Notes on Prealgebra Lesson 10-3: Draw Triangles with Given Conditions

Key Concepts

• The lesson focuses on drawing triangles given their side lengths and angle measures.
• Essential question: How can you determine when it is possible to draw a triangle given certain conditions?

Solve and Discuss

• Problem: Kane has four pieces of wood to build a triangular garden.
• Key Principle: To form a triangle, the sum of the lengths of any two sides must be greater than the length of the third side.
  • Example: Pieces of wood with lengths 2, 3, and 5 cannot form a triangle because 2 + 3 = 5.
  • Valid combinations: 2, 3, and 4; 3, 4, and 5; 2, 4, and 5.

Types of Triangles

• Side Combinations That Form a Triangle:
  • All three sides must satisfy the triangle inequality principle.
• Side Combinations That Do Not Form a Triangle:
  • Example: 2, 3, and 5 because two sides equal the third one.

Focus on Math Practices

• Key Rule: A triangle cannot be formed if the sum of two sides is equal to or less than the third side.

Examples

Example 1: Drawing with Given Side Lengths

• Problem: Will students cut the same triangle with sides 6, 8, and 10 inches?
• Solution: Yes, the triangle formed is unique due to the side lengths, creating a specific shape and size.

Example 2: Possible Side Lengths

• Problem: Can Steve make a triangle with pieces 3, 4, and 8?
• Solution: No, because 3 + 4 < 8.
• Conclusion: The sum of the two shorter sides must be greater than the longest side.

Example 3: Side Lengths and Angle Measures

• Problem: Can more than one triangle be drawn with sides 5 and 6 inches, forming a 45° angle?
• Solution: No, since the sides and angle are fixed, only one triangle can be formed.

Example 4: Non-Included Angle

• Problem: Can more than one triangle be drawn with sides 6 and 9 units and a non-included 40° angle?
• Solution: Yes, multiple triangles can be formed as the angle is not between the fixed sides.

Example 5: Angle Measures

• Problem: Is there a unique triangle with angles of 30°, 60°, and 90°?
• Solution: No unique size, as side lengths can vary despite having the same angles.

Try It Problems

• Problem A: Write three side lengths that will form/not form a triangle.
• Solution: Valid if two shorter sides sum to more than the third.
• Problem B: Can a triangle with a 3-inch side and angles 90° and 89° be drawn?
• Solution: Yes, as they will meet because they are not parallel.

Key Concepts Summary

• Unique Triangle Conditions:
  • Sides: Side-Side-Side (SSS)
  • Angle-Side-Angle (ASA)
  • Side-Angle-Side (SAS)
• Non-Unique Conditions:
  • All angles fixed without side lengths being specific.
  • Two sides with a non-included angle.

Conclusion

• Analyze side lengths and angle measures to determine if one unique triangle, more than one unique triangle, or no triangle can be drawn.