L4 · Pythagoras and angles clinic

Learning goal

“I can prove whether three lengths can form a triangle at all (triangle inequality theorem), and I can choose and chain the right theorems - inequality, Pythagoras, converse, angle sum, isosceles - to analyze a triangle problem completely.”

Triangle inequality theorem — the sum of any two side lengths of a triangle is greater than the third side length: a + b > c, a + c > b, b + c > a. In practice only one check can fail: the two SHORTEST sides against the longest. Picture: two short sticks laid end to end along a longer stick - if they cannot reach past its end, no triangle closes.

Worked example

Can sides 4 cm, 5 cm and 8 cm form a triangle? Step 1: Only one check can fail - the two shortest against the longest: 4 + 5 = 9. Step 2: Compare with the longest: 9 > 8. The other two checks (4 + 8 > 5, 5 + 8 > 4) are automatic when this one passes. Step 3: Yes - the three lengths form a triangle. Say it plainly: the theorem demands GREATER THAN. Equal is not enough.

Check yourself — explain it

Sides 5, 5, 10 cannot form a triangle because...

Paired problem — example still visible

Sides 2 cm, 3 cm, 6 cm: 2 + 3 = 5 < 6 -> cannot form a triangle

Faded completion — last step is yours

"Sides 5 cm, 5 cm, 10 cm: two shortest sum to ___; the theorem requires ___ than 10; so the sides ___ form a triangle." -> 10; greater; cannot

Ready to practise?

10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.

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