Question Video: Finding the Measure of an Angle Using the Vertically Opposite and Supplementary Angles’ Relations Mathematics

Video Transcript

Find the measure of angle 𝐴𝑂𝐵, the measure of angle 𝐴𝑂𝐸, and the measure of angle 𝐷𝑂𝐸.

Here we’ve went ahead and labeled where the angles are placed on our diagram. Let’s go ahead and start solving for them. Here we can see our two green angles are vertical. This is angle 𝐷𝑂𝐵 and it’s vertical with the angle that we need, 𝐴𝑂𝐸.

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

Again, the measure of angle 𝐴𝑂𝐸 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 𝐷𝐴, and it intersects with line 𝐸𝐵. And since we have these two intersecting lines, angle 𝐷𝑂𝐵 and angle 𝐴𝑂𝐸 are vertical, and- which means they are congruent.

Let’s go ahead and kind of clean up our diagram. Now let’s try to find our other two angles. And actually it turns out that angle 𝐷𝑂𝐸, the pink one, and angle 𝐴𝑂𝐵, the orange one, those are also vertical. So the two remaining angles that we’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 they’re next to each other and they connect and make one hundred and eighty degrees; they’re supplementary, which means the measure of angle 𝐷𝑂𝐸 plus the measure of angle 𝐴𝑂𝐸 should equal one hundred and eighty degrees, so we can solve. And we will be able to find the measure of angle 𝐷𝑂𝐸 by first substituting one hundred and forty-two degrees in for the measure of angle 𝐴𝑂𝐸.

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 𝐷𝑂𝐸 equals thirty-eight degrees.

As we stated before, the measure of angle 𝐷𝑂𝐸 and measure of angle 𝐴𝑂𝐵 should be the same since they’re vertical, which means the measure of angle 𝐴𝑂𝐵 should also be thirty-eight degrees.

Now we can verify this using the fact that the measure of angle 𝐴𝑂𝐵 plus the measure of angle 𝐴𝑂𝐸 are also supplementary. Again, 𝐴𝑂𝐵 and 𝐴𝑂𝐸, these angles should add to one hundred and eighty degrees because they’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 𝐴𝑂𝐵 is thirty-eight degrees, the measure of angle 𝐴𝑂𝐸 is one hundred and forty-two degrees, and the measure of angle 𝐷𝑂𝐸 is thirty-eight degrees.