Euclidean geometry and proof

Two angles of a triangle measure $47^\circ$ and $68^\circ$. How many degrees is the third angle?

In triangle $ABC$, $\angle A = 50^\circ$ and $\angle B = 75^\circ$. The side $BC$ is extended beyond $C$ to a point $D$. How many degrees is the exterior angle $\angle ACD$?

Which of these always guarantee that two triangles are congruent? (Choose all that apply.)

In the proof that the angles of a triangle sum to $180^\circ$, we draw the line through $C$ parallel to $AB$ and say that the angle it makes with $CA$ equals $\angle A$. Which assumption is essential for that step?

Points $A$, $B$, $C$ lie on a circle with centre $O$, and $B$ is on the same side of $AC$ as $O$. The central angle $\angle AOC$ is $110^\circ$. How many degrees is the inscribed angle $\angle ABC$?

The vertices of quadrilateral $PQRS$ lie on a circle, in this order, and $\angle PQR = 73^\circ$. How many degrees is $\angle RSP$?

How many degrees is each interior angle of a regular polygon with $12$ sides?

The fallacious proof that every triangle is isosceles lets the bisector of the angle at $A$ meet the perpendicular bisector of $BC$ at a point $P$, and draws $P$ inside the triangle. Where does $P$ really lie when $AB \ne AC$?

Which Lean statement correctly says that SSA is not a congruence criterion? (Here ≅ means the triangles are congruent.)

Write circumcentre(a, b, c) for points given as pairs of integers. It should return the circumcentre as a pair of Fractions, or None if the points are collinear. Use the lemma: $O$ is equidistant from $A$ and $B$ exactly when $2(B - A) \cdot O = |B|^2 - |A|^2$. Two such linear equations (for $AB$ and $AC$) determine $O$.