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Magazine Problem 13
Prove that the supplement of a right angle is a right angle.
Problem 14
set up and complete a proof of each statement. If a point on the base of an isosceles triangle is equidistant from the midpoints of the legs, then that point is the midpoint of the base.
Problem 15
set up and complete a proof of each statement. If a point in the interior of an angle (between the sides) is equidistant from the sides of the angle, then the ray joining the vertex of the angle to this point bisects the angle. (Hint: The distance from a point to a line is defined as the length of the perpendicular segment from the point to the line.)
Problem 15
The ratio of the complements of two angles is \(3: 2,\) and the ratio of their supplements is \(9: 8 .\) Find the two original angles.
Problem 16
To the nearest second, what is the first time after 7: 00 that the hands of a clock form a right angle?
Problem 17
\(\triangle \mathrm{ABC}\) has vertices at \(\mathrm{A}=(2,1), \mathrm{B}=(12,3),\) and \(\mathrm{C}=(6,7)\) Write an argument to show that the median from C to \(\overline{\mathrm{AB}}\) is not longer than the altitude from C to \(\overline{\mathrm{AB}}\).
Problem 22
A four-sided figure with two disjoint pairs of consecutive sides congruent is called a kite. The two segments joining opposite vertices are its diagonals. Prove that one of these diagonals is the perpendicular bisector of the other diagonal.
Problem 23
Prove that if each of the three altitudes of a triangle bisects the side to which it is drawn, then the triangle is equilateral.
