Question

1. in $\triangle abc$. if $m\angle a$ is thirteen less than $m\angle c$ and $m\angle b$ is eleven less than four times $m\angle c$, find the measure of each angle.

$m\angle a = x - 13$
$m\angle b = 4x - 11$
$m\angle c = x$
$m\angle a = \underline{\quad\quad}$
$m\angle b = \underline{\quad\quad}$
$m\angle c = \underline{\quad\quad}$

Explanation:

Step1: Recall triangle angle sum

The sum of angles in a triangle is \(180^\circ\), so \(m\angle A + m\angle B + m\angle C = 180^\circ\).

Step2: Substitute angle expressions

Substitute \(m\angle A = x - 13\), \(m\angle B = 4x - 11\), \(m\angle C = x\) into the sum:
\((x - 13) + (4x - 11) + x = 180\).

Step3: Simplify the equation

Combine like terms: \(x - 13 + 4x - 11 + x = 6x - 24\).
So, \(6x - 24 = 180\).

Step4: Solve for \(x\)

Add 24 to both sides: \(6x = 180 + 24 = 204\).
Divide by 6: \(x = \frac{204}{6} = 34\).

Step5: Find each angle

• \(m\angle A = x - 13 = 34 - 13 = 21^\circ\)
• \(m\angle B = 4x - 11 = 4(34) - 11 = 136 - 11 = 125^\circ\)
• \(m\angle C = x = 34^\circ\)

Answer:

\(m\angle A = 21^\circ\)
\(m\angle B = 125^\circ\)
\(m\angle C = 34^\circ\)