Question

in exercises 13 - 15, find the measure of each angle in the triangle. 13. (23x - 16)° (8x + 17)° 14. (9x - 3)° (5x + 25)° (12x - 24)° 15. (12x - 14)° (9x + 20)°

Explanation:

Step1: Recall angle - sum property of a triangle

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

For the first triangle with angles \((22x - 16)^{\circ}\), \((8x+17)^{\circ}\), and \((4x + 4)^{\circ}\):

Step2: Set up the equation

\((22x - 16)+(8x + 17)+(4x + 4)=180\)

Step3: Combine like - terms

\((22x+8x + 4x)+(-16 + 17+4)=180\)
\(34x+5 = 180\)

Step4: Solve for \(x\)

\(34x=180 - 5\)
\(34x=175\)
\(x=\frac{175}{34}\approx5.15\)
The angles are:
\(22x-16=22\times\frac{175}{34}-16=\frac{3850}{34}-16=\frac{3850 - 544}{34}=\frac{3306}{34}\approx97.24^{\circ}\)
\(8x + 17=8\times\frac{175}{34}+17=\frac{1400}{34}+17=\frac{1400+578}{34}=\frac{1978}{34}\approx58.18^{\circ}\)
\(4x + 4=4\times\frac{175}{34}+4=\frac{700}{34}+4=\frac{700 + 136}{34}=\frac{836}{34}\approx24.59^{\circ}\)

For the second triangle with angles \((9x - 3)^{\circ}\), \((5x + 25)^{\circ}\), and \((12x - 24)^{\circ}\):

Step5: Set up the equation

\((9x - 3)+(5x + 25)+(12x - 24)=180\)

Step6: Combine like - terms

\((9x+5x + 12x)+(-3 + 25-24)=180\)
\(26x - 2=180\)

Step7: Solve for \(x\)

\(26x=180 + 2\)
\(26x=182\)
\(x = 7\)
The angles are:
\(9x-3=9\times7-3=63 - 3=60^{\circ}\)
\(5x + 25=5\times7+25=35 + 25=60^{\circ}\)
\(12x - 24=12\times7-24=84 - 24=60^{\circ}\)

For the third triangle with angles \((12x - 14)^{\circ}\), \((9x + 20)^{\circ}\), and \(90^{\circ}\):

Step8: Set up the equation

\((12x - 14)+(9x + 20)+90=180\)

Step9: Combine like - terms

\((12x+9x)+(-14 + 20+90)=180\)
\(21x+96=180\)

Step10: Solve for \(x\)

\(21x=180 - 96\)
\(21x=84\)
\(x = 4\)
The angles are:
\(12x-14=12\times4-14=48 - 14=34^{\circ}\)
\(9x + 20=9\times4+20=36 + 20=56^{\circ}\)
\(90^{\circ}\)

Answer:

First triangle: Approximately \(97.24^{\circ}\), \(58.18^{\circ}\), \(24.59^{\circ}\)
Second triangle: \(60^{\circ}\), \(60^{\circ}\), \(60^{\circ}\)
Third triangle: \(34^{\circ}\), \(56^{\circ}\), \(90^{\circ}\)