Problem 1: Given the three side lengths of a triangle, come up with a formula for the area of the triangle.
Problem 2: Show that the sum of all the interior angles of a polygon with $n$ sides is equal to $\pi(n-2)$ radians.
Problem 3: Show that the medians of a triangle concur. The point at which they concur is called the centroid of the triangle.
Problem 4: Prove the law of cosines.
Problem 5: Prove the law of sines.
Problem 6: Prove the Pythagoream theorem.
Problem 6: Show that opposite sides of a parallelogram are equal in length.
Problem 7: Show that opposite angles of a parallelogram are equal.
Problem 8: Prove the parallelogram law which is also known as the parallelogram identity.
Problem 8: Prove or disprove that if every angle of a quadrilateral is equal to the opposite angle, then the quadrilateral is a parallelogram.
Problem 9: Prove or disprove that if every side of a quadrilateral is equal in length to the opposite side, then the quadrilateral is a parallelogram.
Problem 10: Prove that every isoceles triangle has at least 2 angles that are equal.
Problem 11: Prove that if two angles of a triangle are equal, then the triangle is isoceles.
Problem 12: If you are given the side lengths of a triangle, how would you find the angles?
Problem 13: If you are given the angles of a triangle and one of the side lengths, how would you find the other side lengths?
Problem 14: Prove that the angle bisectors of a triangle concur.
Problem 15: Given the area and side lengths of a triangle, how would you find the inradius?
Problem 16: Prove that a line segment going from a vertex of a triangle to the incenter of the triangle is an angle bisector.
Problem 17: Given the area and side lengths of a triangle, how would you find the circumradius?
Problem 18: On triangle $ABC$, angle $a$ is opposite to side $A$ and angle $b$ is opposite to side $B$. Prove that $A$ is less than $B$ if and only if $a$ is less than $b$.
Problem 19: Show that a line passing through a midpoint of a side of a triangle and the circumcenter is a perpendicular bisector of the one of the sides of the triangle.
Problem 20: Show that every rhombus is a parallelogram.
Problem 21: In a quadrilateral, show that all pairs of adjacent angles are supplementary if and only if the quadrilateral is a parallelogram.
Problem 22: If you are given the lengths of the diagonals of a kite, how would you find the area?
Problem 23: Prove or disprove that the diagonals of a kite are perpendicular.
Problem 24: In a quadrilateral ABCD, let AB = BC in length and CD = DA in length. Show that this is a kite.