The most common form of geometry taught at the high school level is Euclidian geometry. Based on the theories of Euclid, a Greek mathematician, geometry focuses on both defined and undefined terms and sets of assumptions commonly known as theorems or postulates. Euclidian geometry centers on flat space, so it also incorporates plane geometry.
One of the primary goals in geometry is determining congruence of two-dimensional figures known as polygons. Understanding the characteristics of a triangle is imperative to determining congruence. Geometry students learn postulates and theorems that help them identify the relationship between angles and sides of triangles. They use this information to determine congruence. For example, students may utilize a triangle’s side-side-side criteria to find out if two triangles of identical length will also have identical angles. The side-side-side and other geometric combinations of side length and angle measures are taught to help students understand congruence. Eventually, theorems and postulates are used to determine congruence in all types of polygons using triangular sections of larger shapes to find missing side lengths and angles.
Once students identify congruence between figures, they will learn how to determine similarity. Similar figures are scaled versions of each other. The angle measurements between figures remain equivalent while lengths of sides are longer or shorter than the original figure. In determining similarity, students also use ratios and proportions to calculate missing lengths.
The study of the congruence and similarity of right triangles is the foundation for later study in mathematics and real-world application to architecture and construction. In studying right triangles, students learn the basic principles of sine, cosine and tangent. The Pythagorean theorem, a formula used to find the missing side of a right triangle, should also be introduced. These concepts can be used to prove congruence and similarity in right triangles when little information is provided.
The coordinate plane is a fundemental geometric platform. This plane of four quadrants with an x-axis and a y-axis gives students the foundation to develop an understanding of the properties of lines and polygons. Using coordinating planes, students can utilize distance and slope formulas to determine side lengths, proportions, similarity and congruence.
Geometry is often combined with algebra to refresh mathematic concepts and develop more complex problem solving skills. Problems of congruence and similarity can be combined with algebraic equations to establish a deeper understanding of geometric postulates.
- Comstock/Comstock/Getty Images