Download Notes – Part A Section 5 – 1: Bisectors, Medians, and Altitudes

Date: _____________________________
Section 5 – 1: Bisectors, Medians, and Altitudes
Notes – Part A
Perpendicular Lines:
Bisect:
Perpendicular Bisector: a line, segment, or ray that
passes through the __________________ of a side of
a ________________ and is perpendicular to that side
Points on Perpendicular Bisectors
Theorem
5.1:
Any
point
on
the
perpendicular bisector of a segment is
_____________________ from the endpoints
of the _________________.
Example:
Concurrent Lines: _____________ or more lines that intersect at a common
_____________
Point of Concurrency: the point of ___________________ of concurrent lines
Circumcenter:
the point of concurrency of the _____________________
bisectors of a triangle
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Circumcenter
Theorem:
the
circumcenter of a triangle is equidistant
from the ________________ of the
triangle
Example:
Points on Angle Bisectors
Theorem 5.4:
Any point on the angle
bisector is ____________________ from the
sides of the angle.
Theorem 5.5: Any point equidistant from
the
sides
of
an
angle
lies
on
the
____________ bisector.
Incenter: the point of concurrency of the angle ________________ of a triangle
Incenter Theorem: the incenter of a triangle is _____________________ from
each side of the triangle
Example:
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
Example #1: RI bisects ∠SRA . Find the value of x and m∠IRA .

Example #2: QE is the perpendicular bisector of MU . Find the value of m and
the length of ME .

Example #3: EA bisects ∠DEV . Find the value of x if m∠DEV = 52 and
m∠AEV = 6x – 10.
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Example #4: Find x and EF if BD is an angle bisector.
Example #5: In ∆DEF, GI is a perpendicular bisector.
a.) Find x if EH = 19 and FH = 6x – 5.
b.) Find y if EG = 3y – 2 and FG = 5y – 17.
c.) Find z if m∠EGH = 9z.
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Date: _____________________________
Section 5 – 1: Bisectors, Medians, and Altitudes
Notes – Part B
Median: a segment whose endpoints are a ______________ of a triangle
and the ___________________ of the side opposite the vertex
Centroid: the point of concurrency for the ________________ of a triangle
Centroid Theorem: The centroid of a triangle is
located _________ of the distance from a
____________ to the __________________ of
the side opposite the vertex on a median.
Example:
Example #1: Points S, T, and U are the midpoints of DE , EF , and DF ,
respectively. Find x.
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Altitude:
a
segment
from
a
_______________ to the line containing
the
opposite
side
and
_______________________ to the line
containing that side
Orthocenter: the intersection point of the
____________________
Example #2: Find x and RT if SU is a median of ∆RST. Is SU also an
altitude of ∆RST? Explain.
Example #3: Find x and IJ if HK is an altitude of ∆HIJ.
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Date: _____________________________
Section 5 – 2: Inequalities and Triangles
Notes
Definition of Inequality:
For any real numbers a and b, ____________ if and only if there is a
positive number c such that _________________.
Example:
Exterior Angle Inequality Theorem: If an angle is an ________________
angle of a triangle, then its measures is ________________ than the measure
of either of its ________________________ remote interior angles.
Example:
Example #1: Determine which angle has the greatest measure.
Example #2: Use the Exterior Angle Inequality Theorem
to list all of the angles that satisfy the stated condition.
a.) all angles whose measures are less than m∠8
b.) all angles whose measures are greater than m∠2
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Theorem 5.9: If one side of a triangle is ________________ than another
side, then the angle opposite the longer side has a _______________
measure than the angle opposite the shorter side.
Example #3: Determine the relationship between the measures of the given
angles.
a.) ∠RSU , ∠SUR
b.) ∠TSV , ∠STV
c.) ∠RSV , ∠RUV
Theorem 5.10: If one angle of a triangle has a ________________ measure
than another angle, then the side opposite the greater angle is
________________ than the side opposite the lesser angle.
Example #4: Determine the relationship between the lengths of the given
sides.
a.) AE , EB
b.) CE , CD
c.) BC , EC
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Date: _____________________________
Section 5 – 4: The Triangle Inequality
Notes
Triangle Inequality Theorem: The sum of the lengths of any two sides of
a _________________ is _________________ than the length of the third
side.
Example:
Example #1: Determine whether the given measures can be the lengths of
the sides of a triangle.
a.) 2, 4, 5
b.) 6, 8, 14
Example #2: Find the range for the measure of the third side of a triangle
given the measures of two sides.
a.) 7 and 9
b.) 32 and 61
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Theorem 5.12: The perpendicular segment from a ____________ to a line
is the _________________ segment from the point to the line.
Example:
Corollary 5.1: The perpendicular segment from a point to a plane is the
________________ segment from the point to the plane.
Example:
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