Triangle Inequalities

Step 1: Recall the Triangle Inequality Theorem: The sum of any two sides of a triangle must be greater than the third side

Step 2: Check all three combinations: Test 1: 3 + 7 > 12? 10 > 12? NO ✗

Test 2: 3 + 12 > 7? 15 > 7? YES ✓

Test 3: 7 + 12 > 3? 19 > 3? YES ✓

Step 3: Analyze the results: Since 3 + 7 is NOT greater than 12, the Triangle Inequality is violated

Step 4: Conclusion: These three lengths cannot form a triangle The two shorter sides (3 and 7) aren't long enough to reach around the longest side (12)

Answer: No, these cannot form a triangle because 3 + 7 ≯ 12

Step 1: Use Triangle Inequality Theorem: The third side must satisfy:

• 5 + 8 > third side
• 5 + third side > 8
• 8 + third side > 5

Step 2: Simplify each inequality: From first: third side < 13 From second: third side > 3 From third: third side > -3 (always true for positive lengths)

Step 3: Combine the restrictions: third side > 3 AND third side < 13 Therefore: 3 < third side < 13

Step 4: Interpret: The third side must be greater than 3 and less than 13

Step 5: Note the pattern: The third side must be between:

• The difference of the two known sides: |8 - 5| = 3
• The sum of the two known sides: 8 + 5 = 13

Answer: The third side must be greater than 3 and less than 13 (3 < x < 13)

Step 1: Apply Triangle Inequality: The third side must be between the difference and the sum of the two known sides

Step 2: Find the minimum (difference): Minimum = |10 - 6| = 4 AC must be greater than 4

Step 3: Find the maximum (sum): Maximum = 10 + 6 = 16 AC must be less than 16

Step 4: Express the range: 4 < AC < 16

Step 5: Verify with extreme cases: If AC = 4: sides would be 10, 6, 4 Check: 6 + 4 = 10 (not greater than) ✗ If AC = 4.1: 6 + 4.1 = 10.1 > 10 ✓

If AC = 16: sides would be 10, 6, 16 Check: 10 + 6 = 16 (not greater than) ✗ If AC = 15.9: 10 + 6 = 16 > 15.9 ✓

Answer: 4 < AC < 16

Step 1: Apply all three Triangle Inequalities:

1. DE + EF > DF: (2x) + (x + 5) > 12
2. DE + DF > EF: (2x) + 12 > (x + 5)
3. EF + DF > DE: (x + 5) + 12 > (2x)

Step 2: Solve inequality 1: 2x + x + 5 > 12 3x + 5 > 12 3x > 7 x > 7/3 x > 2.33...

Step 3: Solve inequality 2: 2x + 12 > x + 5 x + 12 > 5 x > -7 (Always true for positive x)

Step 4: Solve inequality 3: x + 5 + 12 > 2x x + 17 > 2x 17 > x x < 17

Step 5: Combine all restrictions: x > 7/3 AND x < 17 Therefore: 7/3 < x < 17 Or: 2.33... < x < 17

Step 6: Also ensure all sides are positive: 2x > 0 → x > 0 x + 5 > 0 → x > -5 The binding constraint is x > 7/3

Answer: 7/3 < x < 17 (approximately 2.33 < x < 17)

Step 1: Understand the Hinge Theorem (SAS Inequality): If two sides of one triangle are congruent to two sides of another triangle, but the included angle of the first is larger, then the third side of the first triangle is longer than the third side of the second triangle

Step 2: Identify the given information: Triangle ABC: sides AB = 8, AC = 10, included angle A = 60° Triangle DEF: sides DE = 8, DF = 10, included angle D = 45°

Step 3: Compare the included angles: Angle A = 60° Angle D = 45° Therefore: Angle A > Angle D

Step 4: Apply the Hinge Theorem: Since AB = DE, AC = DF, and angle A > angle D, then BC > EF

Step 5: Conceptual understanding: The larger included angle (60°) "pushes" the opposite side (BC) farther apart than the smaller angle (45°) pushes EF apart

Think of it like opening a door:

• Opening 60° creates a wider gap than opening 45°

Answer: BC > EF (BC is longer than EF)