# Question Video: Completing the Proof for Vertically Opposite Angles Mathematics

Two straight lines, π΄π΅ and πΆπ·, intersect at point πΈ. Fill in the blank: If the angles β π΄πΈπ· and β π΄πΈπΆ are adjacent angles where the ray πΈπΆ βͺ the ray πΈπ· = the line segment πΆπ·, then πβ π΄πΈπΆ + πβ π΄πΈπ· = οΌΏ. Fill in the blank: If the angles β π΄πΈπΆ and β πΆπΈπ΅ are adjacent angles where the ray πΈπ΄ βͺ the ray πΈπ΅ = the line segment π΄π΅, then πβ π΄πΈπΆ + πβ πΆπΈπ΅ = οΌΏ. True or False: We deduce from the two parts above that πβ π΄πΈπ· = πβ πΆπΈπ΅.

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### Video Transcript

Two straight lines, π΄π΅ and πΆπ·, intersect at point πΈ. Fill in the blank. If the angles π΄πΈπ· and π΄πΈπΆ are adjacent angles, where the union of rays πΈπΆ and πΈπ· equals the line segment πΆπ·, then the measure of angle π΄πΈπΆ plus the measure of angle π΄πΈπ· equals what. Fill in the blank. If the angles π΄πΈπΆ and πΆπΈπ΅ are adjacent angles, where the union of rays πΈπ΄ and πΈπ΅ equals the line segment π΄π΅, then the measure of angle π΄πΈπΆ plus the measure of angle πΆπΈπ΅ equals what. True or False: We deduce from the two parts above that the measure of angle π΄πΈπ· equals the measure of angle πΆπΈπ΅.

Letβs begin this question by drawing the two given lines, π΄π΅ and πΆπ·, which intersect at a point πΈ. Notice that we could have drawn any different diagram of the lines π΄π΅ and πΆπ· that intersect at point πΈ, 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 letβs use the first diagram and look at the first part of this question. Here, we need to first identify the angles π΄πΈπ· and π΄πΈπΆ. The second part of this sentence, which tells us that the union of rays πΈπΆ and πΈπ· is the line segment πΆπ·, 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 π΄πΈπ· plus the measure of angle π΄πΈπΆ is equal to 180 degrees. And thatβs the first missing blank completed.

Letβs look at the second part of the question. This time, weβre looking at the angles π΄πΈπΆ and πΆπΈπ΅. Once again, weβre told that the rays πΈπ΄ and πΈπ΅ form one straight-line segment, π΄π΅. And we know that the angle measures on a straight line sum to 180 degrees. So these two angle measures of π΄πΈπΆ and πΆπΈπ΅ will also sum to 180 degrees. So now we have answered the second part of this question.

Letβs look at the final part. In this part, we are considering the angle measures of π΄πΈπ· and πΆπΈπ΅. To help us with this, weβll consider what we discovered in parts one and two. In the first part, we recognized that the measures of angles π΄πΈπΆ and π΄πΈπ· added to give 180 degrees. Letβs label the measure of angle π΄πΈπ· as π₯ degrees and the measure of angle π΄πΈπΆ as π¦ 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 π¦ degrees equals 180 degrees, then we can say that the measure of angle πΆπΈπ΅ is also π₯ degrees.

So, we can say that the statement that the measure of angle π΄πΈπ· equals the measure of angle πΆπΈπ΅ 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 π΄πΈπΆ equals the measure of angle π·πΈπ΅. These vertically opposite angles will both be π¦ degrees.