Download 5-5 The Triangle Inequality (1)

Five-Minute Check (over Lesson 5–4)
CCSS
Then/Now
Theorem 5.11: Triangle Inequality Theorem
Example 1: Identify Possible Triangles Given Side Lengths
Example 2: Standardized Test Example: Find Possible Side
Lengths
Example 3: Real-World Example: Proof Using Triangle
Inequality Theorem
Over Lesson 5–4
State the assumption you would make to start an
indirect proof of the statement.
ΔABC  ΔDEF
A. ABC is a right triangle.
B. A = D
___
___
C. AB = DE
D. ΔABC / ΔDEF
Over Lesson 5–4
State the assumption you would make to start an
indirect
proof of the statement.
___
RS is an angle bisector.
___
A. RS is a perpendicular bisector.
___
B. RS is not an angle bisector.
___
C. R is the midpoint of ST.
D. mR = 90°
Over Lesson 5–4
State the assumption you would make to start an
indirect proof of the statement.
X is a right angle.
A. X is a not a right angle.
B. mX < 90°
C. mX > 90°
D. mX = 90°
Over Lesson 5–4
State the assumption you would make to start an
indirect proof of the statement.
If 4x – 3 ≤ 9, then x ≤ 3.
A. 4x – 3 ≤ 9
B. x > 3
C. x > 1
D. 4x ≤ 6
Over Lesson 5–4
State the assumption you would make to start an
indirect proof of the statement.
ΔMNO is an equilateral triangle.
A. ΔMNO is a right triangle.
B. ΔMNO is an isosceles
triangle.
C. ΔMNO is not an equilateral
triangle.
D. MN = NO = MO
Over Lesson 5–4
Which
is a contradiction to the statement
___statement
___
that AB  CD?
A. AB = CD
B. AB > CD
___
___
C. CD  AB
D. AB ≤ CD
Content Standards
G.CO.10 Prove theorems about triangles.
G.MG.3 Apply geometric methods to solve problems
(e.g., designing an object or structure to satisfy
physical constraints or minimize cost; working with
typographic grid systems based on ratios).
Mathematical Practices
1 Make sense of problems and persevere in solving
them.
2 Reason abstractly and quantitatively.
You recognized and applied properties of
inequalities to the relationships between the
angles and sides of a triangle.
• Use the Triangle Inequality Theorem to
identify possible triangles.
• Prove triangle relationships using the
Triangle Inequality Theorem.
Identify Possible Triangles Given Side Lengths
A. Is it possible to form a triangle with side lengths
1 , 6 __
1 , and 14 __
1 ? If not, explain why not.
of 6 __
2
2
2
Check each inequality.

Answer:
X
Identify Possible Triangles Given Side Lengths
B. Is it possible to form a triangle with side lengths
of 6.8, 7.2, 5.1? If not, explain why not.
Check each inequality.
6.8 + 7.2 > 5.1
14 > 5.1
7.2 + 5.1 > 6.8
12.3 > 6.8 
6.8 + 5.1 > 7.2
11.9 > 7.2 
Since the sum of all pairs of side lengths are greater than
the third side length, sides with lengths 6.8, 7.2, and 5.1
will form a triangle.
Answer: yes
A. yes
B. no
B. Is it possible to form a triangle given the side
lengths 4.8, 12.2, and 15.1?
A. yes
B. no
Find Possible Side
Lengths
In ΔPQR, PQ = 7.2 and QR = 5.2. Which measure
cannot be PR?
A 7
B 9
C 11
D 13
Find Possible Side
Lengths
Read the Test Item
You need to determine which value is not valid.
Solve the Test Item
Solve each inequality to determine the range of values
for PR.
or
n < 12.4
Find Possible Side
Lengths
Notice that n > –2 is always true for any whole number
measure for x. Combining the two remaining
inequalities, the range of values that fit both inequalities
is n > 2 and n < 12.4, which can be written as
2 < n < 12.4.
Find Possible Side
Lengths
Examine the answer choices. The only value that does
not satisfy the compound inequality is 13 since 13 is
greater than 12.4. Thus, the answer is choice D.
Answer: D
In ΔXYZ, XY = 6, and YZ = 9. Which measure cannot
be XZ?
A. 4
B. 9
C. 12
D. 16
Proof Using Triangle Inequality Theorem
TRAVEL The towns of Jefferson, Kingston, and
Newbury are shown in the map below. Prove that
driving first from Jefferson to Kingston and then
Kingston to Newbury is a greater distance than
driving from Jefferson to Newbury.
Proof Using Triangle Inequality Theorem
Abbreviating the vertices as J, K, and N: JK represents
the distance from Jefferson to Kingstown; KN represents
the distance from Kingston to Newbury; and JN the
distance from Jefferson to Newbury.
Answer: By the Triangle Inequality Theorem,
JK + KN > JN. Therefore, driving from Jefferson to
Kingston and then Kingston to Newbury is a greater
distance than driving from Jefferson to Newbury.
Jacinda is trying to run errands
around town. She thinks it is a longer
trip to drive to the cleaners and then
to the grocery store, than to the
grocery store alone. Determine
whether Jacinda is right or wrong.
A. Jacinda is correct,
HC + CG > HG.
B. Jacinda is not correct,
HC + CG < HG.