Conditions on Measurements that Determine a Triangle

Examples, videos, and solutions to help Grade 7 students learn how to draw triangles under different criteria to explore which criteria result in many, a few, or one triangle.

New York State Common Core Math Grade 7, Module 6, Lesson 11

Worksheets for Grade 7

Lesson 11 Student Outcomes

• Students understand that three given lengths determine a triangle, provided the largest length is less than the sum of the other two lengths; otherwise, no triangle can be formed.
• Students understand that if two side lengths of a triangle are given, then the third side length must be between the difference and the sum of the first two side lengths.
• Students understand that two angle measurements determine many triangles, provided the angle sum is less than 180°; otherwise, no triangle can be formed.

Lesson 11 Summary

• Three given lengths determine a triangle, provided the largest length is less than the sum of the other two lengths; otherwise, no triangle can be formed.
• Two angle measurements determine a triangle, provided the sum of the two angle measurements is less than 180° ; otherwise, no triangle can be formed.
• Three given angle measurements do not determine a unique triangle. Scale drawings of a triangle have equal corresponding angle measurements, but corresponding side lengths that are proportional.

Lesson 11 Classwork

Exploratory Challenge 1
a. Can any three side lengths form a triangle? Why or why not?
b. Draw a triangle according to these instructions:

• Draw segment AB of length 10 cm in your notebook.
• Draw segment BC of length 3 cm on one piece of patty paper.
• Draw segment AC of length 5 cm on the other piece of patty paper.
• Line up the appropriate endpoint on each piece of patty paper with the matching endpoint on AB.
• Use your pencil point to hold each patty paper in place, and adjust the paper to form ABC.

c. What do you notice?
d. What must be true about the sum of the lengths of AB and BC if the two segments were to just meet? Use your patty paper to verify your answer.
e. Based on your conclusion for part (c), what if BC = 3 cm as you originally had, but AC = 10 cm in length. Could you form triangle ABC?
f. What must be true about the sum of the lengths of AC and BC if the two segments were to meet and form a triangle?

Exercise 1
Two sides of triangle DEF have lengths of 5 cm and 8 cm. What are all the possible whole-number lengths for the remaining side?

Exploratory Challenge 2
a. Which of the following conditions determine a triangle? Follow the instructions to try and draw triangle ABC. Segment has been drawn for you as a starting point in each case.
i. Choose measurements of ∠A and ∠B for triangle ABC so that the sum of measurements is greater than 180°. Label your diagram. Your chosen angle measurements:
Were you able to form a triangle? Why or why not?
ii. Choose measurements of ∠A and ∠B for triangle ABC so that the measurement of ∠A is supplementary to the measurement of ∠B. Label your diagram.
Your chosen angle measurements:
Were you able to form a triangle? Why or why not?
iii. Choose measurements of ∠A and ∠B for triangle ABC so that the sum of measurements is less than 180°. Label your diagram. Your chosen angle measurements:
Were you able to form a triangle? Why or why not?
b. Which condition must be true regarding angle measurements in order to determine a triangle?
c. Measure and label the formed triangle in part (b) with all three side lengths and the angle measurement for ∠ C. Now, use a protractor, ruler, and compass to draw triangle ABC with the same angle measurements, but side lengths that are half as long.
d. Do the three angle measurements of a triangle determine a unique triangle? Why or why not?

Exercise 2
Which of the following sets of angle measurements determines a triangle?
Choose one example from above that does determine a triangle and one that does not. For each, explain why it does or does not determine a triangle using words and a diagram.

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