Random question - I'm a tutor teaching some year 8 kids on geometry proving (triangle congruency, similarity etc). I have learned this a while ago and remembering not enjoying this topic at all because it's a lot of repetition of writing down reasons etc. I was so bad at it but finally caught up in year 11 because I need to do well in HSC. I'd like to know if anyone who has had teachers who have actually made this topic less boring?? Would love to hear what your teachers did to help make this topic more fun.
Geometric Proving (2 Viewers)
- Thread starter Anh_Ho
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When I was in my Primary and Secondary school, classical (Euclidean) Geometry was an important part of our Maths curriculum. If you are good in this elementary geometry, it will help your other maths greatly. Proofs in school geometry are your first exposure to Deductive Reasoning - i.e. how to reason logically. Many years after my school days, much of Geometry was trimmed down in the curriculum to make way for more "contemporary" topics. In a way, much was lost.
It'd be too long to explain the importance of proofs of congruency and similarity in triangles. Strangely enough, similarity in triangles appear to be harder than congruency, to most students!
It'd be too long to explain the importance of proofs of congruency and similarity in triangles. Strangely enough, similarity in triangles appear to be harder than congruency, to most students!
Yes I noticed similarity seems harder than congruency as well for many students. The majority of students find this deductive process unrelatable because you can see with your eyes that the shapes are congruent 
Why are we trying to prove it.
Why are we trying to prove it.What was the content that got remove if you don't mind me asking.
Above from an old Signpost Year 8 Book. Say we are given a parallelogram ABCD. e start with the definition of the parallelogram, in this case only that it is a quadrilateral with opposite sides, viz AB & DC and AD & BC being parallel. That's all we are given. How do you prove that:
i) AB = DC and AD = BC
ii) angle ABC = angle CDA
??
You must not assume the common properties of a parallelogram, like its opposite sides are equal and its opposite angles are equal. You have not proven these properties yet. You have to establish them. You cannot look at the diagram and say, well the opposite sides are clearly equal or the opposite angles are clearly equal. That's not how facts are established.
So what do we do? We can use congruency of a pair of triangles to establish the desired properties. If you cut out 2 congruent triangles and lay them one on top of the other, matching equal sides and equal angles. The simple properties of congruent triangles you are going to use are: their corresponding sides are equal and their corresponding angles are equal.
Now choose a pair of triangles, ABC & CDA (in exact correspondence). You can easily establish that these 2 triangles are congruent (AAS), using properties of parallel sides AB & DC and AD & BC. Therefore we now know that for congruent triangles ABC & CDA: AB =DC, AD = BC and AC = AC. The 3rd case is irrelevant, but the first two establishes the equality of the opposite sides of the parallelogram. Similarly the 3 corresponding angles of the congruent triangle are equal, viz: CAB = ACD (irrelevant), ACB = DAC (irrelevant) and ABC = CDA (i.e. opposite angles ABC & CDA of the //gram ABCD are equal). We can repeat with congruent triangles BAD & DCB to establish that opposite angles BAD & DCB of //gram ABCD are also equal.
So this is one use of congruency of triangles to prove a property or some properties, in this case, of a //gram.
i) AB = DC and AD = BC
ii) angle ABC = angle CDA
??
You must not assume the common properties of a parallelogram, like its opposite sides are equal and its opposite angles are equal. You have not proven these properties yet. You have to establish them. You cannot look at the diagram and say, well the opposite sides are clearly equal or the opposite angles are clearly equal. That's not how facts are established.
So what do we do? We can use congruency of a pair of triangles to establish the desired properties. If you cut out 2 congruent triangles and lay them one on top of the other, matching equal sides and equal angles. The simple properties of congruent triangles you are going to use are: their corresponding sides are equal and their corresponding angles are equal.
Now choose a pair of triangles, ABC & CDA (in exact correspondence). You can easily establish that these 2 triangles are congruent (AAS), using properties of parallel sides AB & DC and AD & BC. Therefore we now know that for congruent triangles ABC & CDA: AB =DC, AD = BC and AC = AC. The 3rd case is irrelevant, but the first two establishes the equality of the opposite sides of the parallelogram. Similarly the 3 corresponding angles of the congruent triangle are equal, viz: CAB = ACD (irrelevant), ACB = DAC (irrelevant) and ABC = CDA (i.e. opposite angles ABC & CDA of the //gram ABCD are equal). We can repeat with congruent triangles BAD & DCB to establish that opposite angles BAD & DCB of //gram ABCD are also equal.
So this is one use of congruency of triangles to prove a property or some properties, in this case, of a //gram.
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