E. Spatial Sense · Spatial Sense: Angles, Triangles, Transformations and Area

Constructing Triangles from Sides and Angles

E1.1 E2.4 A1About 45 minutes

Learning goals

I can construct a triangle when I am given two sides and the angle between them.
I can construct a triangle when I am given all three side lengths.
I can construct a triangle when I am given one side and the two angles at its ends.
I can explain why some sets of measurements can't make a triangle.

Words to know

construct
To draw a shape exactly, using tools like a ruler and a protractor.

label
To write the measures or letters on a drawing.

vertex
A corner of a shape, where two sides meet. Triangles have 3 vertices, often named A, B and C.

compass
A drawing tool with a sharp point and a pencil, used to draw circles and arcs. Use it only with a grown-up.

Remember from the last lessons

You can measure and draw angles with a protractor, and you can name triangles by their sides and their angles. You also know the angles of a triangle add up to 180°. Today you will put it all together and construct (draw exactly) triangles from a few measurements.

Let's start with a kite

Kai wants to build a triangle-shaped kite. He tells his cousin, "Make the bottom edge 60 cm, the left edge 40 cm, with a 50° angle between them." Is that enough information to cut the right shape? Yes! With just three well-chosen measurements, you can build the whole triangle.

There are three kinds of instructions you will use:

Two sides and the angle between them.
All three sides.
One side and the two angles at its ends.

The pictures in this lesson are smaller than real size, but they are drawn in proportion, so a 6 cm side looks longer than a 4 cm side in the right amount.

Two sides and the angle between them: a 6 cm side, a 50° angle at its left end, and a 4 cm side along the angle's second arm.

Worked example

Example 1: Two sides and the angle between them

Construct a triangle with sides of 6 cm and 4 cm and an angle of 50° between them.

Use a ruler to draw a 6 cm line. Label its ends A and B.
Put the protractor's centre mark on A, with the baseline along AB. Count up from the 0 on AB to 50. Make a dot.
Draw a light line from A through the dot. Measure 4 cm along it from A and mark point C.
Join C to B with the ruler.
Check: measure the angle at A (50°) and the sides AB (6 cm) and AC (4 cm). Then measure BC. It should be about 4.6 cm.

A triangle with sides 6 cm, 4 cm and about 4.6 cm. All its sides are different, so it is scalene.

Three sides: AB is 7 cm. Dots that are all 5 cm from A and dots that are all 4 cm from B meet at point C.

Worked example

Example 2: Three sides

Construct a triangle with sides of 7 cm, 5 cm and 4 cm.

Draw the longest side first: a 7 cm line, AB.
The third corner, C, must be 5 cm from A. Put your ruler's 0 on A and make small dots 5 cm away, in many slightly different directions. They make a curve.
C must also be 4 cm from B. Do the same from B with 4 cm.
The two curves of dots cross at one point. That is C.
Join A to C and B to C. Check each side with the ruler.

A triangle with sides 7 cm, 5 cm and 4 cm. Measure its angles: about 34°, 44° and 102°, so it is an obtuse scalene triangle. (If a grown-up helps, a compass draws the curves faster.)

One side and two angles: an 8 cm side with a 40° angle at the left end and a 60° angle at the right end. The arms meet at the third vertex.

Worked example

Example 3: One side and two angles

Construct a triangle with an 8 cm side, a 40° angle at one end and a 60° angle at the other end.

Draw an 8 cm line AB.
At A, use the protractor to draw a 40° angle. Make the new arm long.
At B, draw a 60° angle on the same side of AB. Make that arm long too.
The two arms cross. That point is C. Erase any extra bits of the arms.
Check: the third angle should be 180 − 40 − 60 = 80°. Measure it!

A triangle with angles 40°, 60° and 80°. All the angles are acute and different, so it is an acute scalene triangle.

When you can't make a triangle

Some measurements just don't work:

Sides: the two shorter sides must add up to more than the longest side. Otherwise they can't reach each other.
Angles: two angles must add up to less than 180°, so there is room for the third angle.

Sides of 2 cm and 3 cm can't reach each other across an 8 cm side, because 2 + 3 = 5 and 5 is less than 8.

Worked example

Example 4: Will it work?

Can you build a triangle with sides of 3 cm, 4 cm and 9 cm?

Find the two shorter sides: 3 cm and 4 cm.
Add them: 3 + 4 = 7.
Compare with the longest side: 7 is less than 9. The short sides can't reach.

No triangle is possible.

Slow and steady

Constructions need care. Use a sharp pencil, hold the ruler firmly, and draw light lines first so you can fix them. If the last side doesn't measure what you expected, don't erase everything. Check one step at a time to find the slip. Being patient and organized is part of being good at math.

Just for fun

Silly what if: Try to build a triangle with sides of 2 cm, 3 cm and 8 cm. The two short sides wave at each other across the gap, but they can never hold hands!

Hands-on activity

Straw and string triangles

You need: drinking straws or strips of paper, a ruler, scissors (with a grown-up), a pencil, a protractor and paper.

Cut straws (or paper strips) to these lengths: 4 cm, 5 cm, 6 cm, 7 cm, 8 cm and 3 cm.
Choose any three and try to make a triangle on the table with their ends touching. Record which sets work and which don't.
For each set that didn't work, add the two shorter lengths. What do you notice?
Pick one set that works. Construct that triangle on paper with a ruler, using the dot-curve method from Example 2.
Then construct a triangle with a 6 cm side and 45° angles at both ends. Measure the third angle. What kind of triangle is it?

(Answer for step 5: the third angle is 90°, so it is a right isosceles triangle.)

Math talk: "Why do the two short sides have to add up to more than the long side?"

Your turn

Construct an equilateral triangle with sides of 5 cm. Describe how.

Construct a triangle with two sides of 5 cm and a 90° angle between them. What kind of triangle is it?

Construct a triangle with a 6 cm side and 30° angles at both ends. Find the third angle. Name the triangle.

Construct a triangle with sides of 6 cm, 8 cm and 10 cm. Measure the biggest angle.

Can you make a triangle with sides 3 cm, 4 cm and 9 cm? Why or why not?

Can you make a triangle with angles of 100° and 90°? Why or why not?

More practice: warm-up

Which tool do you use to draw an angle?

Can sides of 4 cm, 5 cm and 6 cm make a triangle?

Can sides of 2 cm, 2 cm and 5 cm make a triangle?

Two angles of a triangle are 70° and 70°. What is the third?

When you construct a triangle from three sides, which side is easiest to draw first?

More practice: on your own

Construct a triangle with two 7 cm sides and a 60° angle between them. Measure the third side. What do you notice?

Construct a triangle with a 5 cm side, a 90° angle at one end and a 45° angle at the other. Describe the triangle.

Can sides of 3 cm, 5 cm and 8 cm make a triangle? Explain.

A triangle has a 6 cm side with angles of 50° and 60° at its ends. What is the third angle? Name the triangle by its angles and sides.

Rosa drew a 6 cm side and a 50° angle, then a 4 cm side. Her angle is between the 4 cm side and the 6 cm side. Kim drew the 50° angle at the other end of the 6 cm side. Will their triangles be the same size and shape? Explain.

Challenge

Two classmates each construct a triangle with sides 5 cm, 6 cm and 7 cm. Will their triangles be the same size and shape? Explain.

Two classmates each draw a triangle with angles 50°, 60° and 70°, but nobody gives any side lengths. Will their triangles be the same size? Explain.

A triangle has sides of 5 cm and 9 cm. What whole-number lengths could the third side be?

Check your understanding

Quiz

Which set of sides can make a triangle?

You draw a 6 cm side and want 45° angles at both ends. What is the third angle?

To construct a triangle from two sides and an angle, where must the angle be?

Why can't you build a triangle with angles of 95° and 85°?

Which tool helps you find a point that is exactly 5 cm from another point, in different directions?

You construct a triangle with a 5 cm side and 60° angles at both ends. What is it?

For parents and teachers

Big idea: The Ontario curriculum (E1.1) asks Grade 5 students to construct different types of triangles when given side or angle measurements. This uses the protractor skills from expectation E2.4. Children learn that three well-chosen measurements decide a triangle, and that some measurements are impossible.

Common difficulties:

Drawing the second angle on the wrong side of the base. Both angles must open on the same side.
Measuring from the end of the ruler instead of 0. Check that the 0 mark is on the point.
Rushing. Construction rewards slow, careful work with light pencil lines.

Safety: A compass has a sharp point. Use it only with a grown-up, or use the ruler-and-dots method instead. Help with scissors when cutting straws.

How to help: Ask your child to give you construction instructions and see if your triangle matches theirs. Social-emotional learning (A1) is woven in: patience, organization and checking one step at a time.