6. Draw the line segment $\overline{A B}$. Construct a circle for which the line segment $\overline{A B}$ is the diameter. Choose any point $T$ on the constructed circle, different from $A$ and $B$. Determine the measure of the angle $\Varangle A T B$.
6. Construction:

$1^{\circ}$ Draw a line segment $\overline{A B}$.

$2^{\circ}$ Construct the perpendicular bisector of the line segment $\overline{A B}$, point $S$ is the intersection of the bisector and the line segment and is the center of the circle, so construct the circle.

![](https://cdn.mathpix.com/cropped/2024_05_30_e828ce59cd4e17e48768g-2.jpg?height=268&width=249&top_left_y=1402&top_left_x=421)

$3^{\circ}$ Choose a point $T$ on the constructed circle and mark $\angle A T B$ (or $\triangle A B T$).

![](https://cdn.mathpix.com/cropped/2024_05_30_e828ce59cd4e17e48768g-2.jpg?height=291&width=277&top_left_y=1802&top_left_x=404)

If we draw the line segment $\overline{S T}$, then $\triangle A B T$ will be divided into 2 triangles: $\triangle A S T$ and $\triangle S B T$.

![](https://cdn.mathpix.com/cropped/2024_05_30_e828ce59cd4e17e48768g-2.jpg?height=306&width=259&top_left_y=2234&top_left_x=473)

Both of these triangles are isosceles triangles with bases $\overline{A T}$ and $\overline{B T}$ (the legs are radii of the circle with center at point $S$ and diameter $\overline{A B}$).

1 POINT

Let the measures of the angles at the base of these triangles be $x$ and $y$ respectively.

For the isosceles triangle $\triangle A S T$, the measure of the exterior angle is $|\angle B S T|=2 x$. 1 POINT

For the isosceles triangle $\triangle S B T$, the measure of the exterior angle is $|\angle T S A|=2 y$. 1 POINT

![](https://cdn.mathpix.com/cropped/2024_05_30_e828ce59cd4e17e48768g-3.jpg?height=63&width=1548&top_left_y=528&top_left_x=274)

Thus, $x+y=90^{\circ}$. 1 POINT

Since $|\angle A T B|=x+y$, we conclude that the measure of the desired angle is $90^{\circ}$. 2 POINTS

TOTAL 10 POINTS