Question Video: Comparing the Lengths of Sides in a Triangle | Nagwa Question Video: Comparing the Lengths of Sides in a Triangle | Nagwa

Question Video: Comparing the Lengths of Sides in a Triangle Mathematics • Second Year of Preparatory School

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Consider the shown figure. Fill in the blank using >, <, or = : 𝐴𝐡 _ 𝐢𝐷.

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Video Transcript

Consider the shown figure. Fill in the blank using is greater than, is less than, or is equal to. 𝐴𝐡 what 𝐢𝐷.

In this question, we are given a figure containing information about various triangles. And we want to use this figure to compare two lengths in the figure: 𝐴𝐡 and 𝐢𝐷. To do this, we can start by noting that 𝐡𝐷 is the same length as 𝐢𝐷. If we then highlight 𝐴𝐡 on the diagram, we can see that 𝐴𝐡 and 𝐷𝐡 are both sides of triangle 𝐴𝐡𝐷. This means that we can compare the lengths of 𝐴𝐡 and 𝐢𝐷 by comparing the lengths of the sides in triangle 𝐴𝐡𝐷.

We can compare the side lengths in a triangle by using the side comparison theorem in triangles. This tells us that if one side in a triangle is opposite an angle of larger measure than another side, then it is longer. More formally, it tells us that if π‘₯𝑦𝑧 is a triangle and the measure of angle π‘₯ is greater than the measure of angle 𝑦, then 𝑦𝑧 is longer than π‘₯𝑧. Therefore, we can compare the lengths of these two sides by comparing the measures of the angles opposite them in triangle 𝐴𝐡𝐷. These are angles 𝐡𝐴𝐷 and 𝐴𝐷𝐡, as shown.

We can find the measure of angle 𝐴𝐷𝐡 by noting that it combines with angle 𝐡𝐷𝐢 to make a straight angle. So, their measures sum to give 180 degrees. Therefore, the measure of angle 𝐴𝐷𝐡 is equal to 180 degrees minus 92 degrees, which we can calculate is equal to 88 degrees. We can add this onto our diagram.

We now need to find the measure of angle 𝐡𝐴𝐷. To do this, we can note that sides 𝐢𝐷 and 𝐡𝐷 have the same length. So, triangle 𝐡𝐢𝐷 is an isosceles triangle. And we know that in an isosceles triangle, the angles opposite the congruent sides have equal measure. So, angles 𝐷𝐢𝐡 and 𝐷𝐡𝐢 have the same measure.

We can then use the fact that the sum of all of the measures of the internal angles in a triangle is 180 degrees to show that twice the measure of angle 𝐷𝐢𝐡 plus 92 degrees is equal to 180 degrees. We can then solve this equation for the measure of angle 𝐷𝐢𝐡. We subtract 92 degrees from both sides of the equation and divide through by two to get that the measure of angle 𝐷𝐢𝐡 is equal to 44 degrees. We can then add this angle measure onto the diagram to all of the angles congruent to angle 𝐷𝐢𝐡 as shown.

We now have two of the angle measures of triangle 𝐴𝐡𝐷. So, we can use the sum of the internal angle measures in a triangle to find the measure of the third angle. We have that the measure of angle 𝐡𝐴𝐷 is equal to 180 degrees minus 88 degrees minus 44 degrees. We can calculate that this is equal to 48 degrees. And we can add this onto the diagram as shown.

We can now apply the side comparison theorem in triangles to triangle 𝐴𝐡𝐷. We see that side 𝐴𝐡 is opposite an angle with larger measure than side 𝐷𝐡. So, 𝐴𝐡 must be longer than 𝐡𝐷. But remember, we are given in the diagram that 𝐡𝐷 and 𝐢𝐷 are the same length. So, we can also say that 𝐴𝐡 is greater than 𝐢𝐷, which is our final answer.

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