Transcript Triangle Inequalities in One Triangle

Triangle
Inequality
(Triangle Inequality Theorem)
Objectives:





recall the primary parts of a triangle
show that in any triangle, the sum of the lengths
of any two sides is greater than the length of the
third side
solve for the length of an unknown side of a
triangle given the lengths of the other two sides.
solve for the range of the possible length of an
unknown side of a triangle given the lengths of
the other two sides
determine whether the following triples are
possible lengths of the sides of a triangle
Triangle Inequality
Theorem

The sum of the lengths
of any two sides of a
triangle is greater than
the length of the third
side.
AB + BC > AC
AB + AC > BC
AC + BC > AB
A
B
C
Is it possible for a triangle to have
sides with the given lengths?
Explain.
a. 3 ft, 6 ft and 9 ft
 3+6>9
(NO)
b. 5 cm, 7 cm and 10 cm
 5 + 7 > 10
 7 + 10 > 5
(YES)
 5 + 10 > 7
c. 4 in, 4 in and 4 in
 Equilateral: 4 + 4 > 4 (YES)
Solve for the length of an unknown side
(X) of a triangle given the lengths
of the other two sides.
a. 6 ft and 9 ft
 9 + 6 > x, x < 15
 x + 6 > 9, x > 3
 x + 9 > 6, x > – 3
 15 > x > 3
The value of x:
a + b > x > |a - b|
b. 5 cm and 10 cm 15 > x > 5
c. 14 in and 4 in
28 > x > 10
Solve for the range of the possible
value/s of x, if the triples represent the
lengths of the three sides of a triangle.

Examples:
a. x, x + 3 and 2x
b. 3x – 7, 4x and 5x – 6
c. x + 4, 2x – 3 and 3x
d. 2x + 5, 4x – 7 and 3x + 1
TRIANGLE
INEQUALITY
(ASIT and SAIT)
OBJECTIVES:



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
recall the Triangle Inequality Theorem
state and identify the inequalities relating sides and
angles
differentiate ASIT (Angle – Side Inequality Theorem)
from SAIT (Side – Angle Inequality Theorem) and viceversa
identify the longest and the shortest sides of a triangle
given the measures of its interior angles
identify the largest and smallest angle measures of a
triangle given the lengths of its sides
INEQUALITIES RELATING
SIDES AND ANGLES:
ANGLE-SIDE INEQUALITY THEOREM:
 If two sides of a triangle are not
congruent, then the larger angle lies
opposite the longer side.
If AC > AB, then mB > mC.
SIDE-ANGLE INEQUALITY THEOREM:
 If two angles of a triangle are not
congruent, then the longer side lies
opposite the larger angle.
If mB > mC, then AC > AB.
A
C
B
EXAMPLES:
List the sides of each triangle in ascending
O
E
order.
a.
e.
c.
I.
70
61
59
P
N
M
I
U
46
UE, IE, UI
E
JR, RE, JE
d.
P
73
L
ME & EL, ML
PO, ON, PN
b.
J
R
A
42
79
T
AT, PT, PA
31
E
TRIANGLE INEQUALITY
(Isosceles Triangle Theorem)
Objectives:
recall the definition of isosceles
triangle
 recall ASIT and SAIT
 solve exercises using Isosceles
Triangle Theorem (ITT)
 prove statements on ITT
 recall the definition of angle
bisector and perpendicular bisector

Isosceles Triangle:

a triangle with at least
two congruent sides
Parts of an Isosceles :
Base: AC
A
Legs: AB and BC
Vertex angle: B
Base angles: A and C
B
C
Isosceles Triangle Theorem (ITT):

If two sides of a
triangle are congruent,
then the angles
opposite the sides are
also congruent.
If AB  BC,
then A  C.
B
A
C
Converse of ITT:

If two angles of a
triangle are congruent,
then the sides opposite
the angles are also
congruent.
If A  C,
then AB  BC.
B
A
C
Vertex Angle Bisector-Isosceles
Theorem: (VABIT)

The bisector of the
vertex angle of an
isosceles triangle is the
perpendicular bisector of
the base.
If BD is the angle bisector of
A
the base angle of ABC,
then AD  DC and
mBDC = 90.
B
D
C
Examples:
For items 1-5, use the figure on the
right.
1. If ME = 3x – 5 and EL = x + 13,
solve for the value of x and EL.
E
2. If mM = 58.3, find the mE.
3. The perimeter of MEL is 48m, if
EL = 2x – 9 and ML = 3x – 7.
Solve for the value of x, ME and
ML.
4. If the mE = 65, find the mL.
5. If the mM = 3x + 17 and mE
= 2x + 11. Solve for the value of
x, mL and mE.
M
L
Prove the following using a two
column proof.
1. Given: 1  2
Prove: ABC is isosceles
A
B 3
1 5
4 C
6 2
Statements
1. 1  2
2. 1 & 3, 4 & 2
are vertical angles
Reasons
Given
Def. of VA
3. 1  3 and
4  2
4. 2  3
VAT
Subs/Trans
5. 4  3
Subs/Trans
6. AB  AC
CITT
7. ABC is isosceles
Def. of
Isosceles 
Prove the following using a two
column proof.
2. Given: 5  6
Prove: ABC is isosceles
A
B 3
1 5
4 C
6 2
Statements
Reasons
1. 5  6
Given
2. 5 & 3, 4 & 6 Def. of
are linear pairs
linear pairs
3. m5 = m6
Def. of  s
4. m5 + m3 = 180 LPP
m4 + m6 = 180
5. 4  3
Supplement Th.
6. m4 = m3
Def. of  s
7. AB  AC
8. ABC is isosceles
CITT
Def. of
isosceles 
Prove the following using a two
column proof.
3. Given: CD  CE, AD  BE
Prove: ABC is isosceles
Statements
C
A
31
D
2 4
E
B
Reasons
1. CD  CE, AD  BE
Given
2. 1  2
ITT
3. m1 = m2
4. 1 & 3 are LP s
Def.  s
Def. of LP
2 & 4 are LP s
5. m1 + m3 = 180 LPP
m4 + m2 = 180
6. m4 = m3
Supplement Th
7. ADC  BEC
SAS
8. AC  BC
CPCTC
9. ABC is isosceles
Def. of Isos. 
Triangle Inequality
(EAT)
Objectives:
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recall the parts of a triangle
define exterior angle of a triangle
differentiate an exterior angle of a
triangle from an interior angle of a
triangle
state the Exterior Angle theorem (EAT)
and its Corollary
apply EAT in solving exercises
prove statements on exterior angle of a
triangle
Exterior Angle of a
Polygon:
an angle formed by a
side of a  and an
extension of an
adjacent side.
 an exterior angle and
its adjacent interior
angle are linear pair

3
1
2
4
Exterior Angle Theorem:
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The measure of each
exterior angle of a triangle is
equal to the sum of the
measures of its two remote
1 2
interior angles.
m1 = m3 + m4
3
4
Exterior Angle Corollary:
The measure of an
exterior angle of a
triangle is greater than
the measure of either of
its remote interior
angles.
 m1 > m3 and
m1 > m4

3
1
2
4
Examples:
Use the figure on the right
to answer nos. 1- 4.
1.
The m2 = 34.6 and m4 = 51.3,
solve for the m1.
2.
The m2 = 26.4 and m1 =
131.1, solve for the m3 and m4.
3.
4.
The m1 = 4x – 11, m2 = 2x + 1
and m4 = x + 18. Solve for the
value of x, m3, m1 and m2.
If the ratio of the measures of 2
and 4 is 2:5 respectively. Solve
for the measures of the three
interior angles if the m1 = 133.
1
3
2
4
Proving: Prove the statement using a two column proof.
Statements
Given: 4 and 2 are linear
pair. Angles 1, 2 and 3
are interior angles of
ABC
Prove: m4 = m1 + m3
A
1. 4 and 2 are
linear pair. Angles 1,
2 and 3 are interior
angles of ABC
Given
2. m4 + m2 = 180
LPP
3. m1 + m2 + m3 TAST
= 180
4. m4 + m2 = m1 Subs/
+ m2 + m3
Trans
4 B
2
1
Reasons
5. m4 = m1 + m3 APE
3
C