Question

1. find the measures of all the angles:

Explanation:

Step1: Use the triangle angle - sum property

The sum of angles in a triangle is \(180^{\circ}\). For \(\triangle ABC\), we have \(112^{\circ}+5x + 2x+5=180^{\circ}\).
Combine like terms: \(112 + 5+(5x + 2x)=180\), so \(117+7x = 180\).

Step2: Solve for \(x\)

Subtract \(117\) from both sides: \(7x=180 - 117\), so \(7x = 63\).
Divide both sides by \(7\): \(x=\frac{63}{7}=9\).

Step3: Find the measures of \(\angle C\) and \(\angle B\)

For \(\angle C\): Substitute \(x = 9\) into \(5x\), so \(\angle C=5\times9 = 45^{\circ}\).
For \(\angle B\): Substitute \(x = 9\) into \(2x + 5\), so \(\angle B=2\times9+5=18 + 5=23^{\circ}\).

Answer:

\(\angle A = 112^{\circ}\), \(\angle B=23^{\circ}\), \(\angle C = 45^{\circ}\)