Question

problem 1 here are three intersecting lines. determine the missing angle measurements.

Explanation:

Step1: Use vertical - angle property

Vertical angles are equal. The angle opposite the $45^{\circ}$ angle has a measure of $45^{\circ}$.

Step2: Use linear - pair property

A linear pair of angles sums to $180^{\circ}$. Let the angle adjacent to the $80^{\circ}$ angle be $x$. Then $x + 80^{\circ}=180^{\circ}$, so $x = 180^{\circ}-80^{\circ}=100^{\circ}$.

Step3: Use angle - sum property in triangle - like formation

Consider the intersection of the three lines. The sum of angles around a point is $360^{\circ}$, and in a non - overlapping set of angles formed by intersecting lines, we can also use the fact that the third unknown angle (let's call it $y$) and the $45^{\circ}$ and $80^{\circ}$ angles and other known angles are related. Another way is to use the fact that the angle adjacent to the $45^{\circ}$ angle in the relevant angle - combination is $135^{\circ}$ (since it forms a linear pair with $45^{\circ}$). And since the sum of angles where the three lines intersect and are non - overlapping should satisfy angle relations. The third unknown angle can be found using the fact that the sum of angles on one side of a line in the intersection region. The third unknown angle is $55^{\circ}$ because if we consider the angles formed by the intersection of the three lines and use the fact that the sum of angles in a non - overlapping set around the intersection point and the linear - pair and vertical - angle properties.

Answer:

The three missing angles are $45^{\circ}$, $100^{\circ}$, $55^{\circ}$ (in the order as we analyze them based on the given $45^{\circ}$ and $80^{\circ}$ angles in the problem, from the angle opposite $45^{\circ}$, the angle adjacent to $80^{\circ}$, and the remaining non - yet - found angle respectively)