As I develop some of the higher level geometry waypoints for our framework I’m widening my understanding of ‘different geometries’ and am trying to offer a balance of approaches and show how they blend together.
We have many ways to describe the Euclidean plane and objects in the plane. Here’s the case of the isosceles triangle, which shows how our starting definitions can be different yet the same properties deduced (admittedly in different ways).
Now I wouldn’t recommend you start looking at this particular case with students as it is here as a base example for you to consider and compare. But what might be useful is the narrative that goes on to explain the properties of the isosceles triangle: there isn’t one unique explanation, but justifications can take on different ‘lenses’, which is worth thinking about when we listen to students’ responses that aren’t quite the way we’d explain it!
Euclid defined an isosceles triangle as a triangle with two equal sides:
What can we deduce from this using the congruence axioms?
If triangle ABC has |AB| = |AC| then triangles ABC and ACB are congruent by side-angle-side (SAS).
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)
In the two triangles above both left hand sides are equal, both right hand sides are equal and the angle at A is the same angle (since they are the same triangle).
It follows from SAS that if two triangles are congruent then all corresponding angles are congruent. In other words the angle at B is equal to the angle at C, so angles opposite the equal sides are equal.
Now bisect the angle at A and draw in line segment AM.
![](data:image/png;base64,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)
Again by SAS triangles ABM and ACM are congruent; |AB| = |AC|, the line segment AM bisected the angle at A so ∠BAM and ∠CAM are equal, the triangle shared side AM.
Therefore |BM| = |MC|, M is the midpoint of BC and AM is perpendicular to BC. Hence AM is the altitude and median of triangle ABC.
Now another approach is using transformations.
An isosceles triangle is a triangle with at least one line of symmetry. This means that triangle ABC is its own image under reflection in AM.
Logic and my knowledge tells me that this must look like this:
![](data:image/png;base64,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)
This is written as r(△ABC) = △ACB.
So B is reflected to C, r(B) = C, C to B, r(C) = B and A is on the mirror line, r(A) = A
What other properties can we deduce from this?
a) Since r(B) = C and r(A) = A:
• |AB| = |AC|, two sides are of equal length
b) Since r(B) = C, AM is the mirror line, M lies on BC and r(M) = M:
• M is the midpoint of BC
• AM is the perpendicular bisector of BC
• AM is the median and altitude of the triangle ABC
c) Since r(A) = A, r(B) = C, and r(M) = M:
• r(∠CAM) = ∠CBM • ∠CAM = ∠CBM, the two ‘base angles’ are equal
It’s worth comparing these two arguments. How are they similar or different? Do you have a preference? Can you apply these kinds of arguments to explain the properties of a parallelogram?
SOMETHING TO TRY
KS1: Where in the classroom do you sit? How many ways can you describe your position?
KS2: In how many different ways can you describe a square? Think about its vertices, edges, and symmetry.
KS3: How could you find the mid-point of a line segment?
In what ways might your method change if:
· The line segment was drawn on a piece of tracing paper?
· The line segment was drawn on squared paper?
· You had a ruler?
· You had the co-ordinates of the ends of the line segment?
· You had the vector describing the journey from one end of the line segment to the other?
KS4: Here are two definitions of a parallelogram. Show how each leads to some properties of a parallelogram using congruency or transformations.
a) a quadrilateral with parallel opposite sides
HINT: first show angle information using parallel angle rules (extend the sides of the parallelogram), then draw in a diagonal, work on the angles and shared side of the two triangles to show ASA congruency to lead to opposite sides being equal in length
b) a quadrilateral with rotational symmetry order 2
HINT: consider the images of vertices and sides of the parallelogram when it is rotated 180˚, consider special properties of any line rotated about any point 180˚
KS5: Investigate the many proofs of Pythagoras’ theorem including each of the following:
Using similar triangles and scale factors of length and hence area:
![](data:image/png;base64,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)
Using a geometrical diagram and algebraic manipulation, such as investigating the different algebraic representations of the area of this trapezium that is constructed using 3 right angled triangles, two of which are congruent:
![](data:image/png;base64,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)
Using the properties of shearing. Read through the explanation here or search online for an animation.
For further ideas visit https://www.cut-the-knot.org/pythagoras/ |
There’s no shame in not knowing the words… what is a shame is that we are not always confident to stop and ask for their meaning. Wouldn’t it be great if we could all feel comfortable enough to ask for an explanation; after all it’s what we wish the students would do!
Here are a few from this blog that may or may not be of use.
Midpoint: The point in the centre of a line segment or half way between two points.
Altitude of a triangle: the line segment from a triangles vertex to the opposite side, such that the two line segments are perpendicular. Every triangle has three altitudes.
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)
Median of a triangle: the line segment from a triangles vertex to the midpoint of the opposite side. Every triangle has three medians.
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)