Geometry path · G03

Before this lesson: Triangle congruence, Inequality rules and signs

Your goal: Use strict triangle inequalities and explain geometric equality limits.

Start with the idea and one example. Take a break before the written task if you need it. The questions are original teaching exercises, not official past-paper questions.

Triangle inequalities: the key idea

Three positive lengths a,b,c form a nondegenerate triangle exactly when each is smaller than the sum of the other two. Equivalently |a−b|<c<a+b. Equality describes a straight configuration and is excluded. In a triangle, the larger side lies opposite the larger angle, with equal sides opposite equal angles. The distance along a broken path is at least the straight-line distance between its endpoints; equality requires the intermediate points to lie in order on the straight segment.

Triangle ABC with marked points on its sides.ABCDE
D lies inside ABC. Extend AD to E on BC for the broken-path proof.

A worked example

Two sides are 4 and 7. Which positive integer values can the third side take?

The triangle inequality gives |7−4|<c<7+4, so 3<c<11. The integer possibilities are 4,5,6,7,8,9,10: seven values.

Your turn: change one thing

In triangle ABC, AB=5,BC=7,CA=6. Which angle is largest?

Try this on paper before opening the explanation.

Compare your reasoning

The largest side is BC, opposite ∠A. Thus ∠A is largest. Match the angle to the opposite side, not a neighbouring side.

Pause and check

A trap to avoid: Including the degenerate equality case as an ordinary triangle.

Practise and adjust the level

Foundation checks the language; Core applies the method; Stretch asks you to choose or justify an idea. These are levels within this lesson. A session selects six of the nine questions; unused questions allow the level to change. Advanced theory still needs written practice.

Interactive practice loads here. You can also use the complete question set below.

Write a complete argument

Prove that for a point D strictly inside triangle ABC, AD+DB<AC+CB.

Planning hint

List the assumptions and the exact conclusion. Identify the definition or theorem in this lesson that connects them. Explain why its conditions hold before using it.

Read the full solution after your attempt

Extend AD to meet BC at E. In triangle DEB, DB<DE+EB, so AD+DB<AE+EB. Then triangle ACE gives AE<AC+CE. Hence AD+DB<AC+CE+EB=AC+CB. Both inequalities are strict because the involved points form nondegenerate triangles.

My proof notebook

Write on paper, or keep a draft here. Compare your reasoning with the solution only after a real attempt. The checklist is your own review, not an automatic mark.

All nine practice questions

Prefer paper or have JavaScript switched off? The complete question set, hints and solutions are here. Interactive practice uses these same questions in an order chosen from your answers.

1. The third side c with sides a,b must satisfy what?

Foundation

  1. 0<c always suffices
  2. c>|a+b|
  3. |a−b|<c<a+b
  4. c=a+b
Hint

Use both upper and lower triangle bounds.

Answer and reasoning

|a−b|<c<a+b. The two shorter-side inequalities combine to c>|a−b|.

2. Which angle is opposite side BC?

Foundation

  1. ∠C
  2. All three
  3. ∠A
  4. ∠B
Hint

The opposite vertex is not on the side.

Answer and reasoning

∠A. A is the vertex outside the endpoints B,C.

3. With sides 4 and 7, how many integer third sides are possible?

Core

  1. 8
  2. 10
  3. 7
  4. 6
Hint

List integers strictly between 3 and 11.

Answer and reasoning

7. They are 4 through 10 inclusive, seven integers.

4. Can 2,3,5 be a nondegenerate triangle?

Core

  1. No
  2. Yes
  3. Only if right-angled
  4. Only if obtuse
Hint

Check the strict inequality.

Answer and reasoning

No. 2+3=5 creates a degenerate straight configuration.

5. In a triangle with sides 5,6,7, the largest angle faces which side?

Stretch

  1. All angles are equal
  2. 7
  3. 5
  4. 6
Hint

Side and opposite-angle order agree.

Answer and reasoning

7. The longest side lies opposite the largest angle.

6. When can a two-segment path equal the direct distance?

Stretch

  1. The point is outside the line
  2. Only when both lengths are zero
  3. The intermediate point lies on the segment between endpoints
  4. The segments are perpendicular
Hint

Use the equality condition in the triangle inequality.

Answer and reasoning

The intermediate point lies on the segment between endpoints. The path must proceed straight without reversing direction.

7. In a triangle, equal sides face what?

Foundation

  1. Equal angles
  2. Right angles always
  3. Supplementary angles always
  4. Unequal angles
Hint

Use side-angle correspondence.

Answer and reasoning

Equal angles. The isosceles triangle theorem gives equal opposite angles.

8. Sides 5 and 9 are given. Which third side is allowed?

Core

  1. 14
  2. 15
  3. 7
  4. 4
Hint

Require 4<c<14.

Answer and reasoning

7. Only 7 lies strictly inside the valid interval.

9. Why exclude c=a+b?

Stretch

  1. It is never an equality of numbers
  2. The vertices would be collinear
  3. It produces a right triangle
  4. The sides become negative
Hint

The straight path would equal the broken path.

Answer and reasoning

The vertices would be collinear. This is a degenerate triangle of zero area.

Choose your next step

Continue to Triangle area ratios. If this felt difficult, return to a prerequisite above. Every lesson stays open.

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