Geometry Chapter 5 Benedict

Vocabulary Perpendicular Bisector- Segment, ray, line or plane that is perpendicular to a segment at its midpoint. Equidistant- The distance from each point is the same. Perpendicular Bisector Theorem- A point on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.

Extension Below draw a picture of a perpendicular bisector.

Vocabulary Converse of the Perpendicular Bisector Theorem- If a point is equidistant from the endpoint of a segment, then it is on the perpendicular bisector of the segment. Distance from a point to a line- The length of the perpendicular segment from the point to the line. Equidistant from the two lines- When a point is the same distance from one line as it is from another line.

Extension Below draw a picture of the converse of the perpendicular bisector.

Vocabulary Angle Bisector Theorem- If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle. Converse of the Angle Bisector Theorem- If a point is in the interior of an angle and it is equidistant from the sides of the angle, then it lies on the bisector of the angle.

Extension Below draw a picture of the angle bisector theorem.

Extension Below draw a picture of the converse of the angle bisector theorem.

Vocabulary Median of a Triangle- segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side. Centroid of the Triangle- The three medians of the triangle are concurrent, this is the point of concurrency.

Extension Below draw a picture representing the median of the triangle.

Extension Below draw a picture of a triangle and represent the centroid of the triangle.

Vocabulary Altitude of the Triangle- The perpendicular segment from a vertex to the opposite side or to the line that contains the opposite side. Orthocenter of the Triangle- the lines containing the altitudes that are concurrent and intersect at a point.

Extension Below draw a picture representing the altitude of the triangle. Label the orthocenter of the triangle.

Vocabulary Midsegment of a Triangle- The segment that connects the midpoints of two sides of a triangle. Midsegment Theorem- Segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long.

Extension Below draw a picture representing the midsegment theorem.

Triangle Inequalities The largest angle is across from the longest side. The shortest angle is across from the shortest side. Theorem If one side of a triangle is longer than another side, then the angle opposite the longer side is larger than the angle opposite the shorter side. Theorem If one angle of a triangle is larger than another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle.

Triangle Inequalities External Angle Inequality – The measure of an exterior angle of a triangle is greater than the measure of either of the two non-adjacent interior angles. Triangle Inequality AB + BC > AC AC + BC > AB AB + AC > BC

Vocabulary Indirect Proof- proof that you prove a statement is true by first assuming that its opposite is true. Hinge Theorem- If two sides of a triangle are congruent to two sides of another triangle and the included angle of the first is larger than the included angle of the second, then the third side of the first is longer than the third side of the second. Converse of the Hinge Theorem- If two sides of one triangle are congruent to two sides of another triangle, and the third side of the first it longer than the third side of the second, then the included angle of the first is larger than the included angle of the second.

Extension Below draw a picture representation of the Hinge Theorem and the Converse of the Hinge Theorem.