Download Cornell Notes-Chapter 5 - Kenwood Academy High School

Geometry
Chapter 5: Relationships in Triangles
Objective:
5.1 Bisectors,
-Identify and use perpendicular bisectors and angle bisectors in a triangle
Medians and
-Identify and use altitudes and medians in a triangle
Altitudes
Perpendicular bisector
Angle bisector
Median
Altitude
A perpendicular bisector of a side of a triangle is a line, segment, or ray that is
perpendicular to the side and passes through its midpoint.
(Draw a diagram)
A segment, ray, or line which divides an angle into two congruent angles.
(Draw a diagram)
A median is a line segment that connects the vertex of a triangle to the midpoint
of the opposite side.
(Draw a diagram)
An altitude of a triangle is a segment from a vertex to the line containing the
opposite side and is perpendicular to that side.
(Draw a diagram)
1
Examples
Find the value of x and y.
Find the value of x.
Find x.
Find x.
Assignment 5.1 Part I
page 242 #13-16
2
Construction 1:
1. On a sheet of Patty Paper, draw a triangle with three sides of different
lengths. Label the triangle JKL.
2. Fold the paper to construct the angle bisectors of all three angles in
triangle JKL
3. What do you notice?
Concurrent lines
Point of concurrency
Incenter
Two properties of
angle bisectors are:
1) a point is on the angle bisector of an angle if and only if it is equidistant from
the sides of the angle, and
(2) the incenter of a triangle is equidistant from each side of the triangle.
(Incenter Theorem)
Example Find x and EF if BD is an angle bisector.
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Construction 2:
1. On a sheet of Patty Paper, draw a triangle with three sides of different
lengths. Label the triangle MNO.
2. Fold the paper to construct the medians of all three angles in triangle
MNO.
3. What do you notice?
(Copy your diagram to
the right)
Centroid
Construction 3:
1. On a sheet of Patty Paper, draw a triangle with three sides of different
lengths and no right angles. Label the triangle XYZ.
2. Fold the paper to construct the altitudes of all three angles in triangle
XYZ.
3. What do you notice?
(Copy your diagram to
the right)
Orthocenter
4
Construction 4:
1. On a sheet of Patty Paper, draw a triangle with three sides of different
lengths and no right angles. Label the triangle LMN.
2. Fold the paper to construct the perpendicular bisector of all three angles
in triangle XYZ.
3. What do you notice?
Copy your diagram to
the right)
Circumcenter
Two properties of
perpendicular
bisectors are:
(1) a point is on the perpendicular bisector of a segment if and only if it is
equidistant from the endpoints of the segment, and
(2) the circumcenter of a triangle is equidistant from the three vertices of the
triangle. (Circumcenter Theorem)
Examples
Lines a, b, and c are perpendicular
bisectors of triangle PQR and meet
at A. Find x, y, and z.
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Centroid Theorem The centroid of a triangle is located two thirds of the distance from a vertex to
the midpoint of the side opposite the vertex on a median.
Examples
In triangleLMN, P, Q, and R are the
midpoints of LM, MN, and LN,
respectively. Find x, y, and z.
Assignment 5.1 Part II
page 242 #6, 17-26
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Find x, y, and z.
5.2 Inequalities and
Triangles
Objective:
-recognize and apply properties of inequalities to the relationships between
angles and sides of a triangle
Angle-Side If one side of a triangle is longer than another side, then the angle opposite the
Relationships longer side has a greater measure than the angle opposite the shorter side.
If one angle of a triangle has a greater measure than another angle, then the side
opposite the greater angle is longer than the side opposite the lesser angle.
In a triangle, if one side is longer than another side, then the angle opposite the
Examples longer side is larger than the angle opposite the shorter side.
a. For each triangle, list the angles in order from greatest to least.
C
5
6
7
A
b.
B
For each triangle, list the sides in order from longest to shortest.
J
66°
50°
Assignment 5.2 page
252 #29-34, 37-42,46-48
64°
L
7
K
5.4 The Triangle
Inequality
Objective:
-apply the triangle inequality theorem to determine whether a triangle can be
drawn given specific side lengths
Triangle Inequality The sum of the lengths of any two sides of a triangle is greater than the length of
Theorem the third side.
Examples Determine whether it is possible to draw a triangle with sides of the given
measures. Write yes or no.
a. 15, 12, 9
b. 23, 16, 7
The measures of two sides of a triangle are given. Between what two numbers
must the measure of the third side fall?
a. 9 and 15
b. 11 and 20
Assignment 5.4 page
264 #18-32 even
Summary
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