Geometry

Lesson

We know that a triangle is a polygon with $3$3 sides, this means that triangles also have $3$3 interior angles.

The other known fact we have is the the sum of the interior angles of a triangle is $180^\circ$180°.

Let's explore what kind of criteria we need to be explicitly given to be able to draw a unique triangle.

I'll start by asking you some questions, see if from the criteria I give you whether you can draw $1$1, none or many triangles.

How many unique triangles exist with side lengths $3$3 cm and $5$5 cm?

- Do you get the same answer with any $2$2 side lengths?
- If it's not possible to make one, what would you change to be able to make one?
- If there are multiple triangles possible, what do you need to know to be able to create just one unique one?

How many unique triangles exist when two of the angles are $30^\circ$30° and $45^\circ$45°?

- What about with two angles of any measure?
- Is there a requirement on the sizes of the angles?
- If it's not possible to make one, what would you change to be able to make one?
- If there are multiple triangles possible, what do you need to know to be able to create just one unique one?

How many unique triangles exist with an angle of $60^\circ$60° and a side length of $7.5$7.5 cm?

- What about with any one angle or any one side length?
- Is there a requirement on the sizes of the angles, or the length of the sides?
- If it's not possible to make one, what would you change to be able to make one?
- If there are multiple triangles possible, what do you need to know to be able to create just one unique one?

To try and gather our thoughts on this, fill in this table with whether $1$1 unique, more than $1$1 triangle or no triangles exist for the combinations of angles and sides given.

I've filled in the first one for you. With $3$3 angles given there is more than one triangle that can be created. See the image below as an example.

Now you fill in as many others as you can.

0 angle measure | 1 angle measure | 2 angle measures | 3 angle measures | |
---|---|---|---|---|

0 side lengths | many $\triangle$△ 's | |||

1 side length | ||||

2 side lengths | ||||

3 side lengths |

Draw or construct (if possible) the following triangles with the given angle and side measures, if it isn't possible explain why, and what further information you would need to know.

It may help to remind yourself of some of basic constructions for lines and angles here.

a) $\triangle ABC$△`A``B``C`, with $\angle A=30^\circ$∠`A`=30°, $\angle C=50^\circ$∠`C`=50° and length $AB=4$`A``B`=4 cm

b) $\triangle PQR$△`P``Q``R`, with $\angle Q=45^\circ$∠`Q`=45°, $\angle R=60^\circ$∠`R`=60° and length $QR=10$`Q``R`=10 cm

c) $\triangle ABD$△`A``B``D`, with $\angle D=90^\circ$∠`D`=90°, $\angle B=90^\circ$∠`B`=90° and length $BD=2$`B``D`=2 cm

d) $\triangle MNP$△`M``N``P`, with all angles $=75^\circ$=75° and side length $M=5$`M`=5 cm

e) Isosceles triangle $\triangle ABC$△`A``B``C`, with angles $A$`A` and $B$`B` both equal to $30^\circ$30° and the longest side length is $12$12 cm.

f) Equilateral triangle of side length $3.8$3.8 cm.

g) $\triangle XYZ$△`X``Y``Z`, with angles $30^\circ$30°, $60^\circ$60° and $90^\circ$90° respectively. Side lengths are $3$3, $4$4 and $6$6 cm.

(continue on to quadrilaterals)