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Question Video: Completing the Proof for Vertically Opposite Angles Mathematics

Two straight lines, 𝐴𝐡 and 𝐢𝐷, intersect at point 𝐸. Fill in the blank: If the angles ∠𝐴𝐸𝐷 and ∠𝐴𝐸𝐢 are adjacent angles where the ray 𝐸𝐢 βˆͺ the ray 𝐸𝐷 = the line segment 𝐢𝐷, then π‘šβˆ π΄πΈπΆ + π‘šβˆ π΄πΈπ· = οΌΏ. Fill in the blank: If the angles ∠𝐴𝐸𝐢 and ∠𝐢𝐸𝐡 are adjacent angles where the ray 𝐸𝐴 βˆͺ the ray 𝐸𝐡 = the line segment 𝐴𝐡, then π‘šβˆ π΄πΈπΆ + π‘šβˆ πΆπΈπ΅ = οΌΏ. True or False: We deduce from the two parts above that π‘šβˆ π΄πΈπ· = π‘šβˆ πΆπΈπ΅.

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Video Transcript

Two straight lines, 𝐴𝐡 and 𝐢𝐷, intersect at point 𝐸. Fill in the blank. If the angles 𝐴𝐸𝐷 and 𝐴𝐸𝐢 are adjacent angles, where the union of rays 𝐸𝐢 and 𝐸𝐷 equals the line segment 𝐢𝐷, then the measure of angle 𝐴𝐸𝐢 plus the measure of angle 𝐴𝐸𝐷 equals what. Fill in the blank. If the angles 𝐴𝐸𝐢 and 𝐢𝐸𝐡 are adjacent angles, where the union of rays 𝐸𝐴 and 𝐸𝐡 equals the line segment 𝐴𝐡, then the measure of angle 𝐴𝐸𝐢 plus the measure of angle 𝐢𝐸𝐡 equals what. True or False: We deduce from the two parts above that the measure of angle 𝐴𝐸𝐷 equals the measure of angle 𝐢𝐸𝐡.

Let’s begin this question by drawing the two given lines, 𝐴𝐡 and 𝐢𝐷, which intersect at a point 𝐸. Notice that we could have drawn any different diagram of the lines 𝐴𝐡 and 𝐢𝐷 that intersect at point 𝐸, so long as it shows that important line and intersection information. We would still be able to use any such diagram to answer the questions.

So let’s use the first diagram and look at the first part of this question. Here, we need to first identify the angles 𝐴𝐸𝐷 and 𝐴𝐸𝐢. The second part of this sentence, which tells us that the union of rays 𝐸𝐢 and 𝐸𝐷 is the line segment 𝐢𝐷, is really stating the fact that these lines form one straight-line segment. And what do we know about the angles on a straight line? Well, the angle measures on a straight line sum to 180 degrees. And so, the measure of angle 𝐴𝐸𝐷 plus the measure of angle 𝐴𝐸𝐢 is equal to 180 degrees. And that’s the first missing blank completed.

Let’s look at the second part of the question. This time, we’re looking at the angles 𝐴𝐸𝐢 and 𝐢𝐸𝐡. Once again, we’re told that the rays 𝐸𝐴 and 𝐸𝐡 form one straight-line segment, 𝐴𝐡. And we know that the angle measures on a straight line sum to 180 degrees. So these two angle measures of 𝐴𝐸𝐢 and 𝐢𝐸𝐡 will also sum to 180 degrees. So now we have answered the second part of this question.

Let’s look at the final part. In this part, we are considering the angle measures of 𝐴𝐸𝐷 and 𝐢𝐸𝐡. To help us with this, we’ll consider what we discovered in parts one and two. In the first part, we recognized that the measures of angles 𝐴𝐸𝐢 and 𝐴𝐸𝐷 added to give 180 degrees. Let’s label the measure of angle 𝐴𝐸𝐷 as π‘₯ degrees and the measure of angle 𝐴𝐸𝐢 as 𝑦 degrees. In the second part of the question, we identified another pair of angle measures that added to 180 degrees. And since π‘₯ degrees plus 𝑦 degrees equals 180 degrees, then we can say that the measure of angle 𝐢𝐸𝐡 is also π‘₯ degrees.

So, we can say that the statement that the measure of angle 𝐴𝐸𝐷 equals the measure of angle 𝐢𝐸𝐡 is true. And in fact, what we have here is a proof that vertically opposite angles are equal. We could even have continued in this example to demonstrate that the measure of angle 𝐴𝐸𝐢 equals the measure of angle 𝐷𝐸𝐡. These vertically opposite angles will both be 𝑦 degrees.

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