Transcript 5-1 Special Segments in Triangles I. Triangles have four types of special segments:

5-1
Special Segments in Triangles
I. Triangles have four types of
special segments:
A. Perpendicular bisector

Any point on the perpendicular bisector of
a segment is equidistant from the
endpoints of the segment

every triangle has 3 perpendicular bisectors
Examples of perpendicular bisectors
In an acute triangle, the
perpendicular bisectors
meet inside the triangle.
In a right triangle, the perpendicular
bisectors meet on the triangle.
In an obtuse triangle, the
perpendicular bisectors
meet outside the triangle.
Circumcenter

The point where all the perp. Bisectors
meet in a triangle
B. Median
A median of a triangle is a
segment whose endpoints are
a vertex of the triangle and the
midpoint of the opposite side.
D
PROPERTIES OF MEDIANS
• every triangle has 3 medians
• the medians always meet inside the
triangle
Centroid

Point where all the medians meet
C. Altitude
The three altitudes intersect at G,
a point inside the triangle.
In right triangle
ABC, segment A
AB and segment
BC are two of
the altitudes of
the triangle
B
C
Orthocenter

Point where all the altitudes meet
D. Angle bisector
-there are 3 angle bisectors in every triangle.
-any point on an angle bisector of an angle is equidistant
(same distance) from the sides of the angle.
-any point on or in the interior of an angle and equidistant
from the sides of the angle lies on the angle bisector.
-if a triangle is isosceles, the bisector of the vertex angle is
also a median and an altitude.
-The angle bisectors
always meet
inside the triangle.
Incenter

Point where all the angle bisectors meet
1. Given: < F = 80, < E = 30
DG bisects < EDF
Prove: < DGE = 115
F
G
D
E
2. Find x and < 2.
MS is an altitude of  MNQ
< 1 = 3x + 11
< 2 = 7x + 9
M
R
1
Q
2
S
N
3. MS is a median of  MNQ.
QS = 3a -14
SN = 2 a + 1
<MSQ = 7a + 1
Find a.
Is MS an altitude?
M
R
1
Q
2
S
N
4. Always, sometimes, never




3 medians intersect in the interior
3 altitudes intersect at a vertex
3 angle bisectors intersect on the exterior
3 perpendicular bisectors intersect on the exterior
5. Triangle ABC has vertices A(-3, 10) B(9, 2) and C(9, 15)
a. Determine the coordinates of P on segment AB so that
segment CP is
a median of the triangle
b. Determine if segment CP is also an altitude of the
triangle
C
A
B
a.
P is the midpoint of segment AB
(-3 + 9)/2
(10 + 2)/2
The midpoint is P(3, 6)
b.
The slope of segment CP is (6 – 15)/(3 – 9) = 3/2
The slope of segment AB is (2 – 10)/(9 - - 3) = -2/3
Since the slopes are opposite reciprocals,
CP is perpendicular to AB.
B Therefore, segment CP is an altitude
6. Find the indicated information:
a.
A
b.
P
B
Q
S
R
Find PQ if segment PS is
a median of the triangle
QS = x + 5 SR = 3x – 17
And PQ = 2x -8
D
C
Find BD if AC is an altitude and
m<ACD = 3x + 30,
BC = x + 4 and CD = 2x + 8
W
c.
X
Y
Z
Find m<XWZ if m<XWY = 2x – 4 and
m<XWZ = 5x – 12. Segment WY is an angle
bisector.
Solutions to example 6:
a.
Since segment PS is a median, QS = SR. So, x + 5 = 3x – 17.
Then 5 = 2x – 17
22 = 2x
11 = x
Finally PQ = 2x – 8
2(11) – 8
PQ = 14
b. If BC is an altitude, then <ACB is a right angle. Then, 3x + 30 = 90.
3x = 60 x = 20
Finally BD = BC + CD x + 4 + 2x + 8
3x + 12 3(20) +12
BD = 72
c. Since segment WY is an angle bisector, it divides the angle in half.
So m<XWZ = 2m<XWY. Therefore 5x – 12 = 2(2x - 4)
5x – 12 = 4x - 8
x – 12 = -8
x=4
So m<XWZ = 5x – 12 5(4) – 12
m<XWZ = 8
Example 7: Write at least one conclusion that can be made from each of
the following statements.
S
N
L
R
M
a.
b.
c.
Segment SM is an altitude to segment RE
SN = NE
M is equidistant from R and E, and <RMS
is a right angle
E
d. m<ERN = m<SRN
e. segment EL is perpendicular to segment SR
Sample answers:
a.
b.
c.
d.
e.
Segment SM is perpendicular to segment RE
<SMR and <SME are right angles
segment RN is a median and N is the midpoint
Segment SM is a perpendicular bisector
segment SM is a median
segment SM is an altitude
Segment RN is an angle bisector
Segment EL is an altitude and <ELR and
<ELS are right angles
1. The segment that bisects an angle of
the  and has one endpoint at a vertex of
the  and the other endpoint at another
point on the  is called the _________.

2. A _________ is a line or segment that
passes through the midpoint of a side of a
 and is perpendicular to that side.


3. Triangle XYZ has vertices X(-1, 1),
Y(3, 9), and Z(6, -2). Determine the
coordinates of point W on so that is a
median of the triangle.
4. Triangle CPR has vertices C(15, 1),
P(9, 11), and R(2, 1). Determine the
coordinates of point A on so that is a
median of triangle CPR.
5-2 Inequalities for the sides
and angles of a triangle
I. Theorem
 The angle opposite the larger side is
always bigger than the angle opposite the
shorter side of any triangle

II. Theorem

If one angle is larger than another angle,
then the side opposite it is also larger than
the side opposite the smaller angle.
III. Theorem

The perpendicular segment from a point to
a line is the shortest segment from the
point to the line.

Corollary The perp. Segment from a point
to a plane is the shortest segment also.
IV.Examples
1. Refer to the figure in example 1 of
book.
 Given: angle A is greater than angle D
 Prove: BD is greater than AB


2. Draw triangle JKL with J(-4,2) K (4,3) L
(1,-3). List the angles from greatest to
least measure.
5-3 Indirect Proof
I.



Indirect Proof Steps
Assume that the conclusion is false.
Show that this leads to a contradiction of
a known property, rule, etc…
Point this out and since it is false, it
follows that the conclusion must be true.
You are proving that the contrapositive
of the conditional is true, therefore
the original conditional must be true
II. A. Exterior Angle Inequality
Theorem

If an angle is an exterior angle of a
triangle, then its measure is greater
than the measure of either of its
corresponding remote interior angles.
Recall that the sum of the two remote interior angles is
equal to the exterior. Since neither of the interior
angles are zero, the exterior angle will always be
bigger than either of them.
B. Definition of an inequality

For any real numbers a and b, a > b if
and only if there is a positive number c
such that a = b + c.
Properties of Inequalities for
Real Numbers p. 254 in book
Comparison Property:
one of three things has to be true
a<b a>b a=b
Transitive Property:
If a<b and b<c, then a<c
If a>b and b>c, then a>c
Addition and Subtraction Property:
if a>b then a+c>b+c and a-c>b-c
if a<b then a+c<b+c and a-c<b-c
Multiplication and Division Property
if c>o (positive)and a>b then ac>bc and a/c>b/c
if c>o (positive)and a<b then ac<bc and a/c<b/c
if c<o (negative) and a>b then ac<bc and a/c<b/c
if c<o (negative) and a<b then ac>bc and a/c>b/c
REMEMBER:if you multiply or divide an inequality by a
negative number it switches the direction of the inequality
III. Examples

1. Which assumption would you make to
start an indirect proof of the statement
“two acute angles are congruent”.
2. Which assumption would you make to
start an indirect proof of the following
statements?
 Bob took the dog for a walk
 EF is not a perpendicular bisector
 3x = 4 y + 1
 < 1 is less than or equal to < 2

3. Name the property that justifies if a < b,
then a + c < b + c.
 4. Name the property that justifies that if a
is less than b, then a cannot be greater
than b.
 5. Given: 2 y + 8 = 16
Prove: y ≠ 5


6. Given: JKL with side lengths 3, 4, 5
Prove: < K < < L
J
K
L
Refer to page 255 in your
textbook and try 5 –13
(hint for 8 – 10: use the
exterior angle inequality
theorem-the exterior angle is
greater than the remote
interior angles)
Answers to p 255-256
5 -13
5. Assume lines l and m do not intersect at x
6. Assume: If the alt int <‘s formed by two parallel lines
and a transversal are congruent, then the lines are
not parallel.
7. Assume Sabrina did not eat the leftover pizza
8. <3, <7, <5, <6
9. <4 and <8
10. <8 > <7 because <7 is part of a remote interior angle for
the exterior <8
11. Division
12. Addition
13. Transitive
5-4 The Triangle Inequality
I. Theorem
 The sum of the lengths of any two sides of
a triangle is greater than the length of the
third side.

II. Examples

1. If Mrs. Ewing gave Elizabeth four
pieces of tubing measuring 6 m, 7 m, 9 m,
and 16 m, how many different triangles
could she make?

2. What are the possible lengths for the
third side of a triangle with two sides of 8
and 13?

3. How many triangles can be made from
a rope 10 ft long?
5-5 Inequalities with two
triangles
I. Theorem
 SAS Inequality

II.

SSS Inequality
Theorem
5-2 Right Triangles

I. What are the postulates for proving 2
right triangles are congruent?
SAS aka LL
LL means Leg Leg
ASA, AAS aka LA
LA means Leg Angle
AAS aka HA
HA means Hypotenuse Angle
Finally, a new one!
HL
(only for right triangles/in
non-right triangles SSA)
HL means Hypotenuse Leg
1. In the figure,
is the angle
bisector < BAC. Are the
triangles congruent?
Tell which of the
right
triangle methods
will prove the
triangles
congruent.
2. Find x so that the right triangles
are congruent.
3. What do you need to prove by
HA?
4. Name the theorem used to
prove the triangles are congruent.
 5.
Which theorem or postulate states
that if the hypotenuse and an acute
angle of one right triangle are
congruent to the hypotenuse and
corresponding acute angle of another
right triangle, then the triangles are
congruent?
Refer to page 248 in
your textbook and
try problems 6-11
Answers to p 248: 6-11
6. Need to know that <B and <D are right angles
You already know AC = AC
7. Need to know that ST = TU
8. Need to know that LN = QR or NM = RP
9. 2x + 10 = 5x – 8 so 10 = 3x –8 then 18 = 3x and 6 =
x
10. x + 7 = 3x – 5 so 7 = 2x – 5 then 12 = 2x and 6 = x
11. 4x – 26 = 10 then 4x = 36 and x = 9