Question Video: Finding the Measure of an Angle in a Triangle Using the Similarity of Triangles | Nagwa Question Video: Finding the Measure of an Angle in a Triangle Using the Similarity of Triangles | Nagwa

# Question Video: Finding the Measure of an Angle in a Triangle Using the Similarity of Triangles Mathematics • Second Year of Preparatory School

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Determine the length of line segment π΄πΆ

04:48

### Video Transcript

Determine the length of line segment π΄πΆ.

In the figure, we have two triangles: π₯π¦π§ on the left and π΄π΅πΆ on the right. We are asked to find the length of line segment π΄πΆ, which appears in triangle π΄π΅πΆ.

Now we might notice that we only have the length of one other side in this triangle, which wonβt be enough on its own to work out line segment π΄πΆ. We might therefore wonder if the two triangles are similar. We can recall that similar triangles have corresponding angles congruent and corresponding sides in proportion. So if the triangles are similar, this would give us a way to work out any unknown side lengths.

If we want to prove that two triangles are similar, we can either demonstrate that all corresponding angles are congruent or that all pairs of corresponding sides are in proportion. If we look at the triangles, we only have one pair of sides in the triangles given. That wouldnβt be enough to prove they are similar. So letβs consider the angles.

We can observe that the measures of both angles π§ and πΆ are given as 61 degrees. We are given that the measure of angle π΄ is 59 degrees, but we arenβt given the measure of angle π₯. However, letβs see if we can calculate it. We know that the angle measures in a triangle sum to 180 degrees. So the measure of angle π₯ will be 180 degrees subtract the sum of 61 degrees and 60 degrees, as these are the two remaining angles in the triangle. This gives us 59 degrees. So the measure of angle π₯ is equal to the measure of angle π΄.

Next, letβs consider the measure of angle π¦, which is given as 60 degrees. Its corresponding angle in triangle π΄π΅πΆ will be angle π΅. And the measure of angle π΅ is equal to 180 degrees subtract the sum of 59 degrees and 61 degrees, which is 60 degrees. And so the measures of angle π¦ and π΅ are equal.

We have therefore demonstrated that we have three pairs of corresponding angles congruent, so the triangles are similar. Writing this similarity relationship carefully, remembering to make sure that the corresponding vertices are given in the correct place in the similarity relationship, we can say that triangle π₯π¦π§ is similar to triangle π΄π΅πΆ. We can now use this to help us work out the unknown side length of line segment π΄πΆ.

We were given the lengths of two corresponding sides: π§π¦ and πΆπ΅. We could write the proportion of the sides as πΆπ΅ over π§π¦. Now the side π₯π§ is corresponding to the side π΄πΆ that we need to calculate. And because the triangles are similar, their sides are in proportion, so πΆπ΅ over π§π¦ equals π΄πΆ over π₯π§. We can then fill in the measurements for the lengths. So we have 22.8 over 12 equals π΄πΆ over 12.1.

We can simplify the left-hand side first to 11.4 over six and then multiply both sides by 12.1, which gives us the answer that the length of line segment π΄πΆ is 22.99 centimeters.

For a good, approximate check on the answer, we can consider the sides. In triangle π₯π¦π§, sides π§π¦ and π₯π§ are 12 and 12.1 centimeters, respectively, which are quite close in value, and side π₯π§ is slightly longer. We would expect that sides πΆπ΅ and π΄πΆ would follow the same pattern. That is, they are quite close in value with the side π΄πΆ being slightly longer. And 22.99 centimeters would seem like an appropriately sized value. So π΄πΆ is 22.99 centimeters.

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