Question

a. what is m∠qtr? show your work.

Explanation:

Step1: Find the exterior - interior angle relationship

The exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. Consider the outer - most triangle with exterior angle of \(110^{\circ}\). Let's first find the non - labeled interior angle adjacent to the \(110^{\circ}\) exterior angle. It is \(180 - 110=70^{\circ}\) (linear pair of angles).

Step2: Use angle - sum property of a triangle

In the triangle with angles \(45^{\circ}\), \(74^{\circ}\) and \(\angle QTR\), and the third angle we just found (\(70^{\circ}\)). The sum of the interior angles of a triangle is \(180^{\circ}\).
We know that \(45 + 74+\angle QTR+70 = 180\).
Combining like terms: \(189+\angle QTR = 180\).
\(\angle QTR=180-(45 + 74+70)\)
\(\angle QTR = 180 - 189=- 9\) which is wrong. Let's use another approach.
Let's consider the angles in terms of the larger geometric figure.
We know that the sum of angles around a point is \(360^{\circ}\).
Let's assume we use the property of exterior angles of a polygon.
We know that the exterior angle of a triangle at a vertex is equal to the sum of the two non - adjacent interior angles.
Let's consider the angles formed by the lines.
We know that the sum of angles in a triangle is \(180^{\circ}\).
Let's find the angle adjacent to \(110^{\circ}\) which is \(70^{\circ}\).
We use the fact that for the triangle with angles \(45^{\circ}\), \(74^{\circ}\) and \(\angle QTR\)
We know that \(\angle QTR=180-(45 + 74)\) (using the angle - sum property of a triangle where we consider the non - overlapping part of the angles related to the triangle formed by the given angles)
\(\angle QTR = 61^{\circ}\)

Answer:

\(61^{\circ}\)