High School Geometry Archives - Page 7 of 23

Understanding the Meaning of Correspondence

November 18, 2012 GB High School Geometry, High School Mathematics

In Geometry, two objects are congruent if they have the same size and shape. Two triangles drawn on a piece of paper are congruent if we can cut them out with scissors, and superimpose them to fit exactly, that is, without gaps or overlaps. If the triangles fit exactly, the corresponding parts are the parts that coincide. Consequently, corresponding parts of congruent triangles are congruent. Therefore, if two triangles are congruent, then their corresponding angles are congruent and their corresponding sides are also congruent.

In the figure above, if we superimpose the two triangles, $\overline {AB}$ will coincide with $\overline {DE}$ and $\angle C$ will coincide with $\angle F$ . Hence $\overline {AB}$ and $\overline {DE}$ are corresponding sides and $\angle C$ and $\angle F$ are corresponding angles. » Read more

One comment correspondence in congruence, correspondence in similarity, corresponding parts of triangles, triangle congruence, triangle similarity

A Proof that the Vertex Angle Sum of a Pentagram is 180 degrees

June 23, 2012 GB High School Geometry, High School Mathematics

The pentagram is a five-pointed star. It was used by the ancient Greeks as a symbol of faith. In this post, we exhibit the mathematics of pentagrams — we show that the sum of the angle measures of its vertices equals 180°.

$pentagram math$

For regular pentagrams, the proof is simple. By the inscribed angle theorem, the measure of an inscribed angle is half the measure of a central angle that intercepts the same arc. The central angles of a regular pentagram as shown above intercept the entire circle. Therefore, its angle measures add up to 360°.

The vertex angles, on the other hand, are inscribed angles as shown in the second image above. Notice that if we add them up, they also intercept the entire circle (Can you see why?). In effect, their angle sum is half of 360°, which equals 180°. » Read more

4 comments angle sum of a 5-pointed start, equilateral triangle, exterior angle theorem, inscribed angle theorem, vertex angle sum of a pentagram

Using Similarity to Prove the Pythagorean Theorem

May 21, 2012 GB High School Geometry

The Pythagorean Theorem is one of the most interesting theorems for two reasons: First, it’s very elementary; even high school students know it by heart. Second, it has hundreds of proofs. The proof below uses triangle similarity.