Question

abc is a scalene triangle. ( mangle a=(2x + 8)^{circ} ), ( mangle b=(2x - 2)^{circ} ), and ( mangle c=(3x - 8)^{circ} ). find the measure of the largest angle.
a ( 26^{circ} )
b ( 50^{circ} )
c ( 60^{circ} )
d ( 70^{circ} )
e ( 80^{circ} )

Explanation:

Step1: Use the triangle - angle sum property

The sum of the interior angles of a triangle is \(180^{\circ}\). So, \((2x + 8)+(2x-2)+(3x - 8)=180\).
Simplify the left - hand side: \(2x+8 + 2x-2+3x - 8=(2x+2x + 3x)+(8-2 - 8)=7x-2\).
The equation becomes \(7x-2 = 180\).

Step2: Solve for \(x\)

Add \(2\) to both sides of the equation \(7x-2=180\): \(7x-2 + 2=180 + 2\), which gives \(7x=182\).
Divide both sides by \(7\): \(x=\frac{182}{7}=26\).

Step3: Find the measures of each angle

For \(\angle A\): \(m\angle A=(2x + 8)^{\circ}\). Substitute \(x = 26\), then \(m\angle A=(2\times26+8)^{\circ}=(52 + 8)^{\circ}=60^{\circ}\).
For \(\angle B\): \(m\angle B=(2x-2)^{\circ}\). Substitute \(x = 26\), then \(m\angle B=(2\times26-2)^{\circ}=(52-2)^{\circ}=50^{\circ}\).
For \(\angle C\): \(m\angle C=(3x - 8)^{\circ}\). Substitute \(x = 26\), then \(m\angle C=(3\times26-8)^{\circ}=(78-8)^{\circ}=70^{\circ}\).

Answer:

D. \(70^{\circ}\)