 In this video, we're going to use what we know about arombus, all the properties that we just talked about, and answer some questions. So in number one, we are looking at the measure of angle HRM to be 82 degrees. So that's this angle here, this is 82 degrees. And we want to find each of these other angle measures. So the first one, letter A, is the measure of angle MOH. And so because we know that opposite angles are congruent in arombus, if angle HRM is 82 degrees, then also angle MOH has to be 82 degrees, because that's this angle here. The next one, angle RHO, so angle RHO is this angle up here. And because we know that consecutive angles are supplementary, that means that if we just take 180 degrees and subtract the 82 degrees that's down here, then we will figure out what this angle is. So 180 minus 82 is 98 degrees. Going on to find the measure of angle 3. Measure of angle 3, we can figure out, because we just learned that in arombus, the diagonals bisect the opposite angles. So that means 82 degrees is being bisected by this diagonal. So I can find the measure of angle 3 by taking 82 degrees and dividing it by 2. And we get 41 degrees. The measure of angle 4 has to be equal to the measure of angle 3 because, again, angle O is being bisected. And because we know that both of these angles are 82 degrees, the same reasoning applies for angle 4. We would take 82 divided by 2 and we would get 41 degrees. The measure of angle 6, you have to make sure that you know that arombus has perpendicular diagonals. So as soon as we see the diagonals here, we know that 90 degree angles have been formed in the middle. So that means that angle 6 has to be 90 degrees. And then lastly, the measure of angle 1, we have to go back to what we found here. The measure of angle Ro was 98 degrees. And because its angle Ro is bisected by the diagonal, again, we would just take 98 and divide by 2 to get the measure of angle 1. And so that would equal 49 degrees. In this example, we are given the measure of angle 1 is 3x plus 10. The measure of angle 5 is 5x minus 20. So the first thing we need to do is find the value of x. Well because we know that the opposite angles are congruent and we know that the opposite angles have been bisected, that means that angles 1 and 5 have to be equal. So I'm going to set up an equation 3x plus 10 has to equal 5x minus 20. I'm going to solve this for x. So subtract 3x on both sides. We get 10 equals 2x minus 20. I'm going to add 20 to both sides so we get 30 equals 2x. And when we divide, we end up with the answer of x equals 15. Now the other part of this problem asks us to figure out the measure of angle 2. And so again, we come back to our picture here and we see that angles 1 and 2 have to be congruent. And because we know that the angle 1 is equal to 3x plus 10, I can go ahead and find the measure of angle 2 by using the same 3x plus 10, but I'm going to plug 15 in for x. So 3 times 15 is 45 and if we add 10, we get 55 degrees. In this third example, it tells us that the measure of angle 6 is 5x minus 15 and we want to find the value of x. Well, in order to do this problem, you have to remember the property that says in a rhombus diagonals bisect, I'm sorry, not bisect, diagonals are perpendicular to one another. And so therefore, we know that all of these angles here in the middle have to be 90 degrees. And so to find x, I'm going to take the 5x minus 15 and set it equal to 90. Add 15 to both sides. We get 5x equals 105. And when we divide by the 5, we get x equals 21.