Reference / Curriculum / 3B / Chapter 12

3B · Chapter 12 · Lesson 5

Properties of Triangles

Reason about equal sides/angles in drawn triangles.

MATH-3B-C12-L05Independent and tutor-supported

Today's mathematical question

A triangle has side lengths 3, 3, and 5 units. Classify it by its side lengths.

Learning target

Reason about equal sides/angles in drawn triangles.

Before starting: Prior lesson: Triangles. Try the warm-up independently.

Prerequisite · 3 questions

Check earlier skills first

These are earlier ideas you need for today; revisit them if they are difficult.

  1. P1

    A triangle has side lengths 5, 5, and 7 units. Classify it by its side lengths.

    Reveal explanation

    Answer: isosceles

    Isosceles means exactly two equal sides. These lengths satisfy the triangle inequality.

  2. P2

    A triangle has side lengths 8, 8, and 10 units. Classify it by its side lengths.

    Reveal explanation

    Answer: isosceles

    Isosceles means exactly two equal sides. These lengths satisfy the triangle inequality.

  3. P3

    A triangle has side lengths 9, 9, and 9 units. Classify it by its side lengths.

    Reveal explanation

    Answer: equilateral

    Equilateral means three equal sides. These lengths satisfy the triangle inequality.

Learn and understand

Side lengths help us describe triangles. Three equal sides form an equilateral triangle; exactly two equal sides form an isosceles triangle; and three different lengths form a scalene triangle. These names describe side relationships, not the direction of the drawing. For sides 4, 4, and 6, the triangle is isosceles because two sides match. Lengths 2, 3, and 8 cannot form a triangle because the shorter two cannot reach across the longest.

Represent it three ways

  1. Concrete: Build equal pieces, measures, counters or shapes with real objects when useful.
  2. Pictorial: Use equal-length marks on the sides and sort several triangles. Try a rotated example to check that orientation does not change the category.
  3. Abstract: Write the matching number sentence, calculation, comparison or mathematical description. Explain each number's unit.

Important vocabulary

equilateral
three equal sides
isosceles
at least two equal sides (here exactly two is used to distinguish from equilateral)
scalene
no equal sides

Fully worked examples and checks

Try each new understanding check before revealing the explanation.

Worked example 1

Explore, calculate, verify

A triangle has side lengths 3, 3, and 5 units. Classify it by its side lengths.

  1. Identify the whole, the units, and what the question asks.
  2. Isosceles means exactly two equal sides. These lengths satisfy the triangle inequality.
  3. Check that the answer uses the right units and makes sense.

Answer: isosceles

Check your understanding 1

A triangle has side lengths 7, 7, and 9 units. Classify it by its side lengths.

Reveal answer and explanation

isosceles — Isosceles means exactly two equal sides. These lengths satisfy the triangle inequality.

Worked example 2

Explore, calculate, verify

A triangle has side lengths 10, 10, and 10 units. Classify it by its side lengths.

  1. Identify the whole, the units, and what the question asks.
  2. Equilateral means three equal sides. These lengths satisfy the triangle inequality.
  3. Check that the answer uses the right units and makes sense.

Answer: equilateral

Check your understanding 2

A triangle has side lengths 4, 4, and 6 units. Classify it by its side lengths.

Reveal answer and explanation

isosceles — Isosceles means exactly two equal sides. These lengths satisfy the triangle inequality.

Worked example 3

Explore, calculate, verify

A triangle has side lengths 12, 12, and 14 units. Classify it by its side lengths.

  1. Identify the whole, the units, and what the question asks.
  2. Isosceles means exactly two equal sides. These lengths satisfy the triangle inequality.
  3. Check that the answer uses the right units and makes sense.

Answer: isosceles

Check your understanding 3

A triangle has side lengths 9, 10, and 11 units. Classify it by its side lengths.

Reveal answer and explanation

scalene — Scalene means no equal sides. These lengths satisfy the triangle inequality.

Guided · 5 questions

Guided practice

Keep the model and earlier examples visible while practising.

  1. G1

    A triangle has side lengths 13, 13, and 13 units. Classify it by its side lengths.

    Reveal explanation

    Answer: equilateral

    Equilateral means three equal sides. These lengths satisfy the triangle inequality.

  2. G2

    A triangle has side lengths 12, 13, and 14 units. Classify it by its side lengths.

    Reveal explanation

    Answer: scalene

    Scalene means no equal sides. These lengths satisfy the triangle inequality.

  3. G3

    A triangle has side lengths 5, 5, and 5 units. Classify it by its side lengths.

    Reveal explanation

    Answer: equilateral

    Equilateral means three equal sides. These lengths satisfy the triangle inequality.

  4. G4

    A triangle has side lengths 14, 14, and 14 units. Classify it by its side lengths.

    Reveal explanation

    Answer: equilateral

    Equilateral means three equal sides. These lengths satisfy the triangle inequality.

  5. G5

    A triangle has side lengths 3, 3, and 3 units. Classify it by its side lengths.

    Reveal explanation

    Answer: equilateral

    Equilateral means three equal sides. These lengths satisfy the triangle inequality.

Independent · 8 questions

Independent practice

Work these without hints and explain the method used. Write answers on paper.

  1. I1

    A triangle has side lengths 8, 9, and 10 units. Classify it by its side lengths.

    Reveal explanation

    Answer: scalene

    Scalene means no equal sides. These lengths satisfy the triangle inequality.

  2. I2

    A triangle has side lengths 4, 5, and 6 units. Classify it by its side lengths.

    Reveal explanation

    Answer: scalene

    Scalene means no equal sides. These lengths satisfy the triangle inequality.

  3. I3

    A triangle has side lengths 4, 4, and 4 units. Classify it by its side lengths.

    Reveal explanation

    Answer: equilateral

    Equilateral means three equal sides. These lengths satisfy the triangle inequality.

  4. I4

    A triangle has side lengths 7, 8, and 9 units. Classify it by its side lengths.

    Reveal explanation

    Answer: scalene

    Scalene means no equal sides. These lengths satisfy the triangle inequality.

  5. I5

    A triangle has side lengths 6, 7, and 8 units. Classify it by its side lengths.

    Reveal explanation

    Answer: scalene

    Scalene means no equal sides. These lengths satisfy the triangle inequality.

  6. I6

    A triangle has side lengths 6, 6, and 8 units. Classify it by its side lengths.

    Reveal explanation

    Answer: isosceles

    Isosceles means exactly two equal sides. These lengths satisfy the triangle inequality.

  7. I7

    A triangle has side lengths 10, 11, and 12 units. Classify it by its side lengths.

    Reveal explanation

    Answer: scalene

    Scalene means no equal sides. These lengths satisfy the triangle inequality.

  8. I8

    A triangle has side lengths 10, 10, and 12 units. Classify it by its side lengths.

    Reveal explanation

    Answer: isosceles

    Isosceles means exactly two equal sides. These lengths satisfy the triangle inequality.

Challenge · 2 questions

Challenge and error analysis

Go beyond arithmetic: justify a claim, correct reasoning, or connect two representations.

  1. C1

    A triangle has side lengths 3, 3, and 4. Explain why it is isosceles but not equilateral.

    Reveal explanation

    Answer: Isosceles

    It has exactly two equal sides. An equilateral triangle requires all three sides equal.

  2. C2

    Can three line segments of lengths 2, 3, and 8 units make a triangle? Explain.

    Reveal explanation

    Answer: No

    The two shorter lengths total only 5, which cannot reach across a segment of length 8.

Exit · 3 questions

Exit check

Solve independently, then return on another day to check retention.

  1. E1

    A triangle has side lengths 13, 14, and 15 units. Classify it by its side lengths.

    Reveal explanation

    Answer: scalene

    Scalene means no equal sides. These lengths satisfy the triangle inequality.

  2. E2

    A triangle has side lengths 9, 9, and 11 units. Classify it by its side lengths.

    Reveal explanation

    Answer: isosceles

    Isosceles means exactly two equal sides. These lengths satisfy the triangle inequality.

  3. E3

    A triangle has side lengths 11, 11, and 13 units. Classify it by its side lengths.

    Reveal explanation

    Answer: isosceles

    Isosceles means exactly two equal sides. These lengths satisfy the triangle inequality.

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