Question

determine the measure of each indicated angle. note: the figure is not drawn to scale.
note: solving a quadratic equation will result in two possible answers. always make sure the values of ( x ) produce positive angles.
( x=)
( m angle b a c ):
( m angle a c b ):
( m angle a b c ):

Explanation:

Step1: Use the exterior angle theorem

The exterior angle theorem states that an exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. So, \(x^{2}+2x = 99\).

Step2: Rearrange the equation

Rearrange \(x^{2}+2x = 99\) to the standard quadratic form \(ax^{2}+bx + c=0\). We get \(x^{2}+2x - 99=0\).

Step3: Factor the quadratic equation

Factor \(x^{2}+2x - 99\). We need two numbers that multiply to \(-99\) and add up to \(2\). The numbers are \(11\) and \(-9\). So, \((x + 11)(x - 9)=0\).

Step4: Solve for \(x\)

Set each factor equal to zero:

• \(x+11 = 0\) gives \(x=-11\)
• \(x - 9=0\) gives \(x = 9\)

Since angles must be positive, we discard \(x=-11\).

Step5: Find \(m\angle BAC\)

\(m\angle BAC=x^{2}\). Substitute \(x = 9\), so \(m\angle BAC=9^{2}=81^{\circ}\).

Step6: Find \(m\angle ABC\)

\(m\angle ABC = 2x\). Substitute \(x = 9\), so \(m\angle ABC=2\times9 = 18^{\circ}\).

Step7: Find \(m\angle ACB\)

Since \(\angle ACB\) and the \(99^{\circ}\) angle are supplementary (they form a linear pair), \(m\angle ACB=180^{\circ}-99^{\circ}=81^{\circ}\).

Answer:

\(x = 9\)
\(m\angle BAC:81^{\circ}\)
\(m\angle ACB:81^{\circ}\)
\(m\angle ABC:18^{\circ}\)