Question

in the triangle, suppose that ( mangle d=(6x - 3)^{circ}), ( mangle e=(4x + 7)^{circ}), and ( mangle f=x^{circ}).
(a) write an equation to find ( x). make sure you use an = sign in your answer.
equation: (square)
(b) find the degree measure of each angle.
( mangle d=square^{circ})
( mangle e=square^{circ})
( mangle f=square^{circ})

Explanation:

Step1: Use the triangle - angle sum theorem

The sum of the interior angles of a triangle is \(180^{\circ}\). So, \(x+(4x + 7)+(6x-3)=180\).
Simplify the left - hand side: \(x+4x+7 + 6x-3=(1 + 4+6)x+(7 - 3)=11x + 4\).
The equation is \(11x+4 = 180\).

Step2: Solve the equation for \(x\)

Subtract \(4\) from both sides of the equation \(11x+4=180\):
\(11x+4-4=180 - 4\), which gives \(11x=176\).
Divide both sides by \(11\): \(x=\frac{176}{11}=16\).

Step3: Find the measure of each angle

• For \(\angle F\): \(m\angle F=x^{\circ}\), so \(m\angle F = 16^{\circ}\).
• For \(\angle E\): \(m\angle E=(4x + 7)^{\circ}\). Substitute \(x = 16\), then \(m\angle E=4\times16+7=64 + 7=71^{\circ}\).
• For \(\angle D\): \(m\angle D=(6x-3)^{\circ}\). Substitute \(x = 16\), then \(m\angle D=6\times16-3=96-3 = 93^{\circ}\).

Answer:

(a) Equation: \(x+(4x + 7)+(6x-3)=180\) (or \(11x + 4=180\))
(b) \(m\angle F = 16^{\circ}\), \(m\angle E=71^{\circ}\), \(m\angle D=93^{\circ}\)