91Ó°ÊÓ

PROVING A THEOREM Prove the Converse of the Perpendicular Bisector Theorem (Thm. 6.2\()\) . (Hint: Construct a line through point \(C\) perpendicular to \(\overline{A B}\) at point \(P . )\) Given \(\mathrm{CA}=\mathrm{CB}\) Prove Point \(\mathrm{Clies}\) on the perpendicular bisector of \(\overline{\mathrm{AB}}\) .

In Exercises 31–36, complete the statement with always, sometimes, or never. Explain your reasoning. A median is _______ the same line segment as a perpendicular bisector.

PROVING A THEOREM Use a congruence theorem to prove each theorem. a. Angle Bisector Theorem (Thm. 6.3\()\) b. Converse of the Angle Bisector Theorem (Thm. 6.4 )

Explain why the hypotenuse of a right triangle must always be longer than either leg.

In Exercises 31–36, complete the statement with always, sometimes, or never. Explain your reasoning. An altitude is _______ the same line segment as an angle bisector.

Is it possible to decide if three side lengths form a triangle without checking all three inequalities shown in the Triangle Inequality Theorem (Theorem 6.11\() ?\) Explain your reasoning.

THOUGHT PROVOKING The postulates and theorems in this book represent Euclidean geometry. In spherical geometry, all points are on the surface of a sphere. A line is a circle on the sphere whose diameter is equal to the diameter of the sphere. In spherical geometry, is it possible for two lines to be perpendicular but not bisect each other? Explain your reasoning.

In Exercises 31–36, complete the statement with always, sometimes, or never. Explain your reasoning. The centroid is ________ formed by the intersection of the three medians.

Compare an altitude of a triangle with a perpendicular bisector of a triangle.

PROOF Where is the circum center located in any right triangle? Write a coordinate proof of this result.