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# 5.7: Chapter 5 Review

Difficulty Level: At Grade Created by: CK-12

Keywords, Theorems and Postulates

Midsegment
A line segment that connects two midpoints of adjacent sides of a triangle.
Midsegment Theorem
The midsegment of a triangle is half the length of the side it is parallel to.
Perpendicular Bisector Theorem
If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.
Perpendicular Bisector Theorem Converse
If a point is equidistant from the endpoints of a segment, then the point is on the perpendicular bisector of the segment.
Point of Concurrency
When three or more lines intersect at the same point.
Circumcenter
The point of concurrency for the perpendicular bisectors of the sides of a triangle.
Concurrency of Perpendicular Bisectors Theorem
The perpendicular bisectors of the sides of a triangle intersect in a point that is equidistant from the vertices.
Angle Bisector Theorem
If a point is on the bisector of an angle, then the point is equidistant from the sides of the angle.
Angle Bisector Theorem Converse
If a point is in the interior of an angle and equidistant from the sides, then it lies on the bisector of the angle.
Incenter
The point of concurrency for the angle bisectors of a triangle.
Concurrency of Angle Bisectors Theorem
The angle bisectors of a triangle intersect in a point that is equidistant from the three sides of the triangle.
Median
The line segment that joins a vertex and the midpoint of the opposite side (of a triangle).
Centroid
The point of concurrency for the medians of a triangle.
Concurrency of Medians Theorem
The medians of a triangle intersect in a point that is two-thirds of the distance from the vertices to the midpoint of the opposite side.
Altitude
A line segment from a vertex and perpendicular to the opposite side.
Orthocenter
The point of concurrency for the altitudes of triangle.
Theorem 5-9
If one side of a triangle is longer than another side, then the angle opposite the longer side will be larger than the angle opposite the shorter side.
Converse of Theorem 5-9
If one angle in a triangle is larger than another angle in a triangle, then the side opposite the larger angle will be longer than the side opposite the smaller angle.
Triangle Inequality Theorem
The sum of the lengths of any two sides of a triangle must be greater than the length of the third.
SAS Inequality Theorem
If two sides of a triangle are congruent to two sides of another triangle, but the included angle of one triangle has greater measure than the included angle of the other triangle, then the third side of the first triangle is longer than the third side of the second triangle.
SSS Inequality Theorem
If two sides of a triangle are congruent to two sides of another triangle, but the third side of the first triangle is longer than the third side of the second triangle, then the included angle of the first triangle is greater in measure than the included angle of the second triangle.
Indirect Proof
When the conclusion from a hypothesis is assumed false (or opposite of what it states) and then a contradiction is reached from the given or deduced statements.

## Review

If \begin{align*}C\end{align*} and \begin{align*}E\end{align*} are the midpoints of the sides they lie on, find:

1. The perpendicular bisector of \begin{align*}\overline{FD}\end{align*}.
2. The median of \begin{align*}\overline{FD}\end{align*}.
3. The angle bisector of \begin{align*}\angle FAD\end{align*}.
4. A midsegment.
5. An altitude.
6. Trace \begin{align*}\triangle FAD\end{align*} onto a piece of paper with the perpendicular bisector. Construct another perpendicular bisector. What is the point of concurrency called? Use this information to draw the appropriate circle.
7. Trace \begin{align*}\triangle FAD\end{align*} onto a piece of paper with the angle bisector. Construct another angle bisector. What is the point of concurrency called? Use this information to draw the appropriate circle.
8. Trace \begin{align*}\triangle FAD\end{align*} onto a piece of paper with the median. Construct another median. What is the point of concurrency called? What are its properties?
9. Trace \begin{align*}\triangle FAD\end{align*} onto a piece of paper with the altitude. Construct another altitude. What is the point of concurrency called? Which points of concurrency can lie outside a triangle?
10. A triangle has sides with length \begin{align*}x + 6\end{align*} and \begin{align*}2x - 1\end{align*}. Find the range of the third side.

## Texas Instruments Resources

In the CK-12 Texas Instruments Geometry FlexBook, there are graphing calculator activities designed to supplement the objectives for some of the lessons in this chapter. See http://www.ck12.org/flexr/chapter/9690.

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Date Created:
Feb 22, 2012