Unique Triangles—Two Sides and a Non-Included Angle

Video solutions to help Grade 7 students learn understand that two sides of a triangle and an acute angle, not included between the two sides, may not determine a unique triangle.

New York State Common Core Math Grade 7, Module 6, Lesson 12

Lesson 12 Student Outcomes

• Students understand that two sides of a triangle and an acute angle, not included between the two sides, may not determine a unique triangle.
• Students understand that two sides of a triangle and a angle (or obtuse angle), not included between the two sides, determine a unique triangle.

Lesson 12 Summary

• A triangle drawn under the condition of two sides and a non-included angle, where the angle is acute, does not determine a unique triangle. This condition determines two non-identical triangles.
• Consider a triangle correspondence △ABC ↔ △XYZ that corresponds to two pairs of equal sides and one pair of equal, non-included angles. If the triangles are not identical, then △ABC can be made to be identical to △XYZ by swinging the appropriate side along the path of a circle with a radius length of that side.
• A triangle drawn under the condition of two sides and a non-included angle, where the angle is 90° or greater, does determine a unique triangle.

Lesson 12 Classwork

Exploratory Challenge
1. Use your tools to draw △ABC, provided AB = 5 cm, BC = 3 cm, and ∠A = 30°. Continue with the rest of the problem as you work on your drawing.
a. What is the relationship between the given parts of △ABC?
b. Which parts of the triangle can be drawn without difficulty? What makes this drawing challenging?
c. A ruler and compass are instrumental in determining where C is located.
• Even though the length of AC is unknown, extend the ray AC in anticipation of the intersection with BC.
• Draw segment BC with length 3 cm away from the drawing of the triangle.
• Draw a circle with center B and a radius equal to BC, or 3 cm.
d. How many intersections does the circle make with AC? What does each intersection signify?
e. Complete the drawing of △ABC.

2. Now attempt to draw this triangle: draw △DEF, provided DE = 5 cm, EF = 3 cm, and ∠F = 90°.
a. How are these conditions different from those in Exercise 1, and do you think the criteria will determine a unique triangle?
b. What is the relationship between the given parts of △DEF?
c. Describe how you will determine the position of DE.
d. How many intersections does the circle make with FE?
e. Complete the drawing of △DEF. How is the outcome of △DEF different from that of △ABC?

3. Now attempt to draw this triangle: draw △JKL, provided KL = 8 cm, KJ = 4 cm, and ∠J = 120°. Use what you drew in Exercises 1 and 2 to complete the full drawing.

4. Review the conditions provided for each of the three triangles in the Exploratory Challenge, and discuss the uniqueness of the resulting drawing in each case.

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