Question

1. find the measure of the angle indicated. 15) find (mangle s). 16) find (mangle h).

Explanation:

1. For problem 14:

• Explanation:
• Step 1: Use the exterior - angle property of a triangle
• The exterior - angle of a triangle is equal to the sum of the two non - adjacent interior angles. In \(\triangle BCD\), the exterior angle \(\angle DBV\) is given by \(\angle DBV=\angle BCD+\angle BDC\). So, \(20x + 5=(9x - 2)+40\).
• Step 2: Solve the equation for \(x\)
• First, simplify the right - hand side of the equation: \(20x+5 = 9x - 2+40\), which becomes \(20x+5 = 9x + 38\).
• Subtract \(9x\) from both sides: \(20x-9x + 5=9x-9x + 38\), so \(11x+5 = 38\).
• Then subtract 5 from both sides: \(11x+5 - 5=38 - 5\), giving \(11x = 33\).
• Divide both sides by 11: \(x=\frac{33}{11}=3\).
• Step 3: Find the measure of the indicated angle
• We want to find the measure of the exterior angle \(\angle DBV\). Substitute \(x = 3\) into the expression for \(\angle DBV\): \(m\angle DBV=20x + 5=20\times3+5=60 + 5=65^{\circ}\).

1. For problem 15:

• Explanation:
• Step 1: Use the exterior - angle property of a triangle
• In \(\triangle RST\), the exterior angle \(\angle JRT\) is equal to the sum of the two non - adjacent interior angles \(\angle S\) and \(\angle T\). So, \(140=(3x + 4)+(8x + 4)\).
• Step 2: Solve the equation for \(x\)
• First, simplify the right - hand side of the equation: \(140=3x + 4+8x + 4\), which becomes \(140 = 11x+8\).
• Subtract 8 from both sides: \(140 - 8=11x+8 - 8\), so \(132 = 11x\).
• Divide both sides by 11: \(x=\frac{132}{11}=12\).
• Step 3: Find the measure of \(\angle S\)
• Substitute \(x = 12\) into the expression for \(\angle S\): \(m\angle S=3x + 4=3\times12+4=36 + 4=40^{\circ}\).

1. For problem 16:

• Explanation:
• Step 1: Use the exterior - angle property of a triangle
• In \(\triangle GFH\), the exterior angle \(\angle DFG\) is equal to the sum of the two non - adjacent interior angles \(\angle G\) and \(\angle H\). So, \(14x + 1=89+(5x - 7)\).
• Step 2: Solve the equation for \(x\)
• First, simplify the right - hand side of the equation: \(14x + 1=89+5x - 7\), which becomes \(14x + 1=5x + 82\).
• Subtract \(5x\) from both sides: \(14x-5x + 1=5x-5x + 82\), so \(9x+1 = 82\).
• Subtract 1 from both sides: \(9x+1 - 1=82 - 1\), giving \(9x = 81\).
• Divide both sides by 9: \(x=\frac{81}{9}=9\).
• Step 3: Find the measure of \(\angle H\)
• Substitute \(x = 9\) into the expression for \(\angle H\): \(m\angle H=5x - 7=5\times9 - 7=45 - 7=38^{\circ}\).

Answer:

1. \(65^{\circ}\)
2. \(40^{\circ}\)
3. \(38^{\circ}\)