Angles and triangles
We can recall that the angle sum of a triangle will always be $180^\circ$180° as seen in the triangle below.
We can also take any of the triangles sides and extend it to create an exterior angle.
The size of an exterior angle is always equal to the sum of the internal angles on the opposite side. We can see the exterior angle created by extending one of the sides.
There are many exterior angles we make with a single triangle. The exterior angles will always equal the sum of the internal angles on the opposite sides.
Worked example
Consider the triangle below.
Think: The exterior angle is equal to the sum of the two opposite interior angles. We can construct an equation and then solve for $x$x.
Do:
| $\left(5x\right)+\left(4x-10\right)$(5x)+(4x−10) | $=$= | $\left(8x+5\right)$(8x+5) |
Using the exterior angle |
| $9x-10$9x−10 | $=$= | $8x+5$8x+5 |
Combining like terms |
| $9x-8x$9x−8x | $=$= | $5+10$5+10 |
Moving $x$x to one side |
| $x$x | $=$= | $15$15 |
Evaluating |
We can now substitute this value into the expression $8x+5$8x+5 to find the size of the exterior angle.
| $8x+5$8x+5 | $=$= | $8\times\left(15\right)+5$8×(15)+5 |
Substituting $x=15$x=15 |
| $=$= | $120+5$120+5 |
|
|
| $=$= | $125$125 |
Evaluating |
So the exterior angle is $125^\circ$125°.
Reflect: We can also calculate the other two angles in the triangle to be $75^\circ$75° and $50^\circ$50°.
The exterior angle of a triangle is equal to the sum of the two opposite interior angles.
Angles and sides
In any triangle the longest side will always be opposite the largest angle. Same with the smallest side and the smallest angle.
If we increase one side while keeping the other two sides the same size the side that is getting longer will also have the opposite angle get bigger. A triangle that has two sides that are the same length means the opposite angles must also be equal.
Impossible triangles
Exploration
Consider the side lengths given below:
Can these sides be arranged into a triangle?
These three side lengths can not be made into a triangle with these side lengths no matter how we arrange them. If we replace one of the sides with a longer side it is now possible to make these three sides into a triangle.
We can compare the two sets of sides to see what is the defining difference.
Worked example
Will three sides of length $5$5, $6$6, and $12$12, make a triangle?
Think: If each pair of sides is greater than the length of the other side then it is possible.
Do: Create a table to compare the side lengths:
| Side | Sum of remaining sides | |
|---|---|---|
| $5$5 | $6+12$6+12 | $18$18 |
| $6$6 | $5+12$5+12 | $17$17 |
| $12$12 | $5+6$5+6 | $11$11 |
The first two rows meet the condition, but the final row does not. One of the sides is longer than the other two sides combined, so the triangle is impossible.
Reflect: For scalene triangles we only have to check that the smallest two sides are bigger than the largest side, in this case $5+6<12$5+6<12 so the triangle is impossible.
For a triangle to be possible with all three sides, the combined length of each pair of sides is longer than the remaining side.
|
|
$m+n>p$m+n>p $m+p>n$m+p>n $n+p>m$n+p>m |
Practice questions
Question 1
Solve for the value of $x$x in the diagram below.
Enter each line of working as an equation.
A triangle with each vertices labeled with angle measures and marked with a colored arc. The angle at the top vertex measures $\left(x\right)$(x) and colored red. The angle at the bottom-right vertex measures $70^\circ$70° and colored blue. The angle at the bottom-left vertex measures $35^\circ$35° and colored yellow.
Question 2
Solve for the value of $x$x in the diagram below.
Enter each line of working as an equation.
