6.5 Indirect Proof and Inequalities in One Triangle with Work
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6.5 Indirect Proof and Inequalities in One Triangle with work 1 Three data sets black • Centriod • Circumcenter • Incenter • Mdpt • Vertex • I • Orthocenter Median Altitude Perpendicular Bisector • POC is located inside the triangle • POC is located inside, outside, or on the triangle Three data sets black • Centriod • Circumcenter • Incenter Angle Bisector • Mdpt • Vertex • I • Orthocenter Median Altitude Perpendicular Bisector • POC is located inside the triangle • POC is located inside, outside, or on the triangle
6.5 Indirect Proof and Inequalities in One Triangle with Work
6.5 Indirect Proof and Inequalities in One Triangle with
work6.5 Indirect Proof and Inequalities in One Triangle with work
1
• POC is located inside, outside,
or on the triangle
Three data sets black
• POC is located inside, outside,
or on the triangle
6.5 Indirect Proof and Inequalities in One Triangle with work
2
Pull out your activity sheet from yesterday!
Dec 81:01 PM
Roll three die. Record the numbers from least to greatest in the chart.
Use the straws to make a triangle and record results in chart below.
PART 1: Can any 3 different side
lengths make a triangle?
6.5 Indirect Proof and Inequalities in One Triangle with work
3
2)
List the measures that did not form triangles.
3)
List all possible conjectures based on data gathered.
4)
Write a sentence or more about what the BIG IDEA of the
experiment might be.
5) What side lengths would make a triangle if two of its
side lengths were 3 and 10? (hint: write as an inequality)
Dec 81:17 PM
A triangle is formed by three segments, but not
every set of three segments can form a triangle.
Determine if a triangle can have side lengths of: 8, 13, 21 .
Explain.
6.5 Indirect Proof and Inequalities in One Triangle with work
4
Dec 81:25 PM
The figure shows the approximate distances
between cities in California. What is the
range of distances from San Francisco to
Oakland?
Example 1: Travel Application
Dec 811:21 AM
PART 2: Inequalities Activity
for Sides and Angles of Triangles
Triangle inequality side lengths and angle measures.gsp
What conjecture can you make about the relationship of the sides and angles?
You need a paper, ruler, and protractor. Draw a triangle of any size on the back of the paper and
label A, B, C! Then determine the side and angle measures using your construction tools. Return
them to the tin when you are done and answer the three questions at the bottom.
6.5 Indirect Proof and Inequalities in One Triangle with work
5
Next day warmup Inequalities of Sides
Tell whether a triangle can have sides with the
given lengths. A. 4,10,7 B. 12,2,9
C. 3, 1.1, 1.7
Find the possible lengths of 3rd side:
• 28 and 23 • 4 and 19 •
3.07 and 1.89
Nov 123:47 PM
Examples 2 & 3 •
List the sides of ΔABC in
order from shortest to longest.
x=12 m<A=56, m<B=61, m<C=63
so BC<AC<ABA
B
C
(3x + 20)o
(2x + 37)o
(4x + 15)o
• List the angles of ΔKLM
in order from least to greatest
if the perimeter of ΔKLM
is 47 units.
x + 4
6.5 Indirect Proof and Inequalities in One Triangle with work
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Feb 221:53 PM
Dec 81:29 PM
Examples: 4 & 5: Using the Hinge Theorem •
Compare m∠BAC and m∠DAC.
•
Find the range of values for k.
6.5 Indirect Proof and Inequalities in One Triangle with work
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Dec 81:33 PM
John and Luke leave school at the same time. John
rides his bike 3 blocks west and then 4 blocks north.
Luke rides 4 blocks east and then 3 blocks at a
bearing of N 10º E. Who is farther from school?
Explain.
Example 6: Travel Application
Dec 148:27 AM
Answer: AD, BD, AB, BC, CD
The five speed tubes of this mountain bike frame form two
triangles. List the 5 tubes in order from shortest to longest.
Explain your answer.
Example 7: Application
6.5 Indirect Proof and Inequalities in One Triangle with work
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Nov 123:47 PM
Again Extra Example
• Which side of ΔRTU is the longest?
•
Name the side of ΔUST that is the longest.
•
If TU is an angle bisector, which side of ΔRST is the longest?
Triangle Midsegment.gsp
6.5 Indirect Proof and Inequalities in One Triangle with work
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A: 29, 35, 37, 41, 43, 45, 50 53
B: 7, 11, 15, 19, 23, 29, 35, 37, 41, 43, 50 53
C: 7, 11 23 (o), 29, 35, 37, 50 53
50.
52.
Dec 81:27 PM
Ray wants to place a chair so it is 10 ft from his
television set. Can the other two distances
shown be 8 ft and 6 ft? Explain.
hw check problems
When the swing ride is at full speed, the
chairs are farthest from the base of the
swing tower. What can you conclude about
the angles of the swings at full speed
versus low speed? Explain.
Answer: The ∠ of the swing at full speed is
greater than the ∠ at low speed because the
length of the triangle on the opposite side is
the greatest at full swing.
Find the range of values for z.
6.5 Indirect Proof and Inequalities in One Triangle with work
10
Midsegment Review •
MP is a midsegment. LM = , MP = ,
and NO = .
Find: X = ____ LN = ___
MP = ___
Nov 123:47 PM
Another Extra Example •
Draw triangle with vertices F(0,0), R(3,0),
and I(0,4). List the angles in order from least
measure to greatest measure.
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