# GEOMETRY RESOURCES

### Module 1: Transformations and Symmetry

In Module 1, students use rigid geometric transformations (reflection, rotation, translation) to understand important properties of geometric shapes. They will also use geometric transformations to understand ideas of algebra (namely, slopes of parallel or perpendicular lines).

By the end of this module, students should be able to do the following:

• Reflect a shape on a grid (on graph paper) across a line to the precise new location.
• Rotate a shape on a grid (on graph paper) around a point to the precise new location.
• Translate a shape on a grid (on graph paper) to the precise new location.
• Identify a sequence of transformations that will match an original image (“preimage”) to a transformed/new image (“image”).
• Write an equation of a line that is parallel to a give/know line.
• Write an equation of a line that is perpendicular to a give/known line.
• Identify features of geometric shapes (mainly polygons). These features include lines of symmetry, centers of rotation, congruent angles, congruent sides, properties of diagonals, etc.

Module 1 Resources:

### Module 2: Congruence, Construction and Proof

In Module 2, students use compass and straight edge constructions to further understand properties of shapes as well as understand the minimum conditions required to ensure two triangles are congruent (matching in size and shape).

By the end of this module, students should be able to do the following:

• Complete a number of geometric constructions (construct a rhombus, a square, parallelograms, equilateral triangles, inscribed hexagons, parallel lines, perpendicular bisector, perpendicular line through a given point, angle bisector, etc.)
• Identify whether two triangles are congruent using SAS, ASA, AAS, and SSS Triangle Congruence.
• Identify corresponding parts of congruent shapes.

Module 2 Resources:

### Module 3: Geometric Figures

In Module 3, students develop their logic and reasoning skills by creating geometric proofs. Students will use flow diagrams and two-column proof formats to prove things about angles, lines, triangles, and quadrilaterals. Students will review and use geometric constructions and triangle congruence properties (AAS, SAS, SSS, ASA) from HS Math 1.

By the end of this module, students should be able to do the following:

• Construct a logical sequence of statements that flow from beginning assumptions to correct justified conclusions.
• Construct proofs in a variety of formats, including flow diagrams and two-column proofs.
• Use the triangle congruence properties (AAS, SAS, SSS, ASA) to formally prove other conjectures.
• Prove theorems about triangles, including the sum of the interior angles of a triangle is 180 degrees.
• Prove theorems about triangles, including base angles of isosceles triangles are congruent.
• Prove theorems about lines and angles, including properties of perpendicular bisectors of a line.
• Know what an altitude of a triangle is.
• Know what a median of a triangle is.
• Know what an angle bisector of a triangle is.
• Know what a perpendicular bisector is.

Module 3 Resources:

### Module 4: Similarity & Right Triangle Trigonometry

In Module 4, students use the concepts of similarity and dilation from 8th grade to develop new properties of similar shapes, especially triangles. Most importantly, students use similarity to create the world of Trigonometry and establish the definition of the three main trig ratios – sine, cosine, and tangent. After memorizing the trig ratios, students will also learn to use the inverse of the trig ratios as well as develop some properties/identities of the trig ratios.

By the end of this module, students should be able to do the following:

• Determine if two shapes are similar.
• Determine the scale factor (or zoom factor or ratio) between two similar shapes.
• Find missing side lengths and angles of similar shapes.
• Write a sine equation for a given right triangle (both with numeric values and with symbolic values).
• Write a cosine equation for a given right triangle (both with numeric values and with symbolic values).
• Write a tangent equation for a given right triangle (both with numeric values and with symbolic values).
• Use the inverse sine, inverse cosine, and inverse tangent to find missing angles in right triangles.
• Use trig and trig inverse to solve right triangles.
• Solve application problems (word problems) involving right triangles using trigonometry.

Module 4 Resources:

### Module 5: Circles A Geometric Perspective

In Module 5, students explore properties and shapes found within circles. This includes similarity, chords, secant lines, tangent lines, central angles, inscribed angles, circumscribed angles, inscribed polygons, areas of sectors and much more!

By the end of this module, students should be able to do the following:

• Solve problems involving central angles, inscribed angles, and circumscribed angles in circles.
• Solve problems involving chords, secant lines, and tangent lines within circles.
• Find the area of circles and smaller areas of sectors.
• Find the circumference of circles and smaller arc lengths.
• Define a radian angle measure using circles.

Module 5 Resources:

### Module 6: Connecting Algebra and Geometry

In Module 6, students work with geometric shapes on the coordinate plane to connect ideas from geometry to ideas found in algebra. Specifically, students use the geometric Pythagorean Theorem while on the coordinate plane to create the algebraic Distance Formula for coordinates.

By the end of this module, students should be able to do the following:

• Calculate the distance between two points on the coordinate plane.
• Calculate the perimeter and/or area of a geometric shape drawn on the coordinate plane.
• Prove the slope criteria for parallel lines (same slope) and perpendicular lines (opposite reciprocal slope).
• Classify shapes that are drawn on the coordinate plane by connecting algebraic and geometric properties.
• Transform algebraic functions by translating up or down.

Module 6 Resources:

### Module 7: Modeling with Geometry

In Module 7, students examine complex 3-dimensional objects and generate them by revolving common 2-dimensional areas. Students also revisit the trigonometric ratios and add a few special right triangles to their triangle repertoire. Lastly, students learn to work with non-right triangles by using the Law of Sines and the Law of Cosines.

By the end of this module, students should be able to do the following:

• Revolve a given area around either the x-axis or the y-axis.
• Find the volume of common 3-dimensional shapes and use those common shapes to find the volume of compound shapes.
• Know the side lengths of the two special right triangles: 30-60-90 Right triangle and the 45-45-90 Right triangle.
• Use the Law of Sines to find missing sides and angles of a non-right triangle.
• Use the Law of Cosines to find missing sides and angles of a non-right triangle.

Module 7 Resources:

### Module 8: Probability

In Module 8, students examine probabilities in many different situations. They will use Tree Diagrams, Venn Diagrams, and Two-Way Tables to determine probabilities, conditional probabilities, and compound probabilities.

By the end of this module, students should be able to do the following:

• Find a probability from data given in either a Tree Diagram, a Venn Diagram, or a Two-Way Table.
• Find a conditional probability from data given in either a Tree Diagram, a Venn Diagram, or a Two-Way Table.
• Find a compound probability from data given in either a Tree Diagram, a Venn Diagram, or a Two-Way Table.
• Construct a Tree Diagram, a Venn Diagram, or a Two-Way Table for a given situation.

Module 8 Resources: