Transcript Document 7196254

Chapter 5:
Relationships in
Triangles
Lesson 5.1
Bisectors, Medians, and
Altitudes
Perpendicular Bisector
Definition
Facts to Know
Point of
Concurrency
A line, segment or
ray that passes
through the
midpoint of the
opposite side and
is perpendicular to
that side
Any point on a
perpendicular
bisector is
equidistant from
the endpoints
Circumcenter:
The point where 3
perpendicular
bisectors intersect
- the circumcenter
is equidistant from
all vertices of the
triangle
Example
A
E
B
D
BD = CD
AD  BC
E is the circumcenterAE = BE = CE
C
Median
Definition
Facts to Know
Point of
Concurrency
A segment that
goes from a
vertex of the
triangle to the
midpoint of the
opposite side
The median splits
the opposite side
into two congruent
segments
Centroid:
The point where 3
medians intersect
Example
A
E
B
Small = 1/3 median
Big = 2/3 median
2 x small = big
D
BD = CD
E is the centroidED = 1/3 AD
AE = 2/3 AD
2 ED = AE
C
Angle Bisector
Definition
Facts to Know
Point of
Concurrency
A line, segment,
or ray that
passes through
the middle of an
angle and
extends to the
opposite side
Any point on an
angle bisector is
equidistant from the
sides of the triangle
Incenter:
The point where 3
angle bisectors
intersect
Example
A
E
F
G
-the incenter is
equidistant from
all sides of the
triangle
B
D
BAD =  CAD
G is the incenterEG = FG
C
Altitude
Definition
A segment that
goes from a
vertex of the
triangle to the
opposite side
and is
perpendicular to
that side
Facts to Know
Point of
Concurrency
Example
Orthocenter:
The point where 3
altitudes intersect
A
B
D
AD  BC
C
C.
Find the measure of EH.
A.
B.
Find QS.
Find WYZ.
In the figure, A is the circumcenter of ΔLMN.
Find x if mAPM = 7x + 13.
In the figure, point D is the incenter of ΔABC.
What segment is congruent to DG?
In ΔXYZ, P is the centroid and YV = 12. Find YP and
PV.
In ΔLNP, R is the centroid and
LO = 30. Find LR and RO.
Lesson 5.2
Inequalities and Triangles
Foldable
Fold the paper into three sections (burrito
fold) Then fold the top edge down about ½
an inch
 Unfold the paper and in the top small
rectangles label each column…

Exterior Angle Inequality
•Exterior Angle
=
Remote Int. + Remote Int.
-The
exterior angle is
greater than either of the
remote interior angles by
themselves
rem. Int. < ext.
Ex:
Inequality with Sides
Inequality with Angles
-The
biggest side is across
from the biggest angle
-The smallest side is across
from the smallest angle
-The biggest angle is across
from the biggest side/ the
smallest angle is across from
the smallest side
Ex:
Ex:
List the angles of ΔABC in order from smallest to
largest.
List the sides of ΔABC in order from
shortest to longest.
___
___the relationship between the
What
is
lengths of RS and ST?
What is the relationship between the
measures of A and B?
Lesson 5.4
The Triangle Inequality
Triangle Inequality Theorem

The sum of the lengths of any two sides of
a triangle is greater than the length of the
third side
Triangle Inequality Theorem
Problems

Determine if the measures
given could be the sides of
a triangle.

16, 17, 19
16 + 17 = 33 yes, the sum of the
two smallest sides is larger
than the third side
 6, 9, 15
6 + 9 = 15 no, the sum of the
two smallest sides is equal to
the other side so it cannot be
a triangle

Find the range for the
measure of the third side
given the measures of two
sides.

7.5 and 12.1
12.1- 7.5 < x < 12.1 + 7.5
4.6 < x < 19.6
 9 and 41
41-9 < x < 41 + 9
32 < x < 50
Determine
whether it is possible to
form a triangle with side lengths 5, 7,
and 8.
Is it possible to form a triangle with the given side
lengths of 6.8, 7.2, 5.1? If not, explain why not.
Find
the range for the measure of the
third side of a triangle if two sides
measure 4 and 13.
In ΔPQR, PQ = 7.2 and QR = 5.2. Which measure
cannot be PR?
A 7
B 9
C 11
D 13
Lesson 5.3
Indirect Proof
Steps to Completing an Indirect
Proof:
Assume that ______________ (the
conclusion is false)
 Then _______________ (show that the
assumption leads to a contradiction) This
contradicts the given information that
________________.
 Therefore, __________________ (rewrite
the conclusion) must be true.

B. State the assumption you would make to start an
indirect proof for the statement 3x = 4y + 1.
Example Indirect Proof
Given: 5x < 25
Prove: x < 5
1. Assume that
x
2. Then x= 9
And 5(9)= 45

5.
45> 25
This contradicts the given info that 5x < 25
3. Therefore,
x < 5 must be true.
Example Indirect Proof
Given: m is not parallel to n
Prove: m  3 m 2
1. Assume that
m
3
n
2
m  3 = m 2
2. Then, angles 2 and 3 are alternate interior angles
When alternate interior angles are congruent then the
lines that make them are parallel.
This contradicts the given info that m is not parallel to n
3. Therefore, m  3  m  2 must be true.
an indirect proof to show that if –
2x + 11 < 7, then x > 2.
Given: –2x + 11 < 7
Prove: x > 2
Write
Write an indirect proof.
Given: ΔJKL with side lengths 5, 7, and 8 as shown.
Prove: mK < mL
Lesson 5.5
Inequalities Involving Two
Triangles
On the other side of the foldable from Lesson 2 (3 column chart)
SAS Inequality Theorem
(Hinge Theorem)
-When 2 sides of a triangle
are congruent to 2 sides of
another triangle, and the
included angle of one
triangle is greater than the
included angle of the other
triangle…
Then, the side
opposite the larger angle is
larger than the side
opposite the smaller angle
SSS Inequality Theorem
-When
2 sides of a triangle are
congruent to 2 sides of
another triangle, and the 3rd
side of a triangle is greater
than the 3rd side of the other
triangle…
Then, the angle opposite
the larger side is larger than
the angle opposite the smaller
side
Examples:
Ex:
Ex:
Ex:
A. Compare the measures AD and BD.
B. Compare the measures ABD and BDC.
ALGEBRA Find the range of possible values for a.
Find the range of
possible values of n.