Question

#s 13 - 20 use the triangle angle sum theorem and your understanding of algebra to solve for x and use the resultant x value to state the value of angle a.
13)
14)
15)
16)
17)
18)
19)
20)
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Explanation:

Step1: Recall triangle - angle sum theorem

The sum of the interior angles of a triangle is 180°.

Step2: Set up an equation for problem 13

For the triangle with angles \(7x + 5\), \(5x+10\) and \(45^{\circ}\), we have the equation \((7x + 5)+(5x + 10)+45=180\).
First, combine like - terms: \(7x+5x+5 + 10+45=180\), which simplifies to \(12x+60 = 180\).
Then, subtract 60 from both sides: \(12x=180 - 60=120\).
Finally, divide both sides by 12: \(x = 10\).
Angle \(A=7x + 5=7\times10+5=75^{\circ}\).

Step3: Set up an equation for problem 14

For the triangle with angles \(3x-3\), \(6x + 8\) and \(90^{\circ}\), we have \((3x-3)+(6x + 8)+90=180\).
Combine like - terms: \(3x+6x-3 + 8+90=180\), which simplifies to \(9x+95 = 180\).
Subtract 95 from both sides: \(9x=180 - 95 = 85\).
Then \(x=\frac{85}{9}\).
Angle \(A=3x-3=3\times\frac{85}{9}-3=\frac{85}{3}-3=\frac{85 - 9}{3}=\frac{76}{3}\approx25.33^{\circ}\).

Step4: Set up an equation for problem 15

For the triangle with angles \(59 + x\), \(x + 88\) and \(41^{\circ}\), we have \((59 + x)+(x + 88)+41=180\).
Combine like - terms: \(x+x+59 + 88+41=180\), which simplifies to \(2x+188 = 180\).
Subtract 188 from both sides: \(2x=180 - 188=-8\).
Divide both sides by 2: \(x=-4\).
Angle \(A=59 + x=59-4 = 55^{\circ}\).

Step5: Set up an equation for problem 16

For the triangle with angles \(x + 57\), \(x + 52\) and \(85^{\circ}\), we have \((x + 57)+(x + 52)+85=180\).
Combine like - terms: \(x+x+57 + 52+85=180\), which simplifies to \(2x+194 = 180\).
Subtract 194 from both sides: \(2x=180 - 194=-14\).
Divide both sides by 2: \(x=-7\).
Angle \(A=x + 52=-7+52 = 45^{\circ}\).

Step6: Set up an equation for problem 17

For the right - triangle with angles \(x + 38\), \(x + 68\) and \(90^{\circ}\), we have \((x + 38)+(x + 68)+90=180\).
Combine like - terms: \(x+x+38 + 68+90=180\), which simplifies to \(2x+196 = 180\).
Subtract 196 from both sides: \(2x=180 - 196=-16\).
Divide both sides by 2: \(x=-8\).
Angle \(A=x + 38=-8+38 = 30^{\circ}\).

Step7: Set up an equation for problem 18

For the triangle with angles \(4x + 6\), \(5x-5\) and \(80^{\circ}\), we have \((4x + 6)+(5x-5)+80=180\).
Combine like - terms: \(4x+5x+6-5 + 80=180\), which simplifies to \(9x+81 = 180\).
Subtract 81 from both sides: \(9x=180 - 81 = 99\).
Divide both sides by 9: \(x = 11\).
Angle \(A=4x + 6=4\times11+6=44 + 6=50^{\circ}\).

Step8: Set up an equation for problem 19

For the triangle with angles \(5x + 1\), \(6x\) and \(135^{\circ}\), we have \((5x + 1)+6x+135=180\).
Combine like - terms: \(5x+6x+1 + 135=180\), which simplifies to \(11x+136 = 180\).
Subtract 136 from both sides: \(11x=180 - 136 = 44\).
Divide both sides by 11: \(x = 4\).
Angle \(A=5x + 1=5\times4+1=21^{\circ}\).

Step9: Set up an equation for problem 20

For the triangle with angles \(4x\), \(5x\) and \(135^{\circ}\), we have \(4x+5x+135=180\).
Combine like - terms: \(9x+135 = 180\).
Subtract 135 from both sides: \(9x=180 - 135 = 45\).
Divide both sides by 9: \(x = 5\).
Angle \(A=4x=4\times5 = 20^{\circ}\).

Answer:

Problem 13: \(x = 10\), Angle \(A = 75^{\circ}\)
Problem 14: \(x=\frac{85}{9}\), Angle \(A=\frac{76}{3}\approx25.33^{\circ}\)
Problem 15: \(x=-4\), Angle \(A = 55^{\circ}\)
Problem 16: \(x=-7\), Angle \(A = 45^{\circ}\)
Problem 17: \(x=-8\), Angle \(A = 30^{\circ}\)
Problem 18: \(x = 11\), Angle \(A = 50^{\circ}\)
Problem 19: \(x = 4\), Angle \(A = 21^{\circ}\)
Problem 20: \(x = 5\), Angle \(A = 20^{\circ}\)