Summative assessment task, final question.
In the above, many angles may be related to alpha and beta. Lines that appear parallel and perpendicular are exactly so. Label all angles as values relative to alpha and beta, include also gamma as and also the supplement of a given angle by appending a star, e.g. Some angles cannot be directly determined using the existing angle rules. However, these all may be found using a composition of angle rules. Below we determine this relationship, before using it here.
In the above, a parallel line has been constructed through the vertex of a triangle.
What steps were used in constructing the parallel line through the vertex?
Why does these steps necessarily produce parallel lines?
Move the point C around, does it still seem that the equalities hold?
Why are the labelled angles equal, what angle relationship gives this?
What does this imply about the angles of a triangle? Express this in common language and symbolically.
Given this new rule, fill in the values of all angles on the diagram, noting that perpendicular lines divide a rotation into quarters. As a short hand notation for complement that may be useful use:
and equivalent for beta and gamma. Note that this may also be used with the supplement notation introduced earlier. (The supplement of the complement is: )
Write out a proof of the triangle angle relationship using parallel angle results. You may refer to the above diagram, but should include discussion about any construction required from the basic triangle considered.
How would you write out a proof using the definition of translation, parallel lines and of the degree? Do not write such a proof, merely indicate the additional steps required.
Now that you have worked through a guided proof of the triangle angle sum relationship, consider how you could add in additional lines to determine the angles within a quadrilateral.
Solution for Quadrilateral
Construct generic quadrilateral with points ABCD. Draw in line BD. Identify that this produces two triangles. However, triangles are known to have an angle sum of 180 degrees, thus twice this is 360 degrees. Since a line is defined by two points, for any quadrilateral ABCD there exists one and only one such line BD, thus the quadrilateral can always be divided in this way. Therefore any quadrilateral has an angle sum of 360 degrees.
Solution for angles
This solution contains sufficient labelled angles at each point to uniquely specify all angles, once angle at a point and perpendicular lines angle relationships are added.