WEBVTT
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Which inequality is satisfied by this figure?
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(A) The measure of angle π΅ is less than the measure of angle π΅π΄πΆ, which is less than the measure of angle πΆ.
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(B) The measure of angle π·π΄πΆ is less than the measure of angle π΅, which is less than the measure of angle πΆ.
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(C) The measure of angle π΅π΄πΆ is less than the measure of angle πΆ, which is less than the measure of angle π·π΄πΆ.
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Or (D) the measure of angle π·π΄πΆ is less than the measure of angle π΅, which is less than the measure of angle π΅π΄πΆ.
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And finally (E) the measure of angle πΆ is less than the measure of angle π΅, which is less than the measure of angle π΅π΄πΆ.
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For us to be able to order these angles, weβll need to find the measures of a few of the missing angles.
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Currently, we donβt know the measure of angle πΆ or the measure of angle π΅π΄πΆ.
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We should see that angle π΅π΄πΆ and angle π·π΄πΆ make a straight line.
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If these two angles together make a straight line, they are supplementary angles and theyβll add together to be 180 degrees.
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If we plug in the measure for angle π·π΄πΆ, which we know is 92 degrees, then the measure of angle π΅π΄πΆ plus 92 degrees equals 180 degrees.
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And if we subtract 92 degrees from both sides of this equation, then we find that the measure of angle π΅π΄πΆ equals 88 degrees.
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We can add that to our figure.
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And from there, we recognize that we have triangle π΄π΅πΆ.
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And in a triangle, the three angles must add up to 180 degrees.
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So we say the measure of angle π΅ plus the measure of angle π΅π΄πΆ plus the measure of angle πΆ must equal 180 degrees.
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Angle π΅ is 52 degrees, angle π΅π΄πΆ is 88 degrees, and we want to find angle πΆ.
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If we add 52 plus 88, we get 140 degrees.
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To find the measure of angle πΆ, we then need to subtract 140 degrees from both sides of our equation to show that the measure of angle πΆ is 40 degrees.
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And then we can add that back to our figure.
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What we can do now is list the angles that we know in order from least to greatest.
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Our smallest angle is angle πΆ, which measures 40 degrees, followed by the measure of angle π΅, which is 52 degrees, followed by the measure of angle π΅π΄πΆ, which is 88 degrees.
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And the largest of the angles we see in this figure is angle π·π΄πΆ, which is 92 degrees.
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Using this compound inequality, we can see which of the answer choices is true.
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And only option (E) list the angles in correct order, which says the measure of angle πΆ is less than the measure of angle π΅, which is less than the measure of angle π΅π΄πΆ.