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Please have your Chapter 10 Test out and ready for corrections. You may click through for explanations. Correct your work in a different color than what you wrote in on the test. Question 1 In this problem, you just needed to check your notes, if you had them with you. If not, you really should’ve known which of these were supplementary, because the rest were congruent. These are all congruent angles. These are supplementary angles. Question 2 AIA’s AEA’s Oh, wait, 2 and 8 were a pair in there? Question 3 What angles are in the same position? Top Right Top Left Bottom Right Top Right Top Left Bottom Left The same position! Top Right! Question 4 The easiest thing to do is actually put 49 degrees into the angle. Angle 6 is vertical with angle 8, so they’re congruent. 131ᵒ 49ᵒ 49ᵒ 131ᵒ 131ᵒ 49ᵒ 49ᵒ 131ᵒ 180-49= Angle 2 is corresponding with angle 6, so they’re congruent. Angle 4 is vertical with angle 4, so they’re congruent. Angle 7 is linear with angle 8, so they’re supplementary. Angle 5 is vertical with angle 7, so they’re congruent. Angle 1 is corresponding with angle 5, so they’re congruent. Angle 3 is vertical with angle 1, so they’re congruent. Question 5 13+26= 39° 3(3) = 39° What kinds of angles are these? (vertical angles) So are they congruent or supplementary? (congruent) Since they’re congruent, make both angles equal to each other: 3𝑥 = 𝑥 + 26 -x -x 2𝑥 = 26 2 2 𝑥 = 13 subtract x from both sides to move it all to one side divide by 2 to get x by itself Question 6 x = 30° 5(30) = 150° What kinds of angles are these? (linear angles) So are they congruent or supplementary? (supplementary) Since they’re supplementary, they add up to 180: 𝑥 + 5𝑥 = 180 6𝑥 = 180 6 6 𝑥 = 30 combine your x’s (your “like” terms) divide by 6 to get x by itself Question 7 45˚+45˚≠180˚ 50˚+50˚≠180˚ 90˚+90˚=180˚ This means 2 CONGRUENT angles. This means they also add up to 180˚. (They have the same measure.) Question 8 This means 2 CONGRUENT angles. (Their measures will be equal.) m<1=m<2 2𝑥 + 11 = 5𝑥 − 25 −2𝑥 −2𝑥 11 = 3𝑥 − 25 But before you go getting all +25 +25 circle happy and thinking it’s A… 36 = 3𝑥 3 3 𝒎 < 𝟐 = 𝟓𝒙 − 𝟐𝟓 12 = 𝑥 𝒎 < 𝟐 = 𝟓 𝟏𝟐 − 𝟐𝟓 𝒎 < 𝟐 = 𝟔𝟎 − 𝟐𝟓 𝒎 < 𝟐 = 𝟑𝟓˚ They want the actual angle measure, not what x is. move all the x’s to one side by subtraction Question 9 If this angle is 57°… Then this angle must also be 57° (because the two angles are AEA’s) …and that angle is 90° (see the right angle marking?)… And if that angle *is* 57°… Then this angle is 33°. And that angle is supplementary with x, so 33+x=180.... x =147 180-57-90=33 …and all together, they form a triangle, which has a total of 180°… Question 10 First of all… …your answer will be one of these. Again, not quite what we need. 2 −𝑦 = 12 − 𝑥 5 All we need to do to get this equation into y=mx+b form is change the sign of y. But what we do to one side of the equation, we do to everything on the other side of the equation. 2 𝑦 = −12 + 𝑥 5 𝒔𝒍𝒐𝒑𝒆 = 𝟐 𝟓 Not quite in y=mx+b form, but we can get there! 2𝑦 = −5𝑥 − 9 2 2 2 divide everything by 2 to get y by itself −5 9 𝑦= 2 𝑥−2 don’t reduce – slope is best left as a fraction! −𝟓 𝒔𝒍𝒐𝒑𝒆 = 𝟐 These slopes are negative reciprocals, so the lines are perpendicular. Question 11 First of all… …your answer will be one of these. Again, not quite what we need. −2𝑦 = −6𝑥 + 5 -2 -2 -2 divide by -2 to get y by itself 5 𝑦 = 3𝑥 − 2 𝒔𝒍𝒐𝒑𝒆 = 𝟑 Not quite in y=mx+b form, but we can get there! 3𝑦 = 𝑥 − 12 3 3 3 divide everything by 3 to get y by itself 1 𝑥 1𝑥 𝑦 = 3𝑥 − 4 do remember that x is the same as 1x so = 3 3 𝟏 𝒔𝒍𝒐𝒑𝒆 = 𝟑 These slopes are reciprocals, but not negative reciprocals, so the lines are just neither. Question 12 X +5 Y X How do you find the slope of two coordinates? 𝚫𝒚 𝚫𝒙 Y +5 or 𝒚𝟐 −𝒚𝟏 𝒙𝟐 −𝒙𝟏 𝒚𝟐 − 𝒚𝟏 −6 − 4 −6 − 4 −10 = = = = −2 𝒙𝟐 − 𝒙𝟏 3 − (−2) 3+2 5 (add 5 to your x coordinates) C’ (8, -6) D’ (3, 4) You could calculate this or remember what you learned about slopes after translating. 𝒚𝟐 − 𝒚𝟏 −6 − 4 −10 = = = −2 𝒙𝟐 − 𝒙𝟏 8−3 5 The lines are parallel, because the slopes are the same. If you made a simple mistake with your math, like thinking 3 – (-2) = -5 or 1 or -1, perhaps one of your best strategies for next time is to borrow a calculator. If you mixed up which is x and which is y, keep making a note of it on your notes, worksheets, etc. Question 14 This is the only option that we know for sure. Because the triangles have the same shape, and proportional sides, they’re similar. If they’d had lengths, we might have had a chance at choosing c or d. Are they congruent? The same size? nope.com Are they similar? They’re the same shape, not the same size… so maybe. Well… they could be. But there’s nothing to tell us they are FOR SURE. So we’ll have to say no. And like option C, without measurements, there’s nothing to tell us this FOR SURE. Question 15 X, Y P X, Y X, Y X, Y If you noticed the slopes were the same, you might’ve marked parallel lines. Q R S However, if you borrowed a scratch paper to graph the two lines, they’re actually like… the same line (otherwise known as… COINCIDENTAL LINES)!!! To figure out how the lines are related, we’ll have to find their slopes. −5 10 = −1 2 slope of QS: 𝒙𝟐−𝒙𝟏 = 10−(−2) = 10+2 = −6 12 = slope of PR: 𝒚𝟐 −𝒚𝟏 𝒙𝟐 −𝒙𝟏 𝒚 −𝒚 𝟐 𝟏 = 4−9 4−(−6) 1−7 = 4−9 4+6 = 1−7 The slopes of PR and QS are the same! −1 2 Question 16 If this angle is 65°… If this angle is 70°… 70° 45° 70° 65° 65° 65° 115° 70° Let’s do the same for our 45° angles. 45° 65° 65° 45° 70° 65° 45° 70° If this angle is 70°… 115° …then so are these. If this angle is 65°… 45° 65° 65° 45° 70° 180-65=115 …then so are these. …then so are these. 70° 110° 110° 70° That really only leaves a few angles. …then so are these. 180-70=110 BOOM!!! DONE. Remember that these angles are linear pairs with the angles they touch, which means they add up to 180°. Admittedly, there was a lot here. If you notice from the point value, you needed to find 23 angles. Start by labeling the angles you do know. Then, label their vertical angles. Remember, vertical angles are congruent. It comes down to corresponding angles after that (angles that are in the same position). Because of all the parallel lines, we know our corresponding angles are congruent. Please turn your corrected test in to your teacher and continue on to the next activity.