CONCEPT PREVIEW In each figure, find the measures of the numbered...

Question

CONCEPT PREVIEW In each figure, find the measures of the numbered angles, given that lines m and n are parallel. ​​

Accepted Answer

Hello, everyone. We are given a figure. And in the figure, the lines P and Q are parallel, we want to find the angles denoted by the numerals. So to begin with knowing P and Q are parallel is very important to us. So we can use certain other theorems. I am going to make a list of all of the numbers we need to solve for. We need to solve for angle one angle two, angle three, angle four, angle five, angle six, angle seven, angle eight, angle nine and angle 10. So I'm gonna mark what I know on my figure as I find it out. So starting up on line P, we have an angle that is 44 degrees. This angle appears to be a vertical angle with angle one, recalling that vertical angles are congruent angle, one would measure 44 degrees. So let's see, what else do I know from here? Um Not much yet. So let's see what else I can fill in. I'm going down to line Q angle six, angle six is supplementary with the angle 133 degrees. So this means that angle six plus degrees equals 100 80. So I can subtract 133 from 180 and find our measure of angle six and that measure is 47 degrees. So angle six is 47 degrees. I noticed that angle six and angle 10 are vertical angles. So if angle six is 47 degrees, angle 10 is also 47 degrees within that little area. The angle 133 degrees is a vertical angle with angle nine. So again, vertical angles are congruent. So angle nine would measure 133 degrees. So from here, let's see what else I can find. So knowing that P and Q are parallel, if I look at the line that goes from the bottom to the top and increases from left to right, this would mean the angle 133 degrees and angle four are supplementary and if they are supplementary, they add up to 100 80 degrees. And we actually have already done that subtraction. So angle four is also 47 degrees and looking at vertical angles, angle three is a vertical angle with angle four. So angle three is also 47 degrees continuing on there. I notice that angle one, angle two and angle three form a straight line, meaning they will add up to 180 degrees. So to find the measure of angle two, I can do 180 degrees minus 44 degrees and I get 136 minus 47 degrees, which leaves me with 89 degrees. So that means angle two must be 89 degrees. And I notice angle five and angle two are vertical angles. So angle five is also 89 degrees. So the angles we're left to solve for are seven and eight which are vertical angles. So when we find one, we'll find the other. So looking at the line that decreases from left to right. As a transversal cutting our parallel lines, P and Q I can say that the degree 44 degree angle and angle seven are alternate interior angles and that alternate interior angles are congruent. So angle seven is 44 degrees an angle seven and angle eight are vertical angles. So angle eight would also be 44 degrees. So in the end, I have angle one is 44 degrees. Angle two is 89 degrees. Angle three is 47 degrees. Angle four is 47 degrees. Angle five is 89 degrees. Angle six is 47 degrees. Angle seven is 44 degrees. Angle eight is 44 degrees. Angle nine is 133 degrees. Angle 10 is 47 degrees. Have a nice day.

CONCEPT PREVIEW In each figure, find the measures of the numbered angles, given that lines m and n are parallel.

Key Concepts

Parallel Lines and Transversals

Angle Relationships

Angle Notation and Measurement

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