Question

in the triangle, suppose that (mangle d=(x + 7)^{circ}), (mangle e=(6x + 2)^{circ}), and (mangle f=(2x)^{circ}). (a) write an equation to find (x). make sure you use an \=\ sign in your answer. equation: (b) find the degree - measure of each angle. (mangle d=square^{circ}) (mangle e=square^{circ}) (mangle f=square^{circ})

Explanation:

Step1: Use angle - sum property of a triangle

The sum of the interior angles of a triangle is 180°. So, we add the given angle measures and set it equal to 180°.
$(x + 7)+(6x + 2)+(2x)=180$

Step2: Combine like - terms

Combine the x - terms and the constant terms on the left - hand side.
$x+6x + 2x+7 + 2=180$
$9x+9 = 180$

Step3: Solve for x

Subtract 9 from both sides of the equation:
$9x+9-9=180 - 9$
$9x=171$
Divide both sides by 9:
$x=\frac{171}{9}=19$

Step4: Find the measure of each angle

For $\angle D$:
$m\angle D=(x + 7)^{\circ}=(19 + 7)^{\circ}=26^{\circ}$
For $\angle E$:
$m\angle E=(6x + 2)^{\circ}=(6\times19+2)^{\circ}=(114 + 2)^{\circ}=116^{\circ}$
For $\angle F$:
$m\angle F=(2x)^{\circ}=(2\times19)^{\circ}=38^{\circ}$

Answer:

(a) Equation: $(x + 7)+(6x + 2)+(2x)=180$
(b)
$m\angle D = 26^{\circ}$
$m\angle E = 116^{\circ}$
$m\angle F = 38^{\circ}$