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Warm Up – use last week’s
blue paper
• Solve for the value of x.
Segment DE is a midsegment.
1.
2.
x
3. Given AC = 42, CB = 46, AB = 48.
D, E, F are midpoints.
Find the perimeter of triangle DEF.
Investigation 1 – What is the shortest path from A to B?
• Each person in your group should do each
construction. Compare results when you finish.
• You will need: a compass and a straightedge
• Follow directions from the sheet given to you at the
beginning of class.
• You should have been able to construct
∆CAT, but not ∆FSH. Why? Discuss
your results with others. State your
observation as your next conjecture.
• Triangle Inequality Conjecture
The sum of the lengths of any two sides
greater
of triangle is _____________
the length
of the third side.
Investigation 2 – Where Are the Largest and Smallest Angles?
• You will need: a ruler, a
protractor
• Each person should draw a
different scalene triangle for this
investigation. Some group
members should draw acute
triangle, and some should draw
obtuse triangles.
1. Measure the angles in your triangle.
Label the angle with greatest measure
∠L, the angle with second greatest
measure ∠M, and the smallest angle
∠S.
2. Measure the three sides. Label the
longest side l, the second longest side
m, and the shortest side s.
3. Which side is opposite ∠L? ∠M? ∠S?
Discuss your results with others. Write a
conjecture that states where the largest and
smallest are in a triangle, in relation to the
longest and shortest sides.
• Side-Angle Inequalities Conjecture
In a triangle, if one side is longer than
another side, then the angle opposite
Larger than the angle opposite
the longer side is ______________
the shorter side
Investigation 3 – Exterior Angles of a Triangle
• You will need: a straightedge, patty paper.
Each person should draw a different scalene
triangle for this investigation. Some group
members should draw acute, some should
draw obtuse, and some should draw right
triangles.
1. On your paper, draw a triangle, ∆ABC.
Extend segment AC beyond C and
label a point D outside the triangle on
ray AC. Label the angles as shown.
b
a
c x
2. Copy the two remote interior angles,
∠A and ∠B, onto patty paper to show
their sum.
a
b
3. How does the sum of a and b compare
with x? Use your patty paper from
Step 2 to compare.
4. Discuss your results with your group. State
your observations as a conjecture.
Triangle Exterior Angle
Conjecture
• The measure of an exterior angle of a
triangle ___________________
equals to the sum of the two non-adjacent
interior angles
Say What???
The early Egyptians used to make triangles by
using a rope with knots tied at equal intervals. Each
vertex of the triangle had to occur at a knot.
Suppose you had a rope with exactly 10 knots
making 9 equal lengths as shown below. How many
different triangles could you make?
PLAN:
Let x, y, and z be the length of each side.
Check every possible combination of
x + y + z = 9 to see how many can be made
into triangles.
A table can help us keep track of the
y
z
Triangle?
combinations. x
1
1
1
1
2
2
1
2
3
4
2
3
7
6
5
4
5
4
x
1
1
1
1
2
2
3
y
1
2
3
4
2
3
3
z
7
6
5
4
5
4
3
Triangle?
no
no
no
yes
no
yes
yes
Ch. 5.5 Inequalities in One
Triangle
Students will
compare side and
angle measures in
a triangle.
Theorem 5.10
• If one side of a triangle is longer than
another side, then the angle opposite
the longer side is larger than the
angle opposite the shorter side.
Theorem 5.11
• If one angle of a triangle is larger
than another angle, then the side
opposite the larger angle is longer
than the side opposite the smaller
angle.
600
800
Practice Problem
• Name the largest side of the triangle
(Note: Triangle is not drawn to scale.)
Exterior Angle Inequality
Theorem
• The measure of an exterior angle of a
triangle is greater than the measure of
either of the two nonadjacent interior
angles.
m1 > mA and m1 > mB
A
1
C
B
Triangle Inequality Theorem
(5.13)
• The sum of the lengths of any two sides
of a triangle is greater than the length of
the third side.
A
• AB + BC > AC
• AC + BC > AB
• AB + AC > BC
C
B
Examples:
1. Find all possible values of x if
K
12 + x
20-x
J
15
L
2. The measures of three sides of a triangle are given.
Use the Triangle Inequality Theorem to find all possible
values of x.
8, 10 - x, 3 + x
Now you try…
The measures of three sides of a triangle are given.
Use the Triangle Inequality Theorem to find all possible
values of x.
1.
2x + 5, 4x, 30-x
2. 5x - 3, 2x +7, 3x
5<x<11 2/3
X>5/3
• Hinge Theorem
• All investigations are from Discovering
Geometry – An Investigative Approach by
Michael Serra, Key Curriculum Press.