Paper Folding: How to Construct an Equiangular Triangle

Paper folding or origami can be used to create intricate and stunning designs.  In addition, paper folding can also be used to teach or learn mathematics. In this post, we use a square piece of paper to construct an equiangular triangle. After the construction, we prove that the triangle is equiangular.

Steps in Constructing an Equiangular Triangle 

1. Cut a square piece of paper. For the sake of discussion, we label the square ABCD.

2. Fold at the center so that AB coincides with CD . Crease well, then unfold.

paper folding

3. Select vertex and fold so that A falls on the center line and the crease passes through D. Unfold. » Read more

Semiregular Tessellations: Adventurous Ideas for Floor Tiling

If you are planning to tile your your floor and want something different from the usual tiling which is usually made up of rectangles or squares, the following semiregular tessellations might give you some ideas.

Octagon Square Tessellations

In the first figure, the tessellation is made up of squares and octagons (8-sided polygons). » Read more

SSS Congruence Theorem and Its Proof

Many high textbooks consider the congruence theorems (SSS Congruence Theorem, SAS Congruence Theorem, ASA Congruence Theorem) as postulates. This is because their proofs are complicated for high school students.  However, let us note that strictly speaking, in Euclidean Geomtery (the Geometry that we learn in high school), there are only five postulates and no others. All of other postulates mentioned in textbooks aside from these five are really theorems without proofs.

In this post, we are going to prove the SSS Congruence Theorem. Recall that the theorem states that if three corresponding sides of a triangle are congruent, then the two triangles are congruent.

Before proving the SSS Congruence theorem, we need to understand several concepts that are pre-requisite to its proof. These concepts are isometries particulary reflection and translation, properties of kites, and the transitive property of congruence. If you are familiar with these concepts, you can skip them and go directly to the proof. » Read more

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