Question

1. in the figure, m∠1 = m∠2 = 22 and m∠3 = m∠4 = 123. from this, you can conclude that m∠tsr

35.
45.
33.
22.

Explanation:

Step1: Recall angle - sum property of a triangle

The sum of the interior angles of a triangle is 180°. In \(\triangle TKL\), we know that \(\angle 3+\angle 1+\angle TKL = 180^{\circ}\), and in \(\triangle TLR\), \(\angle 4+\angle 2+\angle TLR=180^{\circ}\). Also, note that \(\angle TSR\) can be found using the angle - sum property of \(\triangle STR\).

Step2: First, find the non - given angles in the sub - triangles

In \(\triangle TKL\), since \(\angle 1 = 22^{\circ}\) and \(\angle 3=123^{\circ}\), then \(\angle TKL=180^{\circ}-\angle 1 - \angle 3=180^{\circ}-22^{\circ}-123^{\circ}=35^{\circ}\). In \(\triangle TLR\), since \(\angle 2 = 22^{\circ}\) and \(\angle 4 = 123^{\circ}\), then \(\angle TLR=180^{\circ}-\angle 2-\angle 4=180^{\circ}-22^{\circ}-123^{\circ}=35^{\circ}\).

Step3: Consider \(\triangle STR\)

In \(\triangle STR\), we want to find \(\angle TSR\). Let's assume we use the fact that the sum of angles in \(\triangle STR\) is 180°. We know that the angles at \(T\) and \(R\) and \(S\) sum up to 180°. Another way is to use the property of angles in the figure. Since \(\angle 1=\angle 2 = 22^{\circ}\) and \(\angle 3=\angle 4 = 123^{\circ}\), we can also note that \(\angle TSR\) can be found as follows:
The sum of angles around a point is 360°. Consider the angles at the intersection of the lines in the figure. But a simpler way is to use the angle - sum property of a triangle. In \(\triangle STR\), we know that the sum of interior angles is 180°. Let's assume we know that the non - relevant angles formed by the intersection of lines do not affect the calculation of \(\angle TSR\) in the big triangle.
We know that \(\angle TSR=35^{\circ}\) because of the angle - sum relationships in the sub - triangles and the overall triangle.

Answer:

35