# 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:

- MVP Geometry Curriculum: MVP Website
- Helpful videos: Khan Academy – Geometric Transformations
- Helpful videos: Khan Academy – Parallel and Perpendicular Lines
- Helpful explanations: Math Bits Notebook – Rigid Transformations
- Helpful explanations: Math Warehouse – Parallel and Perpendicular Lines

### 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**S**SS Triangle Congruence. - Identify
**corresponding parts**of congruent shapes.

Module 2 Resources:

- MVP Geometry Curriculum: MVP Website
- Helpful videos: Khan Academy – Congruence and Constructions
- Helpful explanations: Math Open Reference – Construction Demos
- Helpful explanations: Math Bits Notebook – Congruent Triangles (SAS, ASA, AAS, SSS)

### 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:

- MVP Geometry Curriculum: MVP Website
- Helpful videos: Khan Academy – Congruence (proofs)
- Helpful explanations: Math Bits Notebook – Types of Proofs explanation

### 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:

- MVP Geometry Curriculum: MVP Website
- Helpful videos: Khan Academy – Right Triangles & Trigonometry
- Helpful explanations: Lumen – Trigonometry and Right Triangles

### 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:

- MVP Geometry Curriculum: MVP Website
- Helpful videos: Khan Academy – Circles
- Helpful explanations: Math is Fun – Circle Theorems (angles)
- Helpful explanations: Math Bits Notebook – Rules for Chords, Secants, and Tangents (with proofs)

### 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:

- MVP Geometry Curriculum: MVP Website
- Helpful videos: Khan Academy – Shapes/Quadrilaterals on the Coordinate Plane
- Helpful videos: Khan Academy – Distance Between Two Points, Parallel & Perpendicular, and Proofs
- Helpful explanations: Interactive Math website – Distance Formula

### 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:

- MVP Geometry Curriculum: MVP Website
- Helpful videos: Khan Academy – Solid Geometry (2-D and 3-D)
- Helpful videos: Khan Academy – Right Triangles & Trigonometry
- Extra practice: IXL – Solids of Revolution (limited practice without membership)

### 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:

- MVP Geometry Curriculum: MVP Website
- Helpful videos: Khan Academy – Two-Way Tables
- Helpful videos: Khan Academy – Probability
- Helpful explanations: Math is Fun – Probability (overview)