Question

find the measure of the angle indicated. 15) find m∠s. 16) find m∠h. 17) find m∠fab. 18) find m∠ydc.

Explanation:

15) Find \(m\angle S\)

Step1: Use exterior - angle property

The exterior - angle of a triangle is equal to the sum of the two non - adjacent interior angles. In \(\triangle RST\), the exterior angle at \(R\) is \(140^{\circ}\), so \((3x + 4)+(8x + 4)=140\).
\[3x+4 + 8x+4=140\]
\[11x+8 = 140\]

Step2: Solve for \(x\)

Subtract 8 from both sides: \(11x=140 - 8=132\). Then divide both sides by 11: \(x=\frac{132}{11}=12\).

Step3: Find \(m\angle S\)

Substitute \(x = 12\) into the expression for \(m\angle S\): \(m\angle S=3x + 4\). So \(m\angle S=3\times12 + 4=36 + 4=40^{\circ}\).

16) Find \(m\angle H\)

Step1: Use exterior - angle property

In \(\triangle GFH\), the exterior angle at \(F\) gives the equation \((5x - 7)+89=14x + 1\).
\[5x-7 + 89=14x + 1\]
\[5x+82=14x + 1\]

Step2: Solve for \(x\)

Subtract \(5x\) from both sides: \(82=14x + 1-5x\), so \(82 = 9x+1\). Then subtract 1 from both sides: \(9x=82 - 1 = 81\). Divide both sides by 9: \(x = 9\).

Step3: Find \(m\angle H\)

Substitute \(x = 9\) into the expression for \(m\angle H\): \(m\angle H=5x - 7\). So \(m\angle H=5\times9-7=45 - 7=38^{\circ}\).

17) Find \(m\angle FAB\)

Step1: Use exterior - angle property

In \(\triangle ABC\), the exterior angle at \(A\) gives the equation \((3x + 2)+55=13x-3\).
\[3x+2+55=13x - 3\]
\[3x + 57=13x-3\]

Step2: Solve for \(x\)

Subtract \(3x\) from both sides: \(57=13x-3 - 3x\), so \(57 = 10x-3\). Then add 3 to both sides: \(10x=57 + 3=60\). Divide both sides by 10: \(x = 6\).

Step3: Find \(m\angle FAB\)

Substitute \(x = 6\) into the expression for \(m\angle FAB\): \(m\angle FAB=13x-3\). So \(m\angle FAB=13\times6-3=78 - 3=75^{\circ}\).

18) Find \(m\angle YDC\)

Answer:

1. \(m\angle S = 40^{\circ}\)
2. \(m\angle H = 38^{\circ}\)
3. \(m\angle FAB = 75^{\circ}\)
4. \(m\angle YDC = 140^{\circ}\)