Geometry
Unit 5
Relationships in Triangles
Geometry
Chapter 5 – Relationships in Triangles
***In order to get full credit for your assignments they must me done on time and you must SHOW ALL WORK. ***
1.____ (5-1) Bisectors, Medians, and Altitudes – Page 235 1-13 all
2. ____ (5-1) Bisectors, Medians, and Altitudes – Pages 243-244 11-22 all
3. ____ (5-1) Bisectors, Medians, and Altitudes – 5-1 Practice Worksheet
4. ____ (5-2) Inequalities and Triangles – Pages 252-253 17-25, 29-34, 37-43, 46, 47
5. ____ (5-2) Inequalities and Triangles – 5-2 Practice Worksheet
6.____ (5-4) The Triangle Inequality – Pages 264-266 14-36 even, 57, 58
7. _____ Chapter 5 Review WS
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 _____________________
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 #1: RI
uur
bisects ∠SRA. Find the value of x and m IRA∠ .
Example #2: QE
suur
is the perpendicular bisector of MU. Find the value of m and the length of ME.
Example #3: EA
uuur
bisects ∠DEV . Find the value of x if m DEV∠ = 52 and
m AEV∠ = 6x – 10.
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.
Section 5 –
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,
respectively. Find x.
Date: _____________________________
1: Bisectors, Medians, and Altitudes Notes – Part B
segment whose endpoints are a ______________ of a triangle and the ___________________ of the side opposite the vertex
the point of concurrency for the ________________ of a triangle The centroid of a triangle
_______ of the distance from a ____________ to the __________________ of the side opposite the vertex on a median.
S, T, and U are the midpoints of DE EF,
Date: _____________________________
segment whose endpoints are a ______________ of a triangle and the
the point of concurrency for the ________________ of a triangle
,
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.
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
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,
Name ________________________ Period ____________ Chapter 5 (5.4)
Use your paper strips to determine whether a triangle can be formed. Complete the following chart using the correct values.
Orange = 2 inches Yellow = 3 inches Blue = 4 inches Green = 5 inches Side
measure
Trial 1 Trial 2 Trial 3 Trial 4 Trial 5 Trial 6 Trial 7 First side Second side Third side Is it a triangle?
What can you conclude from the data in the table above?
Complete the following sentence:
In order to have a triangle, the sum of two smallest sides must be _____________________________________________________ _________________.
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.
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.
