a A shape has a line of symmetry if we can draw a line through the figure and reflect one half onto the other in that line.
B
b
In Part A, we found that a triangle with 2 congruent sides has one line of symmetry. What is the next logical step?
C
c What types of quadrilaterals are there?
A
a Isosceles triangle
B
b Equilateral triangle
C
c Non-rhombic, non-rectangular Parallelogram
Practice makes perfect
a A shape has a line of symmetry if we can draw a line through the figure and reflect one half onto the other in that line. For this to be true, whatever is on the one half of the line must be identical to what's on the other half. For scalene triangles, there is no line of symmetry as none of the sides are identical and can therefore not map onto each other when folding it.
What about an isosceles triangle where two sides are identical? if we draw a line through the vertex angle, the result is two identical triangles. Since they are identical, and they share the line of symmetry as a side, we are able to fold one on to the other.
Note that we are not able to draw any other line of symmetry and get two congruent triangles.
Therefore, the triangle she is thinking about is an isosceles triangle.
b From Part A, we know that an isosceles triangle, which has two congruent sides, has one line of symmetry. What about if we make all three sides equal and create an equilateral triangle? Then, we can draw three lines of symmetry, one through each vertex of the triangle. In each case, the line of symmetry creates two identical triangles.
Therefore, the triangle she is thinking about is an equilateral triangle.
c If a shape maps onto itself when rotating it by 180^(∘) or less, it has rotational symmetry. For a quadrilateral to have rotational symmetry, the opposite sides of the quadrilateral must be parallel and congruent. This limits the quadrilateral to a parallelogram. However, parallelograms come in many forms.
Rectangles and rhombi have a minimum of two lines of symmetry. A non-rhombic, non-rectangular parallelogram does not have any lines of symmetry. Since all parallelograms have rotational symmetry, Camille must be thinking of an non-rhombic, non-rectangular parallelogram.
