Proof Problems
Simple worked example
In the above a transversal cuts parallel lines.
It can be seen that the labelled angles are equal. What name is given to this pair of angles? Why are they equal? How would you prove that such angles are equal for all situations rather than just this special case of an eighth of a rotation?
Multi-step worked example
What angle relationships have been used to find each angle?
Why do these relationships work?
How would you prove these relationships work in general?
Challenge Question: Exploring triangles with Parallel Lines
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?
Why are the labelled angles equal?
What does this imply about the angles of a triangle?
How would you write out a proof of this relationship using parallel angle results?
How would you write out a proof using the definition of translation, parallel lines and of the degree?
Multi-Step Open Problem Including higher order thinking
Alternative question types:
1. Diagram has value of supplied, Find value of , labelling intermediate points as appropriate.
2. In the above, the two angles may be related to each other in a simple way. Find the relationship between the two angles, labelling intermediate points as appropriate.
Continuing questions:
What angle relationships did you use?
Could you use different parallel line angle relationships to solve this problem? Why or why not?
Do so for one other parallel line angle relationship.
Challenge Problem:
Can you use all 3 different parallel line angle relationships? (Ask for additional diagram copies if required)
Which angle relationship(s) is(are) easiest to use, that is, the most obvious relationship(s)? Why?
Which angle relationship(s) is(are) hardest to use, that is, the least obvious relationship(s)? Why?
Which angle relationship(s) require the least steps?
Extension Problem:
Can you see a short-cut relationship that uses less steps based on the fact the diagram contains a pair of intersecting parallel lines?
Can you prove this relationship holds in general? Why?
The internal shape defined by such an intersection is a parallelogram, and such results may be termed angles on a parallelogram relationships.