Question

in the triangle, suppose that ( mangle h=(4x - 9)^{circ}, mangle i=(3x - 3)^{circ} ), and ( mangle j=x^{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 h=square^{circ} )
( mangle i=square^{circ} )
( mangle j=square^{circ} )

Explanation:

Step1: Use the triangle - angle sum theorem

The sum of the interior angles of a triangle is \(180^{\circ}\). So, \(m\angle H+m\angle I + m\angle J=180^{\circ}\).
Substitute \(m\angle H=(4x - 9)^{\circ}\), \(m\angle I=(3x - 3)^{\circ}\), and \(m\angle J=x^{\circ}\) into the equation:
\((4x - 9)+(3x - 3)+x=180\).
Simplify the left - hand side:
\(4x-9 + 3x-3+x=(4x + 3x+x)-(9 + 3)=8x-12\).
So the equation is \(8x-12 = 180\).

Step2: Solve the equation \(8x-12 = 180\) for \(x\)

Add \(12\) to both sides of the equation:
\(8x-12+12=180 + 12\), which gives \(8x=192\).
Divide both sides by \(8\):
\(x=\frac{192}{8}=24\).

Step3: Find \(m\angle H\)

Substitute \(x = 24\) into \(m\angle H=(4x - 9)^{\circ}\).
\(m\angle H=(4\times24-9)^{\circ}=(96 - 9)^{\circ}=87^{\circ}\).

Step4: Find \(m\angle I\)

Substitute \(x = 24\) into \(m\angle I=(3x - 3)^{\circ}\).
\(m\angle I=(3\times24-3)^{\circ}=(72 - 3)^{\circ}=69^{\circ}\).

Step5: Find \(m\angle J\)

Substitute \(x = 24\) into \(m\angle J=x^{\circ}\).
\(m\angle J=24^{\circ}\).

Answer:

(a) Equation: \(8x-12 = 180\)
(b) \(m\angle H = 87^{\circ}\), \(m\angle I=69^{\circ}\), \(m\angle J = 24^{\circ}\)