answer.

If two triangles are similar, then one is just an enlargement of the other. And it means that corresponding pairs of angles in the two triangles β so thatβs angles which are in the same position in the two triangles β are equal.

To find the ratio which is equivalent to π΄π΅ over πΈπ·, we need to work out which side on the smaller triangle is corresponding with π΄π΅ and which side on the larger triangle is corresponding with πΈπ·. We can do this by looking at the angles of the two triangles.

First, we note that angle π΄πΆπ΅ is equal to angle π·πΆπΈ, as theyβre vertically opposite angles. These angles formed by the intersection of two straight lines. And we know that vertically opposite angles are equal.

Next, we see that the lines π΄π΅ and πΈπ· are parallel, which is whatβs indicated by these arrows here. This means that the angles π΄π΅πΆ and π·πΈπΆ are alternate angles in parallel lines. Theyβre enclosed within a Z-shape. Alternate angles are equal, so we have that angle π΄π΅πΆ equals angle π·πΈπΆ.

Next, we can deduce that angle πΆπ΄π΅ and angle πΆπ·πΈ are also equal. And this is because the angle sum in a triangle is always 180 degrees. So if the other two angles between these triangles are equal, then the third angle must also be equal in order to make the sum 180 degrees.

Now that weβve identified which angles in the two triangles correspond to one another, we can work out which sides correspond to each other by looking at their relative position between the angles. The side π΄π΅ is between the angles marked in pink and orange. So the side that corresponds to this on the other triangle is the side πΈπ·. This means that the ratio weβve been given in the question of π΄π΅ over πΈπ· is actually just the ratio of corresponding sides in these triangles, which means we need to look at which of the four options is actually a ratio of corresponding sides.

The side π΄πΆ is between the pink and green angles. So on the other triangle, that corresponds to the side πΆπ·. So the final pair of corresponding sides between the two triangles, which in each case is between the orange and green angles, is π΅πΆ and πΆπΈ.

If we look at the four options, we see straight away that π΅πΆ over πΆπΈ uses a pair of corresponding sides. And so this will be equivalent to π΄π΅ over πΈπ·. The other three ratios do not use pairs of corresponding sides. For example, if we look at π΅π΄ over πΆπ·, this uses a pink side on the larger triangle but a green side on the smaller triangle. So none of these three ratios are equivalent to π΄π΅ over πΈπ·. Our answer then is π΅πΆ over πΆπΈ.