The Common Core Standards in this group focus primarily on triangles. Students discover the concept of similarity and scale and explore how these concepts apply to real-world situations. The unique nature of triangles also reveals certain mathematical truths about ratios that are very useful in solving a host of problems.

## Similarity

In geometry, similarity refers to triangles that have exactly the same shape but differ in size. Similarity differs from congruence, which describes triangles of identical size and shape. The use of similarity to represent larger objects is commonplace in fields such as engineering and architecture, when someone needs to accurately represent the size of certain objects on a smaller scale.

## Right triangles

Students work extensively with right triangles (triangles with one 90-degree angle). Right triangles are unique in that you can find the length of any one side of the triangle if you know the lengths of the other two sides.

According to the Pythagorean theorem, “the square of the hypotenuse is equal to the sum of the squares of the other two sides,” which can be expressed as c2 = a2 + b2, where c is the hypotenuse (the longest side of the right triangle), and a and b are the other two (shorter) sides.

## Trigonometry

Trigonometry deals with the study and use of ratios involving triangle sides and angles. Students use the trigonometric ratios sine (sin), cosine (cos), and tangent (tan) to solve for missing parts (including a missing side or angle) of a right triangle:

Using trigonometric ratios, you can determine the length of a side of a right triangle without knowing the lengths of the other two sides. You can determine the unknown length of a side given the length of one side and the angle next to it. Take a look for a visual representation of the opposite, adjacent, and hypotenuse sides when labeled for the purposes of practicing using these ratios.

Use To find Given
sine opposite angle & hypotenuse
hypotenuse angle & opposite
angle opposite and hypotenuse