Warm Up – use last week’s blue paper

• Solve for the value of x.

1. 2.

x

3. Given AC = 42, CB = 46, AB = 48. D, E, F are midpoints. Find the perimeter of triangle DEF.

Segment DE is a midsegment.

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 of triangle is _____________ the length of the third side.

greater

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 the longer side is ______________Larger than the angle opposite

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 combinations. x y z

Triangle? 1 1 7 1 2 6 1 3 5 1 4 4 2 2 5 2 3 4 3 3 3

x y z Triangle?

1 1 7

1 2 6

1 3 5

1 4 4

2 2 5

2 3 4

3 3 3

nononoyesnoyes

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.

800

600

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.

m1 > mA and m1 > mB A

BC

1

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.

• AB + BC > AC

• AC + BC > AB

• AB + AC > BC

A

CB

Examples:1. Find all possible values of x if

12 + x 20-x

15J

K

L

2. The measures of three sides of a triangle are given. Use the Triangle Inequality Theorem to find all possiblevalues 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 possiblevalues 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.