### Video Transcript

Find the measure of angle π΄ππ΅, the measure of angle π΄ππΈ, and the measure of angle π·ππΈ.

Here weβve went ahead and labeled where the angles are placed on our diagram. Letβs go ahead and start solving for them. Here we can see our two green angles are vertical. This is angle π·ππ΅ and itβs vertical with the angle that we need, π΄ππΈ.

And we know that vertical angles are congruent, meaning they have the same measure. So if we could find the measure of angle π·ππ΅, we would know the measure of angle π΄ππΈ. So in order to find angle π·ππ΅, 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΄ππΈ is also one hundred and forty-two degrees.

Again, the measure of angle π΄ππΈ 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 πΈπ΅. And since we have these two intersecting lines, angle π·ππ΅ and angle π΄ππΈ are vertical, and- which means they are congruent.

Letβs go ahead and kind of clean up our diagram. Now letβs try to find our other two angles. And actually it turns out that angle π·ππΈ, the pink one, and angle π΄ππ΅, the orange one, those are also vertical. So the two remaining angles that weβre 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 theyβre next to each other and they connect and make one hundred and eighty degrees; theyβre supplementary, which means the measure of angle π·ππΈ plus the measure of angle π΄ππΈ should equal one hundred and eighty degrees, so we can solve. And we will be able to find the measure of angle π·ππΈ by first substituting one hundred and forty-two degrees in for the measure of angle π΄ππΈ.

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·ππΈ equals thirty-eight degrees.

As we stated before, the measure of angle π·ππΈ and measure of angle π΄ππ΅ should be the same since theyβre vertical, which means the measure of angle π΄ππ΅ should also be thirty-eight degrees.

Now we can verify this using the fact that the measure of angle π΄ππ΅ plus the measure of angle π΄ππΈ are also supplementary. Again, π΄ππ΅ and π΄ππΈ, these angles should add to one hundred and eighty degrees because theyβre 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΄ππ΅ is thirty-eight degrees, the measure of angle π΄ππΈ is one hundred and forty-two degrees, and the measure of angle π·ππΈ is thirty-eight degrees.