Question

1. a) solve for x. x = ____________ b) find the measure of each angle. m∠f = __________ m∠h = __________ m∠s = __________ c) classify by angles: __________ classify by sides: ____________

Explanation:

Step1: Use the triangle angle - sum theorem

The sum of the interior angles of a triangle is \(180^{\circ}\). So, \((4x + 2)+6x+(5x - 2)=180\).
Simplify the left - hand side:
\(4x+2 + 6x+5x - 2=180\)
\((4x+6x + 5x)+(2 - 2)=180\)
\(15x=180\)

Step2: Solve for \(x\)

Divide both sides of the equation \(15x = 180\) by \(15\):
\(x=\frac{180}{15}=12\)

Step3: Find the measure of each angle

• For \(\angle F\):

Substitute \(x = 12\) into \(6x\). So, \(m\angle F=6x\).
\(m\angle F=6\times12 = 72^{\circ}\)

• For \(\angle H\):

Substitute \(x = 12\) into \(4x + 2\). So, \(m\angle H=4x+2\).
\(m\angle H=4\times12+2=48 + 2=50^{\circ}\)

• For \(\angle S\):

Substitute \(x = 12\) into \(5x-2\). So, \(m\angle S=5x - 2\).
\(m\angle S=5\times12-2=60 - 2=58^{\circ}\)

Step4: Classify the triangle

• By angles:

Since all angles (\(50^{\circ},58^{\circ},72^{\circ}\)) are less than \(90^{\circ}\), it is an acute triangle.

• By sides:

Since all angles are of different measures, all sides are of different lengths (using the law of sines \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\)), so it is a scalene triangle.

Answer:

a) \(x = 12\)
b) \(m\angle F=72^{\circ}\), \(m\angle H = 50^{\circ}\), \(m\angle S=58^{\circ}\)
c) Classify by angles: acute triangle; Classify by sides: scalene triangle