Students write unknown angle proofs, which does not require any new geometric facts. Rather, writing proofs requires students to string together facts they already know to reveal more information.

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

CK-12's Geometry - Second Edition is a clear presentation of the essentials of geometry for the high school student. Topics include: Proofs, Triangles, Quadrilaterals, Similarity, Perimeter & Area, Volume, and Transformations.

The purpose of this task is to solidify ways of thinking about formal proofs, such as reasoning from a diagram and identifying a sequence of statements that start with given assumptions and lead to a valid conclusion. This task introduces the flow diagram as a way of keeping track of the logical connections between given statements and the conclusions that can be drawn from them.

The purpose of this task is to give students practice in writing proofs to show that the conjectures are true. The specific theorems being examined in this task are:
• Vertical angles are congruent • The measure of an exterior angle of a triangle is the sum of the two remote interior angles
• When a transversal crosses parallel lines, alternate interior angles are congruent, corresponding angles are congruent, and same-side interior angles are supplementary (add to 180°).

In addition to practice writing proofs, the idea of the converse of a statement should be brought up in the discussion of this task. After students prove that alternate interior angles are congruent and corresponding angles are congruent when a transversal crosses parallel lines, they need to examine the converse statements as theorems:
• If corresponding angles are congruent when a transversal crosses two or more lines, then the lines are parallel.
• If alternate interior angles are congruent when a transversal crosses two or more lines, then the lines are parallel.

This task gives students opportunities to practice applying the theorems of this and the previous module. The theorems students will draw upon include:
• Vertical angles are congruent.
• Measures of interior angles of a triangle sum to 180°.
• When transversals cross parallel lines, alternate interior angles are congruent and corresponding angles are congruent.
• A line parallel to one side of a triangle divides the other two sides proportionally.

Students will also apply the Pythagorean theorem to find the missing sides of right triangles, and conversely, to determine if a triangle is a right triangle. The last part of the task allows students to review their “ways of knowing” something is true through inductive and deductive reasoning. Students will collect data about the sums of the measures of the interior angles of quadrilaterals and pentagons. When combined with their knowledge of the sum of the measures of the interior angles of a triangle, students make a conjecture about the sum of the measures of the interior angles of polygons with any number of sides. Students are then asked to use deductive reasoning to prove their conjecture for n-sided polygons.

In this unit, students will review basic geometric vocabulary involving parallel lines, transversals, angles, and the tools used to create and verify the geometric relationships. They will also explore angle relationships formed when lines are cut by a transversal in city planning models

This video provides a proof by contradiction that corresponding angle equivalence implies parallel lines.

Euclid was right, we can’t make much progress in proving statements in geometry without a statement about parallelism. Euclid made an assumption related to parallelism—his frequently discussed and questioned 5th postulate. Non-Euclidean geometries resulted from mathematicians making different assumptions about parallelism. The purpose of this task is to establish some “parallel postulates” for transformational geometry. The authors of CCSS-M suggested some statements about parallelism that they would allow us to assume to be true in their development of the geometry standards: (1) rigid motion transformations “take parallel lines to parallel lines” (that is, parallelism, along with distance and angle measure, is preserved by rigid motion transformations—see 8.G.1), and (2) dilations “take a line not passing through the center of the dilation to a parallel line” (see G.SRT.1a). In this task we develop some additional statements about parallelism for the rigid motion transformations, which we will accept as postulates for our development of geometry: (1) After a translation, corresponding line segments in an image and its preimage are always parallel or lie along the same line; (2) After a rotation of 180°, corresponding line segments in an image and its pre-image are parallel or lie on the same line; (3) After a reflection, line segments in the pre-mage that are parallel to the line of reflection will be parallel to the corresponding line segments in the image. These statements about parallelism will lead to the proofs of theorems about relationships of angles relative to parallel lines crossed by a transversal.

Starting with the location of all Life Flight hospitals in a state, use perpendicular bisectors to draw boundaries between Life Flight regions, which will inform first responders.

Students are provided with information and notice that the original three points given seem to be midpoints of the sides of a newly formed triangle. Their task is to determine how they would prove this conjecture? The mathematics task is intended to be a problem or question that encourages the use of mathematical practices. The dialogue is meant to show how students might engage in the mathematical practices as they work on the task.

This lesson unit is intended to help teachers assess how well students are able to identify and use geometrical knowledge to solve a problem. In particular, this unit aims to identify and help students who have difficulty in: making a mathematical model of a geometrical situation; drawing diagrams to help with solving a problem; identifying similar triangles and using their properties to solve problems; and tracking and reviewing strategic decisions when problem-solving.