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L8 – Exploring Triangles Name_____________________________ 8a – Angle and Side Relationships Per________ Date_________________

Geometry Q2: L8 – Exploring Triangles Handouts

Investigation of Roof Designs

Below are two possible designs for the roof of Grandma’s house. Uncle Bobby wants to know more about the designs and realizes the designs lack measurements. He need to know if the roofs are too steep to build or too flat (could leak). Using your ruler and protractor, measure each angle and each side. Record the measurements on the picture.

Roof Design #1 Roof Design #2

Based on your prior knowledge, what is the sum of the interior angles of a triangle?

Do your measurements agree with that sum?

Compare the length of a side to the measure of the angle opposite that side. Do you notice any patterns? (hint: look at the sizes of the sides and angles compared with the other sides and angles in the triangle).



Investigating Your Patterns

Measure the sides and the angles of the four triangles below. Label each on the sketch. Classify each triangle two ways:

A. By its Angle Name as Acute, Obtuse, or Right (#VOC). B. By its Sides Name as Scalene (all side lengths are different), Isosceles (at least 2 sides are

congruent), and Equilateral (all side lengths are congruent) (#VOC). 1) Angle Name ___________________

Side Name ____________________

2) Angle Name ___________________

Side Name ____________________

3) Angle Name ___________________

Side Name ____________________



4) Angle Name ___________________

Side Name ____________________

Based on your measurements, did the pattern you discovered in Grandma’s roof designs hold with these triangles? Trace one of the triangles onto patty paper. Rip off each vertex, and arrange the vertices so that they are adjacent. What do you notice? Make a conjecture about the sum of the angles. (Tape your patty paper next to your conjecture)



Triangle Angle-Sum Theorem (#THM) Based on your measurements, complete the statement: *The sum of the measures of the interior angles of a triangle is __________________. (#THM) Let’s prove it. Use the diagram below. Given: ∆𝐴𝐵𝐶 and line m ∥ 𝐶𝐴 Prove: 𝑚∠1+𝑚∠2+𝑚∠3 = 180°

Statement Reason



37.32°

130.8°

(𝑥 + 12)°

(3𝑥 + 32)°

(𝑥 +2)°

(𝑥 − 4)°

Using the Triangle Angle-Sum Theorem Fill in any missing interior angle measures. 1. 2. Solve for x in each of the problems. (Hint: Set up an equation to show the sum and solve for x.) 3. 4.

54°

47°

50°

(𝑥)°



Longer Side Inequality Theorem (#THM) Look at the “Investigating Your Patterns” activity from page 1. Make a conjecture from the work regarding the relationship between lengths of the sides and the measures of the angles. In more formal language, (#THM) we can say, *If two sides of a triangle are not ______, then the __________________ angle lies opposite the

__________________ side.

Using the Longer Side Inequality Based on the longer side inequality theorem, list the sides of the triangles in order of longest to shortest. Remember, the longest side must be opposite the __________________ angle.

1. 2.

Based on the longer side inequality theorem, list the angles in order of size from largest to smallest. Remember, the largest angle must be opposite the __________________ side.

3. 4.

A

B C

70°

67° F

E

D126°

36°

A

C B6.04

5.65 5.93

5.79 6.19

7.49

D

F

E

L8 – Exploring Triangles Name_____________________________ 8b – Exterior Angle Theorem Per________ Date_________________


Floor

Angle we need to measure

Front wall

23° 90°

Exterior Angle Theorem

Grandma has a side job for use. She wants her chicken coop repaired. We need to start with a repair to where the roof meets the front wall. We need to build a support that is the exact same angle as that. It’s too dangerous to measure that angle directly, but luckily, we know the angle between the roof and ceiling and the angle between the ceiling and front wall. Below is a sketch of the chicken coop and the angles that we know.

Below is a sketch of a side view Grandma’s old chicken coop.

How can we deduce the measure of the angle we are looking for without actually measuring it? (Complete the next activity to find out.)

Roof

Ceiling



A

B DC

The diagram below shows an exterior angle, its adjacent interior angles, and remote interior angles of a triangle. (#VOC)

Measure the exterior angles and remote interior angles on the below triangles. Compare your measurements with your classmates.

Based on your measurements, describe any patterns you see between the exterior angle and the remote interior angles.



A

B DC

Remote InteriorAngle

Adjacent InteriorAngle

Exterior Angle

Exterior Angle Theorem (#THM) The measure of an ____________________ angle is equal to the __________ of the measures of the

two ________________ ____________________ _________.

Use the figure below to prove the exterior angle theorem. Given: ∆𝐴𝐵𝐶 and exterior angle ∠𝐷𝐶𝐴 Prove: 𝑚∠𝐷𝐶𝐴 = 𝑚∠𝐴 +𝑚∠𝐵

Statement Reason

L8 – Exploring Triangles Name_____________________________ 8c Triangle Inequality Theorem Per________ Date_________________


Creating Triangles

In this exercise, you will create triangle with straws. You should have 6 different length straws of 5, 6, 8, 12, 13, 14 cm. Create triangles with the indicated lengths. Record whether your triangles are acute, obtuse, or right. If it is not possible to create a triangle with the straws, write “not possible”. All the measurements are in cm and the sides are labeled a, b, and c.

1. a=5, b=12, c=13 2. a=5, b=6, c=8 3. a=5, b=6, c=12 4. a=5, b=8, c=12 5. a=5, b=8, c=13 6. a=8, b=12, c=14

7. a=6, b=8, c=13 8. a=6, b=8, c=14 List the side lengths of the triangles below in the possible or not possible categories.

Possible Not Possible Do you notice any patterns? In other words, why were some combinations not possible?



Triangle Inequality Theorem (#THM) The sum of the lengths of any two sides of a triangle must be ___________________ than the length of the ______________ side.

Possible or Not?

Determine whether you could construct a triangle given the following combinations of side lengths. If “not possible”, explain your reasoning.

1) 4, 5, 9 cm 2) 4, 6, 9 cm 3) 7, 13, 8 cm 4) 9, 3, 5 cm

5) Dezmond decides to build a triangular garden. He has three extra 2x4 boards lying in his shed. He measures the boards and finds the lengths to be 4, 10, and 3 ft. a. Will he be able to build a triangular garden without adjusting his boards? Explain your reasoning. b. If Dezmond is only comfortable in working in whole number measurements, what does he have to do to ensure that he is able to build a triangular garden with those boards?

6) Gabby saw Dezmond’s triangular boxes and wants to build her own. She has 6 boards, but no way to cut the boards. Her boards have lengths of 4 ft, 7 ft, 2 ft, 3.5 ft, 6 ft, and 5 ft. What combinations of boards will not work to build a triangle? Explain your reasoning.

M

U S



Young’s Village

Takahashi’s Village

Challenge: Shortest Way – As the Nene Flies Below is a diagram of two villages on a map and a lake between the two villages.

The dotted line represents the flight of a nene between the two villages, and the solid line represents the walking trail. Using the triangle inequality theorem, explain why the flight of the nene is shorter than the walking trail. Extension: Justify the statement “the shortest distance between two points is a straight line.”

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