# Question Video: Finding the Measure of an Angle Using the Vertically Opposite and Supplementary Anglesβ Relations Mathematics • 7th Grade

Find πβ π΄ππ΅, πβ π΄ππΈ, and πβ π·ππΈ.

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### Video Transcript

Find the measure of angle π΄ππ΅, the measure of angle π΄ππΈ, and the measure of angle π·ππΈ.

Here weβve went ahead and labeled where the angles are placed on our diagram. Letβs go ahead and start solving for them. Here we can see our two green angles are vertical. This is angle π·ππ΅ and itβs vertical with the angle that we need, π΄ππΈ.

And we know that vertical angles are congruent, meaning they have the same measure. So if we could find the measure of angle π·ππ΅, we would know the measure of angle π΄ππΈ. So in order to find angle π·ππ΅, we need to add sixty-seven plus seventy-five, which gives us one hundred and forty-two degrees. That means the measure of angle π΄ππΈ is also one hundred and forty-two degrees.

Again, the measure of angle π΄ππΈ is one hundred and forty-two degrees. Now the way that we know that is because we have two intersecting lines, which would be line π·π΄, and it intersects with line πΈπ΅. And since we have these two intersecting lines, angle π·ππ΅ and angle π΄ππΈ are vertical, and- which means they are congruent.

Letβs go ahead and kind of clean up our diagram. Now letβs try to find our other two angles. And actually it turns out that angle π·ππΈ, the pink one, and angle π΄ππ΅, the orange one, those are also vertical. So the two remaining angles that weβre looking for are vertical, which means they should be congruent.

The two angles at the bottom of our diagram make a straight line, which means theyβre next to each other and they connect and make one hundred and eighty degrees; theyβre supplementary, which means the measure of angle π·ππΈ plus the measure of angle π΄ππΈ should equal one hundred and eighty degrees, so we can solve. And we will be able to find the measure of angle π·ππΈ by first substituting one hundred and forty-two degrees in for the measure of angle π΄ππΈ.

Now to solve for our angle, we need to subtract one hundred and forty-two from both sides of the equation. And we get the measure of angle π·ππΈ equals thirty-eight degrees.

As we stated before, the measure of angle π·ππΈ and measure of angle π΄ππ΅ should be the same since theyβre vertical, which means the measure of angle π΄ππ΅ should also be thirty-eight degrees.

Now we can verify this using the fact that the measure of angle π΄ππ΅ plus the measure of angle π΄ππΈ are also supplementary. Again, π΄ππ΅ and π΄ππΈ, these angles should add to one hundred and eighty degrees because theyβre supplementary, so we can verify that this is true. Thirty-eight plus one hundred and forty-two equals one hundred and eighty, which means we had a true statement, so they are supplementary.

So to put this all together, the measure of angle π΄ππ΅ is thirty-eight degrees, the measure of angle π΄ππΈ is one hundred and forty-two degrees, and the measure of angle π·ππΈ is thirty-eight degrees.