Question

mixed exercises
find the value of x. then find the measure of each angle.
15.
16.
17.

1. proof determine whether the following statement is true or false. if false

Explanation:

Step1: Use the triangle angle - sum theorem

The sum of the interior angles of a triangle is \(180^{\circ}\). For the triangle in problem 15, we have the equation \(2x + 3x+4x=180\).

Step2: Solve the equation for \(x\)

Combine like terms: \(9x = 180\). Then \(x=\frac{180}{9}=20\).

Step3: Find the measure of each angle

Substitute \(x = 20\) into each angle expression.

• \(2x=2\times20 = 40^{\circ}\)
• \(3x=3\times20=60^{\circ}\)
• \(4x=4\times20 = 80^{\circ}\)

For problem 16:

Step1: Use the triangle angle - sum theorem

The sum of the interior angles of a triangle is \(180^{\circ}\). The triangle has angles \(x^{\circ}\), \(2x^{\circ}\), and \(90^{\circ}\). So the equation is \(x + 2x+90=180\).

Step2: Solve the equation for \(x\)

Combine like terms: \(3x+90 = 180\). Subtract 90 from both sides: \(3x=180 - 90=90\). Then \(x=\frac{90}{3}=30\).

Step3: Find the measure of each angle

• \(x = 30^{\circ}\)
• \(2x=2\times30 = 60^{\circ}\)
• The right - angle is \(90^{\circ}\)

For problem 17:

Step1: Use the exterior - angle theorem

The exterior - angle theorem states that an exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. So \((5x + 62)=(3x + 47)+37\).

Step2: Solve the equation for \(x\)

Expand the right - hand side: \(5x+62=3x+47 + 37\). Simplify the right - hand side: \(5x+62=3x+84\). Subtract \(3x\) from both sides: \(5x-3x+62=3x-3x + 84\), \(2x+62=84\). Subtract 62 from both sides: \(2x=84 - 62=22\). Then \(x = 11\).

Step3: Find the measure of each angle

• \(5x+62=5\times11+62=55 + 62=117^{\circ}\)
• \(3x + 47=3\times11+47=33+47 = 80^{\circ}\)
• The non - adjacent interior angle is \(37^{\circ}\)

Answer:

For problem 15: \(x = 20\), angles are \(40^{\circ}\), \(60^{\circ}\), \(80^{\circ}\).
For problem 16: \(x = 30\), angles are \(30^{\circ}\), \(60^{\circ}\), \(90^{\circ}\).
For problem 17: \(x = 11\), angles (exterior and non - adjacent interior) are \(117^{\circ}\), \(80^{\circ}\), \(37^{\circ}\).