Students learn why any two triangles that satisfy the ASA or SSS congruence criteria must be congruent.

Students complete proofs requiring a synthesis of the skills learned in the last four lessons.

Students complete proofs requiring a synthesis of the skills learned in the last four lessons.

Students examine the basic geometric assumptions from which all other facts can be derived.
Students review the principles addressed in Module 1.

Students create scale drawings of polygonal figures by the Ratio Method.
Given a figure and a scale drawing from the Ratio Method, students answer questions about the scale factor and the center.

Students create scale drawings of polygonal figures by the Parallel Method.
Students explain why angles are preserved in scale drawings created by the Parallel Method using the theorem on parallel lines cut by a transversal.

Students understand that the Ratio and Parallel Methods produce the same scale drawing and understand the proof of this fact.
Students relate the equivalence of the Ratio and Parallel Methods to the Triangle Side Splitter Theorem: A line segment splits two sides of a triangle proportionally if and only if it is parallel to the third side.

Students divide a line segment into equal pieces by the Side Splitter and Dilation Methods.
Students know how to locate fractions on the number line.

Students understand that similarity is reflexive, symmetric, and transitive.
Students recognize that if two triangles are similar, there is a correspondence such that corresponding pairs of angles have the same measure and corresponding sides are proportional. Conversely, they know that if there is a correspondence satisfying these conditions, then there is a similarity transformation taking one triangle to the other respecting the correspondence.

Students prove the angle-angle criterion for two triangles to be similar and use it to solve triangle problems.

Students indirectly solve for measurements involving right triangles using scale factors, ratios between similar figures, and ratios within similar figures.
Students use trigonometric ratios to solve applied problems.

Students prove the side-angle-side criterion for two triangles to be similar and use it to solve triangle problems.
Students prove the side-side-side criterion for two triangles to be similar and use it to solve triangle problems.

Students state, understand, and prove the Angle Bisector Theorem.
Students use the Angle Bisector Theorem to solve problems.

Students multiply and divide expressions that contain radicals to simplify their answers.
Students rationalize the denominator of a number expressed as a fraction.

It is convenient, as adults, to use the notation “ ” to refer to the value of the square of the sine function. However, rushing too fast to this abbreviated notation for trigonometric functions leads to incorrect understandings of how functions are manipulated, which can lead students to think that is short for “ ” and to incorrectly divide out the variable, “ .”

To reduce these types of common notation-driven errors later, this curriculum is very deliberate about how and when we use abbreviated function notation for sine, cosine, and tangent:

In geometry, sine, cosine, and tangent are thought of as the value of ratios of triangles, not as functions. No attempt is made to describe the trigonometric ratios as functions of the real number line. Therefore, the notation is just an abbreviation for the “sine of an angle” ( ) or “sine of an angle measure” ( ). Parentheses are used more for grouping and clarity reasons than as symbols used to represent a function.

In Algebra II, to distinguish between the ratio version of sine in geometry, all sine functions are notated as functions: is the value of the sine function for the real number , just like is the value of the function for the real number . In this grade, we maintain function notation integrity and strictly maintain parentheses as part of function notation, writing, for example, , instead of .

By pre-calculus, students have had two full years of working with sine, cosine, and tangent as both ratios and functions. It is finally in this year that we begin to blur the distinction between ratio and function notations and write, for example, as the value of the square of the sine function for the real number , which is how most calculus textbooks notate these functions.

Students prove that the area of a triangle is one-half times the product of two side lengths times the sine of the included angle and solve problems using this formula.
Students find the area of an isosceles triangle given the base length and the measure of one angle.

Students find missing side lengths of an acute triangle given one side length and the measures of two angles.
Students find the missing side length of an acute triangle given two side lengths and the measure of the included angle.