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8 Geometry
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Stage 4

8.01 Angles and triangles

Lesson
Worksheet
Practice
Lesson

Angles and triangles

This image shows the angles of a triangle and the angles making a straight line. Ask your teacher for more information.

We can recall that the angle sum of a triangle will always be 180\degree as seen in this triangle.

This image shows a triangle with 3 internal angles and an external angle. Ask your teacher for more information.

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.

This image shows exterior angles of a triangle and the interior angles they equal. Ask your teacher for more information.

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.

Examples

Example 1

Solve for the value of x in the diagram below.

Triangle with angles of 35, 70 and x degrees.
Worked Solution
Create a strategy

Use the angle sum of a triangle to form an equation and solve.

Apply the idea
\displaystyle x+70+35\displaystyle =\displaystyle 180Write the equation
\displaystyle x+70+35-70-35\displaystyle =\displaystyle 180-70-35Subtract 70 and 35 from both sides
\displaystyle x\displaystyle =\displaystyle 75Evaluate

Example 2

Solve for the value of x in the diagram below.

A triangle with interior angles x degrees and 51 degrees, with opposite exterior angle of 107 degrees.
Worked Solution
Create a strategy

Write an equation relating the angles using the exterior angle of a triangle rule.

Apply the idea
\displaystyle x+52\displaystyle =\displaystyle 108Use the exterior angle rule
\displaystyle x\displaystyle =\displaystyle 108-52Subtract 52 from both sides
\displaystyle =\displaystyle 56Evaluate
Idea summary

The exterior angle of a triangle is equal to the sum of the two opposite interior angles.

Triangle with two interior angles and an exterior angle.

Angles and sides

This image shows two triangles showing the largest and smallest sides and angles. Ask your teacher for more information.

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.

Three lines. The orange line is the smallest, the purple line is the middle, the green line is the longest.

Consider these side lengths. Can these sides be arranged into a triangle?

This image shows the attempts to make a triangle. Ask your teacher for more information.

These three side lengths can not be made into a triangle with these side lengths no matter how we arrange them.

This image shows three colored lines made into a triangle. Ask your teacher for more information.

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.

This image shows sets of lines that can form a triangle. Ask your teacher for more information.

We can see that for any side we look at, the other two side lengths when combined are longer.

This image shows sets of lines that cannot form a triangle. Ask your teacher for more information.

For the impossible sides, two of the sides we can choose the other two sides combined are longer. However, looking at the longest side, the two shorter sides combined are still smaller than the longest side. This is the condition which determines whether a triangle is possible or not. For a triangle the combined length of each pair of sides is longer than the remaining side.

This image shows when it is possible and impossible to make a triangle. Ask your teacher for more information.

Examples

Example 3

Will three sides of length 3,\,5, and 9 make a triangle?

Worked Solution
Create a strategy

Create a table to compare the side lengths.

Apply the idea
SideSum of remaining sidesCompare
35+9=145+9>3
53+9=123+9>5
93+5=83+5<9

In the final row the side of 9 is longer than the sum of the other two sides, so the triangle is impossible.

The three sides of length 3,\,5, and 9 cannot make a triangle.

Reflect and check

For scalene triangles we only have to check that the smallest two sides are bigger than the largest side, in this case if 3+5<9 so the triangle is impossible.

Idea summary

For a triangle to be possible with all three sides, the combined length of each pair of sides is longer than the remaining side.

A triangle with sides lengths m, n, and p.

m+n>p \\ m+p>n \\ n+p>m

Outcomes

MA4-17MG

classifies, describes and uses the properties of triangles and quadrilaterals, and determines congruent triangles to find unknown side lengths and angles

MA4-18MG

identifies and uses angle relationships, including those related to transversals on sets of parallel lines

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