91Ó°ÊÓ

Another triangle inequality relationship is given by the Exterior Angle Inequality Theorem. It states: The measure of an exterior angle of a triangle is greater than the measure of either of the non adjacent interior angles. Explain how you know that \(\mathrm{m} \angle 1>\mathrm{m} \angle \mathrm{A}\) and \(\mathrm{m} \angle 1>\mathrm{m} \angle \mathrm{B}\) in \(\triangle \mathrm{ABC}\) with exterior angle \(\angle 1.\)

The length of the base of an isosceles triangle is \(\ell\) . Describe the possible lengths for each leg. Explain your reasoning.

Your classmate claims to have drawn a triangle with one side length of 13 inches and a perimeter of 2 feet. Is this possible? Explain your reasoning.

In what type(s) of triangles can a vertex be one of the points of concurrency of the triangle? Explain your reasoning.

COMPARING METHODS Exercises 49 and \(50,\) state whether you would use perpendicular bisectors or angle bisectors. Then solve the problem. You need to cut the largest circle possible from an isosceles triangle made of paper whose sides are 8 inches, 12 inches, and 12 inches. Find the radius of the circle.

Your friend claims that it is possible for the circumcenter, incenter, centroid, and orthocenter to all be the same point. Do you agree? Explain your reasoning.

The center of gravity of a triangle, the point where a triangle can balance on the tip of a pencil, is one of the four points of concurrency. Draw and cut out a large scalene triangle on a piece of cardboard. Which of the four points of concurrency is the center of gravity? Explain.

Prove that a median of an equilateral triangle is also an angle bisector, perpendicular bisector, and altitude.

Construct an acute scalene triangle. Find the orthocenter, centroid, and circumcenter. What can you conclude about the three points of concurrency?

The endpoints of \(\overline{\mathrm{AB}}\) are given. Find the coordinates of the midpoint M. Then and AB. \(A(-3,5), B(3,5)\)