Question

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

Explanation:

Step1: Recall angle - sum property of a triangle

The sum of the interior angles of a triangle is 180°. Consider triangle $STK$. Let's find the third - angle in triangle $STK$. Let the third angle in $\triangle STK$ be $\angle S$. Then $\angle S+ \angle1+\angle3 = 180^{\circ}$.

Step2: Substitute the given angle values

Given $\angle1 = 22^{\circ}$ and $\angle3=123^{\circ}$. So, $\angle S=180^{\circ}-\angle1 - \angle3=180^{\circ}-22^{\circ}-123^{\circ}=35^{\circ}$.

Step3: Consider triangle $TRL$

In $\triangle TRL$, $\angle4 = 123^{\circ}$ and $\angle2 = 22^{\circ}$. Let the third - angle in $\triangle TRL$ be $\angle R$. Then $\angle R=180^{\circ}-\angle2-\angle4=180^{\circ}-22^{\circ}-123^{\circ}=35^{\circ}$.

Step4: Consider triangle $ST R$

In $\triangle STR$, we know that the sum of interior angles is 180°. Let the unknown angle be $x$. We know two of the non - unknown angles are $\angle S$ and $\angle R$ which we found to be 35° each. So $x = 180^{\circ}-35^{\circ}-35^{\circ}=110^{\circ}$. But it seems the question is not asking for this. If we assume we want to find another related angle. Let's consider the fact that we can use the angle - sum property in a different way. In the figure, if we consider the relationship between the given angles and the angles formed at the intersection of the lines. Since $\angle3$ and $\angle4$ are exterior angles of sub - triangles. Let's assume we want to find an angle which is related to the non - overlapping part of the angles formed. If we consider the fact that in a triangle formed by the intersection of the lines, we know that the sum of angles around a point is 360°. But a more straightforward way is to use the angle - sum property of a triangle. Let's assume we want to find an angle which is part of the basic triangle structure. In $\triangle STK$, we know two angles. In $\triangle TRL$, we know two angles.
If we consider the fact that we can find the angle at the intersection of the non - parallel sides of the two sub - triangles formed. Let's assume we want to find the angle opposite to the side connecting the non - common vertices of the two sub - triangles. Using the angle - sum property of a triangle, we know that in a triangle formed by the intersection of the lines, if we consider the angles $\angle1,\angle2,\angle3,\angle4$. We know that the angle we are looking for can be found as follows:
Let's consider the fact that we can use the property of linear pairs and triangle angles. Since $\angle3$ and $\angle4$ are equal and $\angle1$ and $\angle2$ are equal. We know that the angle we want to find (let's say the angle at the intersection of the non - parallel sides of the two sub - triangles) can be found by considering the fact that in a triangle formed by the intersection of the lines, we have:
The sum of angles in a triangle formed by the intersection of the lines: Let the angle we want to find be $y$.
We know that the sum of angles in a triangle is 180°. We know that the two angles adjacent to the angle $y$ can be related to $\angle1,\angle2,\angle3,\angle4$.
In fact, if we consider the triangle formed by the intersection of the lines, we can use the fact that the angle we want to find is $180^{\circ}-(123^{\circ}-22^{\circ}) = 79^{\circ}$ (this is wrong approach).
Let's use the correct approach:
In $\triangle STK$, we know $\angle1 = 22^{\circ}$ and $\angle3 = 123^{\circ}$. In $\triangle TRL$, $\angle2 = 22^{\circ}$ and $\angle4 = 123^{\circ}$.
We know that the angle we want to find (let's assume it's the angle formed by the non - parallel sides of the two sub - triangles) can be found using the fact that the sum of angles in a triangle is 180°.
If we consider the fact that we can use the property of vertical angles and triangle angles.
Let's assume we want to find the angle at the intersection of the non - parallel sides of the two sub - triangles.
We know that the angle we want to find is $35^{\circ}$. Because if we consider the fact that the two sub - triangles $\triangle STK$ and $\triangle TRL$ are formed in such a way that the angle we are looking for is equal to the third angle of either of the sub - triangles when we use the angle - sum property of a triangle ($180^{\circ}-22^{\circ}-123^{\circ}=35^{\circ}$).

Answer:

35