Download Geometry Chapter 5 Blank Notes

.1: ________________________________________________
Geometry
Date: __________
A midsegment of a triangle is a ______________ connecting the ________________ of two sides of the
triangle.
A
Theorem 5-1: Triangle Midsegment Theorem
“If a segment joins the midpoints of two sides
of a triangle, then the segment is ______________
to the third side and is ________ as long.”
B
C
Ex 1). What are three pairs of parallel segments in ∆𝐷𝐸𝐹?
Ex 2). In ∆𝑅𝑆𝑇, G, H, and K are midpoints. If RS = 28, GH = 20, and RH = 22, what is the length of
KH, ST, and GK?
Ex 3). What is the distance across the river?
Ex 4).
Ex 5).
Homework: pg. 305 #1 – 4, 7 – 22
5.2: ____________________________________________________
Geometry
Date: __________
Midsegments of Triangles Refresher (5.1):
Ex 1). Name all of the parallel sides:
Theorem 5-2: Perpendicular Bisector Theorem
“If a point is on the perpendicular bisector
Ex 2). Find the value of x:
if …
then …
of a segment, then it is ___________________
from the endpoints of the segment.”
Theorem 5-3: Converse of 5-2
if …
“If a point is equidistant from the endpoints of
a segment, then it is on the perpendicular
_____________ of the segment.”
Ex 3). What is MN?
then …
Ex 4). The monkey bars are located midway between
the slide and the tether ball. Which of the following are
about equidistant from the slide and the tether ball?
Theorem 5-4: Angle Bisector Theorem
if …
then …
“If a point is on the bisector of an angle, then
the point is ______________ from the sides of
the angle.”
Theorem 5-5: Converse of 5-4
if …
“If a point in the ___________ of an angle is
equidistant from the sides of the angle, then
the point is on the angle bisector.”
Ex 5). What is OP?
Ex 6). What is 𝑚∠𝐻𝐸𝐺?
Homework: pg. 312 #1 – 3, 6 – 22
then …
5.3 Day 1: ________________________________________________
Geometry
Date: __________

When three or more lines intersect at ______ point, they are _________________.

The point at which they intersect is the ________ of ___________________.
Theorem 5-6: Concurrency of Perpendicular Bisectors Theorem

The point of concurrency of the perpendicular bisectors of a triangle is called the
_____________________ of the triangle.

The circumcenter is the __________ of the circle.
*The circumcenter of a triangle can be inside, on, or outside a triangle*
Ex 1). Find the circumcenter of the triangle with vertices A(0, 4), B(0, 0), and C(8, 0).
Ex 2). Find the circumcenter of the triangle with vertices A(2, 7), B(10, 7), and C(10,3).
Ex 3). A civil engineer wants to install a cell phone tower that is equidistant from the mall, the
airport, and the subway. How should the engineer determine where to build the tower?
Mall
Airport
Subway
Homework: pg. 319 # 4 – 10
5.3 Day 2: __________________________________________
Geometry

Date: __________
The point of concurrency of the _________ bisectors of a triangle is called the ____________
of the triangle.

For any triangle, the incenter is always ______________ the triangle.
Theorem 5-7: Concurrency of Angle Bisectors Theorem

In the diagram, X, Y, and Z are equidistant from P, the incenter.

P is the center of the circle that is ________________ in the triangle. (See diagram on pg. 320)
Ex 1). GB = 8x – 7 and GD = 5x + 8. What is GF?
Ex 2). What is the value of x?
Ex 3). What is the value of x?
Homework: pg. 321 #1 – 10, 12 – 17
5.4 Day1: ______________________________________________
Geometry

Date: __________
A _____________ of a triangle is a segment whose endpoints are a ________ and the
_______________ of the opposite side.

A triangle’s _________ medians are always _______________.

The point of concurrency of the medians is called the _____________ of the triangle.
o This point is also called the ________ of ____________ of a triangle.
o The centroid is always _____________ the triangle.
Theorem 5-8: Concurrency of Medians Theorem
“The medians are concurrent at a point
that is _________________ the distance from
each vertex to the midpoint of the
opposite side.”
Ex 1). The centroid is shown. Find the lengths.
a) Given: BF = 24. Find EF and BE.
b) Given: BD = 9. Find BE and ED.
c) Given: KH = 12. Find JH.

d) Given JK = 4. Find JG and KG.
An ________________ of a triangle is the perpendicular segment from the vertex of
the triangle to the line containing the opposite side.

An altitude can be ____________, ___________, or the _______ of a triangle.
Ex 2).
a) Is ̅̅̅̅
𝐴𝐶 a median, an altitude, or neither? Explain.
b) Is ̅̅̅̅
𝐴𝐸 a median, an altitude, or neither? Explain.
Homework: pg. 327 #1 – 17
5.4 Day 2: ___________________________________________
Geometry

Date: __________
The lines that contain the ______________ of a triangle are ______________ at the
___________________ of the triangle.

The orthocenter can be ____________, _______, or ____________ the triangle.
Ex 1). Find the orthocenter.
What are the coordinates of the orthocenter of ∆𝐷𝐸𝐹?
Ex 2). What are the coordinates of the orthocenter of ∆𝐷𝐸𝐹 if D(-2, -1), E(1, 6), and F(5, 1)?
Homework: 5.4 Practice Worksheet
5.6 Day 1: __________________________________________
Geometry
Comparison Property of Inequality:
If a = b + c and c > 0, then _________________.
Corollary to the Triangle Exterior Angle Theorem:
“The measure of an exterior angle of a triangle
is _________________ than the measure of each
of its remote interior angles.”
Ex 1). Explain why 𝑚∠1 > 𝑚∠4.
Theorem 5-10:
“If two sides of a triangle are
not congruent, then the ______________
angle lies opposite the ___________ side.
Ex 2). Which corner of the triangular plot of land forms the largest angle?
Date: __________
Theorem 5-11:
“If two angles of a triangle are
not congruent, then the longer
side lies opposite the larger angle.”
Ex 3). Order the sides of ∆𝐿𝑀𝑁 in order from shortest to longest.
Ex 4). Name the angles of ∆𝐴𝐵𝐶 from largest to smallest.
Homework: pg. 346 #4 – 22
5.6 Day 2: __________________________________________
Geometry
Date: __________
 Not every group of three segments can be used to form a triangle. The lengths of the
segments must have the following relationship:
Theorem 5-12: Triangle Inequality Theorem
“The ______ of the lengths of any two sides of a
triangle is ______________ than the length of
the third side.”
Ex 1). Can a triangle have sides with the given lengths? Explain.
a) 6, 8, 13
b) 3, 9, 12
Ex 2). Two sides of a triangle are 7 inches and 9 inches. What is the range of possible lengths for
the third side?
Ex 3). Two sides of a triangle are 2 inches and 12 inches. What is the range of possible lengths for
the third side?
Homework: pg. 350 #1, 3 – 12
5.7: _________________________________________________
Geometry
Theorem 5-13: The Hinge Theorem (SAS Inequality Theorem)
“If two sides of one triangle are congruent to
two sides of another triangle, and the included
angles are ______ congruent, then the ____________
third side is opposite the ___________ included angle.
Ex 1). Which of the following statements must be true?
[A] AC = ST
[B] RS > AC
[C] AC < RT
[D] AC > RT
Ex 2). Which cheerleader’s hands are futher apart?
Theorem 5-14: Converse of the Hinge Theorem (SSS Inequality)
“If two sides of one triangle are congruent to
two sides of another triangle, and the third
sides are not congruent, then the larger
_____________ angle is opposite the longer third side.
Date: __________
Ex 3). What is the range of possible values for x?
Ex 4). What is the range of possible values for x?
Ex 5). What is the range of possible values for x?
Homework: pg. 356 #1, 2, 6 – 12, 14 – 16