Question

step angle reason
1 $m\angle adb = 98^{\circ}$ given
2 $m\angle b = 45^{\circ}$ given
3 $m\angle bad = 37^{\circ}$ sum of angles in a triangle
4 $m\angle adc = 82^{\circ}$ linear pair (or triple)
try $m\angle\square i=\square 61^{\circ}$ sum of angles in a triangle

Explanation:

Step1: Recall the sum of angles in a triangle

The sum of the interior angles of a triangle is \(180^{\circ}\).

Step2: Identify the triangle

The triangle in question is \(\triangle ADC\) (since we know \(m\angle ADC = 82^{\circ}\) and we need to find another angle).

Step3: Use the angle - sum formula

Let the unknown angle be \(\angle C\). For \(\triangle ADC\), we have \(m\angle C+m\angle ADC + m\angle CAD=180^{\circ}\). We know \(m\angle ADC = 82^{\circ}\). To find \(m\angle CAD\), note that in \(\triangle ABD\), \(m\angle BAD = 37^{\circ}\) (given). Assuming \(A\) is a single - point vertex for the two triangles \(\triangle ABD\) and \(\triangle ADC\) (from the figure), and if there is no other angle at \(A\) for \(\triangle ADC\) (implied by the problem structure), then \(m\angle CAD\) is related to the sum - of - angles in \(\triangle ADC\). Using the formula \(m\angle C=180^{\circ}-m\angle ADC - m\angle CAD\). Substituting \(m\angle ADC = 82^{\circ}\) and assuming \(m\angle CAD\) (if we consider the problem's use of the sum of angles in a triangle for the unknown angle and the given data) and re - checking the problem's reason as "Sum of angles in a triangle" for the unknown angle calculation.

Answer:

\(\angle C = 61^{\circ}\)