Download Chapter 5 Test (5.1-5.5 Skip 5.4) Section 1: Midsegments of a

Chapter 5 Test (5.1-5.5 Skip 5.4)
Section 1: Midsegments of a Triangle
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Midsegment- a segment connecting the midpoints of two sides of a triangle.
A midsegment is both parallel and half of the third side
Section 2: Bisectors in Triangles
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Perpendicular Bisectors
o A line, segment, or ray that is perpendicular to the segment at its midpoint.
o Perpendicular Bisector Theorem-If a point is on a perpendicular bisector of a
segment, then it is equidistant from the endpoints of the segment.
o Converse of the Perpendicular Bisector Theorem- If a point is equidistant
from the endpoints of a segment, then it is on the perpendicular bisector of
the segment.
o Find two points on the perpendicular bisector
𝑥 +𝑥 𝑦 +𝑦
 Find the midpoint between the two points- ( 1 2 2 , 1 2 2)
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𝑦 −𝑦
Then find the slope between the two points 𝑥2−𝑥1
2
1
Find the slope that is perpendicular (opposite sign/reciprocal) to the
slope that you just found. Use that to find additional points
Angle Bisector
o A ray that divides an angle into two congruent angles.
o Angle Bisector Theorem- If a point is on the bisector of an angle then the
point is equidistant from the sides of the angle.
o Converse of the Angle Bisector Theorem- If a point in the interior of an angle
is equidistant from the sides of the angle then the point is on the angle
bisector.
Section 3: Concurrent Lines, Medians, and Altitudes
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Point of concurrency
o where three or more lines intersect
Concurrent
o when three or more lines intersect in one point.
Perpendicular Bisectors
o Theorem 5-6: The perpendicular bisectors of the sides of a triangle are
concurrent at a point equidistant from the vertices.
o The point of concurrency of the perpendicular bisectors of a triangle is called
the circumcenter.
Angle Bisectors
o Theorem 5-7: The bisectors of the angles of a triangle are concurrent at a
point equidistant from the sides
o The point of concurrency of the angle bisectors of a triangle is called the
incenter of the triangle.
Median
o A segment whose endpoints are a vertex and the midpoint of the opposite
side.
o Theorem 5-8: The medians of a triangle are concurrent at a point that is two
thirds the distance from each vertex to the midpoint of the opposite side.
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o The point of concurrency of the medians is the centroid
Altitude
o A perpendicular segment from vertex to the line containing the opposite side
of a triangle.
o Altitudes can be inside, on or outside the triangle.
o Theorem 5-9: The lines that contain the altitudes of a triangle are concurrent
o The point of concurrency of the altitudes is the orthocenter.
Section 5: Inequalities in Triangles
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Corollary to the Triangle Exterior Angle Theorem-The measure of an exterior angles
of a triangle is greater than the measure of each of its remote interior angles.
Theorem 5-10: If two sides of a triangle are not congruent, then the larger angle lies
opposite the longer side.
Theorem 5-11: If two angles of a triangle are not congruent, then the longest side
lies opposite the larger angle.
Smallest Angles opposite smallest sidebiggest angles opposite biggest sides
Triangle Inequality Theorem- The sum of the lengths of any two sides of a triangle is
greater than the length of the third side
o Determine if the three sides can make a triangle
o Determine the possible third side if given two triangles (answer must be
written as a compound inequality)
Review Problems
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Chapter Review
o Page 297-299 #1-42 skipping any problems that pertain to sec. 4
Chapter Test
o Page 300 #7-24 skipping any problems that pertain to sec. 4
Extra Practice
o Page 724-725 #1-42 skipping any problems that pertain to sec. 4