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
• classify triangles based on side lengths or angles
• construct triangles using a protractor and ruler

Pre-requisite Summary

• Understand that a triangle has $3$ sides, $3$ vertices and $3$ interior angles
• Know that the interior angles of a triangle sum to ${180}^{\circ }$
• 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 $5$ cm, $5$ cm, $7$ cm
b) side lengths $4$ cm, $6$ cm, $8$ cm
c) side lengths $3$ cm, $3$ cm, $3$ cm

Worked Example 3

For each triangle, classify it by interior angles:
a) ${50}^{\circ },{60}^{\circ },{70}^{\circ }$
b) ${90}^{\circ },{35}^{\circ },{55}^{\circ }$
c) ${110}^{\circ },{30}^{\circ },{40}^{\circ }$

Worked Example 4

An isosceles triangle has apex angle ${40}^{\circ }$.
a) State which sides are equal.
b) Explain which two angles are equal.
c) Find the two equal base angles.

Worked Example 5

A labelled triangle $ABC$ has $AB=AC$.
a) State the name of the triangle by side lengths.
b) Identify the apex vertex.
c) State the relationship between $\mathrm{\angle }B$ and $\mathrm{\angle }C$.

Worked Example 6

Construct a triangle using ruler and protractor:
a) Draw base $AB=6$ cm.
b) At $A$, construct an angle of ${50}^{\circ }$.
c) At $B$, construct an angle of ${60}^{\circ }$, mark the intersection as $C$, 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 $6$ cm, $6$ cm, $9$ cm
b) side lengths $5$ cm, $7$ cm, $8$ cm
c) side lengths $4$ cm, $4$ cm, $4$ cm

Problem 3

For each triangle, classify it by interior angles:
a) ${55}^{\circ },{60}^{\circ },{65}^{\circ }$
b) ${90}^{\circ },{42}^{\circ },{48}^{\circ }$
c) ${120}^{\circ },{25}^{\circ },{35}^{\circ }$

Problem 4

An isosceles triangle has apex angle ${30}^{\circ }$.
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 $PQR$ has $PQ=PR$.
a) State the name of the triangle by side lengths.
b) Identify the apex vertex.
c) State the relationship between $\mathrm{\angle }Q$ and $\mathrm{\angle }R$.

Problem 6

Construct a triangle using ruler and protractor:
a) Draw base $XY=7$ cm.
b) At $X$, construct an angle of ${45}^{\circ }$.
c) At $Y$, construct an angle of ${70}^{\circ }$, mark the intersection as $Z$, and classify the completed triangle.

Exercises

Understanding and Fluency

1. Classify each triangle by side lengths:
   a) sides $3,3,5$
   b) sides $4,6,7$
   c) sides $8,8,8$

2. Classify each triangle by side lengths:
   a) sides $9,9,4$
   b) sides $5,6,5$
   c) sides $7,8,9$

3. Classify each triangle by angles:
   a) ${40}^{\circ },{70}^{\circ },{70}^{\circ }$
   b) ${90}^{\circ },{20}^{\circ },{70}^{\circ }$
   c) ${100}^{\circ },{30}^{\circ },{50}^{\circ }$

4. Classify each triangle by angles:
   a) ${60}^{\circ },{60}^{\circ },{60}^{\circ }$
   b) ${90}^{\circ },{45}^{\circ },{45}^{\circ }$
   c) ${95}^{\circ },{35}^{\circ },{50}^{\circ }$

5. For each isosceles triangle, find the missing angles:
   a) apex angle ${36}^{\circ }$
   b) apex angle ${80}^{\circ }$
   c) apex angle ${100}^{\circ }$

6. For each isosceles triangle, find the apex angle:
   a) base angles ${50}^{\circ },{50}^{\circ }$
   b) base angles ${35}^{\circ },{35}^{\circ }$
   c) base angles ${72}^{\circ },{72}^{\circ }$

7. State all possible classifications for each triangle:
   a) sides $5,5,5$
   b) sides $6,6,8$ and angles ${50}^{\circ },{50}^{\circ },{80}^{\circ }$
   c) angles ${90}^{\circ },{35}^{\circ },{55}^{\circ }$ and all side lengths different

8. Construct each triangle using ruler and protractor:
   a) base $5$ cm, base angles ${50}^{\circ }$ and ${60}^{\circ }$
   b) base $6$ cm, base angles ${45}^{\circ }$ and ${45}^{\circ }$
   c) base $7$ cm, base angles ${30}^{\circ }$ and ${80}^{\circ }$

Reasoning

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

10. A student says a triangle with angles ${90}^{\circ },{60}^{\circ },{30}^{\circ }$ is acute. Explain the mistake.

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

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

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

14. A student constructs a triangle with base angles ${70}^{\circ }$ and ${50}^{\circ }$ and then says the third angle is ${70}^{\circ }$. Explain the error.

Problem-solving

15. A triangle has two equal sides and one angle of ${90}^{\circ }$. Classify the triangle by side lengths and by angles.

16. A builder marks a triangular support with side lengths $4$ m, $4$ m and $6$ m. Classify the support by side lengths.

17. A triangular sign has interior angles ${65}^{\circ },{65}^{\circ },{50}^{\circ }$. Classify it by angles and by side lengths.

18. A triangle has one angle of ${110}^{\circ }$ and two equal sides. Classify the triangle and find the two equal angles.

19. Construct a triangle with base $8$ cm and base angles ${55}^{\circ }$ and ${55}^{\circ }$. Classify the triangle after construction.

20. A roof truss is shaped like a triangle with equal sloping sides. The angle at the top is ${44}^{\circ }$. 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 sketch 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 ${180}^{\circ }$

Next: 062. Polygons and Quadrilaterals