Question

Given: SQ and TQ trident angle PQR

measure of angle PQT = 7x+6
measure of angle TQS = -2x +8y-13
measure of angle RQS = 5x +2y
find measure of angle PQR

Answer 1

`10x+10y-7`

Explanation:

We just need to add up all of the angles, PQT, TQS, and RQS to find the measure of angle PQR.

`PQT=7x+6\\TQS=-2x+8y-13\\RQS=5x+2y\\PQT+TQS+RQS=PQR\\7x+6+TQS+RQS=PQR\\7x+6+(-2x+8y-13)+RQS=PQR\\7x+6+(-2x+8y-13)+5x+2y=PQR\\7x+6-2x+8y-13+5x+2y=PQR\\7x-2x+5x+8y+2y+6-13=PQR\\5x+5x+8y+2y+6-13=PQR\\10x+8y+2y+6-13=PQR\\10x+10y+6-13=PQR\\10x+10y-7=PQR`

Answer 2

9514 1404 393

Answer:

123°

Explanation:

Each of the parts of the trisected angle is equal to the others.

∠PQT = ∠TQS

7x +6 = -2x +8y -13

9x -8y +19 = 0 . . . . . . subtract the right-side expression

∠TQS = ∠RQS

-2x +8y -13 = 5x +2y

7x -6y +13 = 0 . . . . . . subtract the left-side expression

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These equations can be solved by any of your favorite methods to give ...

(x, y) = (5, 8)

Then the angle measures are ...

∠PQT = 7x +6 = 7(5) +6 = 41

∠TQS = -2x +8y -13 = -2(5) +8(8) -13 = 64 -23 = 41

∠RQS = 5x +2y = 5(5) +2(8) = 41

The measure of angle PQR is 3·41° = 123°.

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Additional comment

Using Cramer's rule, the solution is ...

x = (-8(13) -(-6)(19))/(9(-6) -7(-8)) = 10/2 = 5

y = (19(7) -13(9))/2 = 16/2 = 8

I prefer a method like this, or the graphical solution, when the numbers don't lend themselves to substitution or elimination.

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Obviously, once we have found angle PQT, we can simply multiply it by 3 to find angle PQR. We chose to compute the values of the other angles as a check on our math.