060. Classifying and Constructing Triangles

Learning Intentions

• To understand that triangles can be classified by their side lengths (scalene, isosceles, equilateral) or by their interior angles (acute, right, obtuse)
• To know that in an isosceles triangle, the angles opposite the apex are equal and the two sides (legs) adjacent to the apex are of equal length
• Identify triangles based on side lengths or angles
• Draw triangles Use a protractor and ruler

Pre-requisite Summary

• Understand that a triangle has sides, vertices and interior angles
• Know that the interior angles of a triangle sum to
• Be able to Identify equal lengths and equal angles from markings in diagrams
• Know the meaning of acute, right and obtuse angles
• Know how to Use a ruler to measure length accurately
• Know how to use a protractor to measure and Draw angles
• Understand that side-length classification and angle classification Describe different properties of the same triangle
• Be able to name vertices and sides in a labelled diagram

Worked Examples

Worked Example 1

a) Define the triangle types scalene, isosceles and equilateral.

b) Define the triangle types acute, right and obtuse.

c) Explain how a triangle can be classified by both side lengths and angles.

Worked Example 2

For each triangle, classify it by side lengths:

a) side lengths cm, cm, cm

b) side lengths cm, cm, cm

c) side lengths cm, cm, cm

Worked Example 3

For each triangle, classify it by interior angles:

a)

b)

c)

Worked Example 4

An isosceles triangle has apex angle .

a) State which sides are equal.

b) Explain which two angles are equal.

c) Solve the two equal base angles.

Worked Example 5

A labelled triangle has .

a) State the name of the triangle by side lengths.

b) Identify the apex vertex.

c) State the relationship between and .

Worked Example 6

Construct a triangle using ruler and protractor:

a) Draw base cm.

b) At , construct an angle of .

c) At , construct an angle of , mark the intersection as , and classify the completed triangle.

Problems

Problem 1

a) Define the triangle types scalene, isosceles and equilateral.

b) Define the triangle types acute, right and obtuse.

c) Explain how a triangle can be classified by both side lengths and angles.

Problem 2

For each triangle, classify it by side lengths:

a) side lengths cm, cm, cm

b) side lengths cm, cm, cm

c) side lengths cm, cm, cm

Problem 3

For each triangle, classify it by interior angles:

a)

b)

c)

Problem 4

An isosceles triangle has apex angle :

a) State which sides are equal.

b) Explain which two angles are equal.

c) Find the two equal base angles.

Problem 5

A labelled triangle has .

a) State the name of the triangle by side lengths.

b) Identify the apex vertex.

c) State the relationship between and .

Problem 6

Construct a triangle using ruler and protractor:

a) Draw base cm.

b) At , construct an angle of .

c) At , construct an angle of , mark the intersection as , and classify the completed triangle.

Exercises

Understanding and Fluency

Exercise 1.

Classify each triangle by side lengths:

a) sides

b) sides

c) sides

Exercise 2.

Classify each triangle by side lengths:

a) sides

b) sides

c) sides

Exercise 3.

Classify each triangle by angles:

a)

b)

c)

Exercise 4.

Classify each triangle by angles:

a)

b)

c)

Exercise 5.

For each isosceles triangle, find the missing angles:

a) apex angle

b) apex angle

c) apex angle

Exercise 6.

For each isosceles triangle, find the apex angle:

a) base angles

b) base angles

c) base angles

Exercise 7.

State all possible classifications for each triangle:

a) sides

b) sides and angles

c) angles and all side lengths different

Exercise 8.

Construct each triangle using ruler and protractor:

a) base cm, base angles and

b) base cm, base angles and

c) base cm, base angles and

Reasoning

Exercise 9.

Explain why an equilateral triangle is also an isosceles triangle under the definition “at least two equal sides”.

Exercise 10.

A student says a triangle with angles is acute. Explain the mistake.

Exercise 11.

Explain why the base angles of an isosceles triangle must be equal.

Exercise 12.

A student says any triangle with two equal angles must be scalene. Explain why this is incorrect.

Exercise 13.

Explain why a triangle can be classified once by sides and once by angles.

Exercise 14.

A student constructs a triangle with base angles and and then says the third angle is . Explain the error.

Problem-solving

Exercise 15.

A triangle has two equal sides and one angle of . Classify the triangle by side lengths and by angles.

Exercise 16.

A builder marks a triangular support with side lengths m, m and m. Classify the support by side lengths.

Exercise 17.

A triangular sign has interior angles . Classify it by angles and by side lengths.

Exercise 18.

A triangle has one angle of and two equal sides. Classify the triangle and find the two equal angles.

Exercise 19.

Construct a triangle with base cm and base angles and . Classify the triangle after construction.

Exercise 20.

A roof truss is shaped like a triangle with equal sloping sides. The angle at the top is . Find the two bottom angles and classify the triangle by sides and angles.

Potential Misunderstandings

• Students may confuse classification by side lengths with classification by angles
• Students may think a triangle can have only one classification in total
• Students may think an equilateral triangle is not isosceles because it has three equal sides rather than two
• Students may confuse the apex angle of an isosceles triangle with one of the equal base angles
• Students may forget that the equal angles in an isosceles triangle are opposite the equal sides
• Students may assume a triangle that looks equal-sided in a Draw must be equilateral without using given information
• Students may misread a protractor scale when constructing angles
• Students may place the protractor centre away from the vertex or align it with the wrong side
• Students may forget that the interior angles of a triangle must add to

Next: 061. Polygons and Quadrilaterals