Triangle Inequality (Triangle Inequality Theorem)

1
TriangleInequality(Triangle Inequality Theorem)
2
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

3
Triangle Inequality Theorem
B

• The sum of the lengths of any two sides of a
  triangle is greater than the length of the third
  side.
• AB BC gt AC
• AB AC gt BC
• AC BC gt AB

C
A
4
Is it possible for a triangle to have sides with
the given lengths? Explain.

• a. 3 ft, 6 ft and 9 ft
• 3 6 gt 9
• b. 5 cm, 7 cm and 10 cm
• 5 7 gt 10
• 7 10 gt 5
• 5 10 gt 7
• c. 4 in, 4 in and 4 in
• Equilateral 4 4 gt 4

(NO)
(YES)
(YES)
5
Solve for the length of an unknown side (X) of a
triangle given the lengths of the other two
sides.
The value of x a b gt x gt a - b

• a. 6 ft and 9 ft
• 9 6 gt x, x lt 15
• x 6 gt 9, x gt 3
• x 9 gt 6, x gt 3
• 15 gt x gt 3
• b. 5 cm and 10 cm
• c. 14 in and 4 in

15 gt x gt 5
28 gt x gt 10
6
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

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TRIANGLE INEQUALITY(ASIT and SAIT)
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OBJECTIVES

• 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 vice-versa
• 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

9
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 gt AB, then m?B gt 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 gt m?C, then AC gt AB.

C
A
B
10
EXAMPLES

1. List the sides of each triangle in ascending
   order.

O
E
a.
c.
e.
R
70?
61?
J
73?
59?
P
N
M
L
31?
JR, RE, JE
ME EL, ML
PO, ON, PN
I
d.
b.
E
A
P
42?
46?
U
E
79?
AT, PT, PA
UE, IE, UI
T
11
TRIANGLE INEQUALITY(Isosceles Triangle Theorem)
12
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

13
Isosceles Triangle
B

• a triangle with at least two congruent sides
• Parts of an Isosceles ?
• Base AC
• Legs AB and BC
• Vertex angle ?B
• Base angles ?A and ?C

A
C
14
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
15
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
16
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 the base angle of
  ?ABC, then AD ? DC and
• m?BDC 90.

B
A
C
D
17
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.
• 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.

E
M
L
18
Prove the following using a two column proof.

• 1. Given ?1 ? ?2
• Prove ?ABC is isosceles

Statements Reasons
1. ?1 ? ?2 Given
2. ?1 ?3, ?4 ?2 are vertical angles
Def. of VA
A
3. ?1 ? ?3 and ?4 ? ?2 VAT
4. ?2 ? ?3 Subs/Trans
5. ?4 ? ?3 Subs/Trans
3
4
B
C
1
5
6
2
6. AB ? AC CITT
7. ?ABC is isosceles Def. of
Isosceles ?
19
Prove the following using a two column proof.

• 2. Given ?5 ? ?6
• Prove ?ABC is isosceles

Statements Reasons
1. ?5 ? ?6 Given
A
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
3
4
B
C
5. ?4 ? ?3 Supplement Th.
1
5
6
2
6. m?4 m?3 Def. of ? ?s
7. AB ? AC CITT
8. ?ABC is isosceles Def. of isosceles ?
20
Prove the following using a two column proof.

• 3. Given CD ? CE, AD ? BE
• Prove ?ABC is isosceles

Statements Reasons
1. CD ? CE, AD ? BE Given
C
2. ?1 ? ?2 ITT
3. m?1 m?2 Def. ? ?s
4. ?1 ?3 are LP ?s Def. of LP ?2
?4 are LP ?s
3
1
2
4
B
A
D
E
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. ?
21
Triangle Inequality(EAT)
22
Objectives

• 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

23
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
24
Exterior Angle Theorem

• The measure of each exterior angle of a triangle
  is equal to the sum of the measures of its two
  remote interior angles.
• m?1 m?3 m?4

3
1
2
4
25
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 gt m?3 and m?1 gt m?4

3
1
2
4
26
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. 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.
4. If the ratio of the measures of ?2 and ?4 is 25
   respectively. Solve for the measures of the
   three interior angles if the m?1 133.

1
3
2
4
27
Proving Prove the statement using a two - column
proof.
Statements Reasons
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?3 180 TAST
4. m?4 m?2 m?1 m?2 m?3 Subs/ Trans
5. m?4 m?1 m?3 APE

• Given ?4 and ?2 are linear pair. Angles 1, 2 and
  3 are interior angles of ?ABC
• Prove m?4 m?1 m?3

B
4
2
1
3
A
C