# High School Geometry Course

**Time4Learning offers an online, interactive high school curriculum that correlates to state standards. **It can be used as a primary homeschool curriculum, a supplement to your current curriculum and as an afterschool or summer skill building program. At the high school level, Time4Learning is organized by courses rather than grade levels, so parents have the option of choosing any four as part of membership.

**The Geometry curriculum is one of five Math courses offered at the high school level. **Geometry is taught using a combination of multimedia lessons, instructional videos, worksheets, quizzes, tests and both online and offline projects. The Geometry course is designed to prepare students for college level mathematics.

**This page includes information about the material covered in the High School Geometry course.**

### Chapter 1 – Introduction to Geometry

**The materials in this chapter introduce and cover the basics of geometry. **It is organized into sections that teach, reinforce and test students on the concepts of points, lines, and planes, measuring and constructing and angle relationships.

**Lessons in this chapter are organized into the following sections:**

**Points, lines, and planes**– Students learn about the different geometric points, lines, planes, as well as the intersection of lines and planes, collinear and noncollinear points and naming conventions.**Measuring and constructing**– Students are taught how to use a compass and a protractor in order to measure and construct segments. Students will be required to bisect segments, and understand the concepts of congruence and congruent segments. An introduction to acute, obtuse and right angles is also included.**Angle relationships**– Students learn about adjacent angles, complementary angles, congruent angles, supplementary angles and vertical angles, and use those angles to solve algebraic problems.

### Chapter 2 – Reasoning and Proofs

**The materials in this chapter introduce and cover reasoning and proofs.** It is organized into sections that teach, reinforce and test students on the concepts of logic and reasoning, classifying polygons and proofs.

**Lessons in this chapter are organized into the following sections:**

**Logic and reasoning**– Students are introduced to three types of reasoning: conjecture, example and counter-example and inductive reasoning. Students are then given examples of inductive reasoning and and taught how to use the process of inductive reasoning as a way to arrive at a reasonable conjecture. Conditional statements, biconditionals, and deductive reasoning are also introduced.**Classifying polygons**– Students learn how to classify polygons by type, including regular and irregular polygons. Students will then be expected to tell the difference between concave and convex polygons, and how to identify lines of symmetry.**Proofs**– Students continue with the study of geometric proofs, such as two-column proofs and paragraph proofs. Students will explore the relationships between angles such as complementary, supplementary and right angles, and learn how to apply special angel relationships to problem situations.

### Chapter 3 – Lines and Angles

**The materials in this chapter introduce and cover lines and angles.** It is organized into sections that teach, reinforce and test students on the concepts of lines and angles, slopes, constructions, perpendicular lines and midpoint and distance formulas.

**Lessons in this chapter are organized into the following sections:**

**Lines and angles**– Students are shown the properties for angle relationships, such as supplementary, complementary, vertical, adjacent and linear pairs, formed by intersecting lines and parallel lines cut by a transversal.**Slopes**– Students are taught how to calculate the slope of a line using the formula for slope that uses the coordinates of two points that are on the line. Students must also find the slope of parallel and perpendicular lines.**Constructions**– Students are required to construct both parallel and perpendicular lines with a compass and a straight edge.**Perpendicular lines**– Students learn about the Perpendiculars to Parallels Theorem and the Two Perpendiculars Theorem.**Midpoint and distance formulas**– Students are taught how to use the distance and midpoint formulas to solve problems and develop coordinate proofs.

### Chapter 4 – Triangles I

**The materials in this chapter introduce and cover triangles. **It is organized into sections that teach, reinforce and test students on the concepts of classifying triangles, triangle congruence and special parts of triangles.

**Lessons in this chapter are organized into the following sections:**

**Classifying triangles**– Students learn how to classify triangles, like acute, equiangular, equilateral, isosceles, obtuse, right and scalene, according to their sides and angles. Students will then use properties of exterior and interior angles to solve problems.**Triangle congruence**– Students are shown the basic triangle congruence postulates and theorems SSS, SAS, ASA, AAS, HL and CPCTC. Students will use these postulates and theorems to complete proofs.**Special parts of triangles**– Students the special features of triangles including midsegments, midpoints, medians, altitude, centroids and orthocenter. Students will then use the properties of these special features to solve problems.

### Chapter 5 – Triangles II

**The materials in this chapter introduce and cover triangles. **It is organized into sections that teach, reinforce and test students on the concepts of indirect proofs, triangle inequalities and similar triangles.

**Lessons in this chapter are organized into the following sections:**

**Indirect proofs**– Students are taught the concepts of indirect proof and proof by contradiction (Modus Tollens and Reductio ad Absurdum) and how to complete them.**Triangle inequalities**– Students will be required to understand the Triangle Inequality Theorem and the Hinge Theorem in order to use them to solve problems.**Similar triangles**– Students will use ratio and proportions and triangle similarity postulates and theorems to prove that two triangles are similar.

In the high school Geometry course, students learn about different types of lines and angles.

In this lesson, students are asked to identify parallel and perpendicular lines both with and without a graph.

### Chapter 6 – Polygons

**The materials in this chapter introduce and cover polygons.** It is organized into sections that teach, reinforce and test students on the concepts of interior angles of polygons, classifying quadrilaterals, confirming a quadrilateral is a parallelogram and similar polygons.

**Lessons in this chapter are organized into the following sections:**

**Interior angles of polygons**– Students will solve problems by applying the properties of the interior angles to certain polygons.**Classifying quadrilaterals**– Students are taught how to classify and chart the following quadrilaterals: parallelograms, rectangles, rhombuses, squares, trapezoids and kites.**Confirming a quadrilateral is a parallelogram**– Students are required to use a set of criteria to determine if a polygon is a quadrilateral or a parallelogram.**Similar polygons**– Students are taught to use ratio and proportion and the properties of similar figures to solve problems involving similar polygons.

### Chapter 7 – Area of Polygons and Circles

**The materials in this chapter introduce and cover area of polygons and circles. **It is organized into sections that teach, reinforce and test students on the concepts of perimeter, circumference, and area, area of polygons, perimeters and areas of similar figures, circles, arcs, and sectors and geometric probability.

**Lessons in this chapter are organized into the following sections:**

**Perimeter, circumference, and area**– Students will solve problems involving rectangles and circles by utilizing area, perimeter and circumference formulas.**Area of polygons**– Students will solve problems involving parallelograms, triangles, trapezoids, rhombuses and kites using the appropriate area formulas.**Perimeters and areas of similar figures**– Students will compare areas of similar figures to determine the effect changing the perimeter has on the area of a figure, and then solve them.**Circles, arcs, and sectors**– Students are required to apply formulas to find the area of sectors and the lengths of any arcs on a circle.**Geometric probability**– Students are taught to understand geometric probability and how to use it to solve a set of problems.

### Chapter 8 – Circles

**The materials in this chapter introduce and cover circles. **It is organized into sections that teach, reinforce and test students on the concepts of tangent lines, arcs and chords, angle relationships in circles, segment relationships in circles, equations of circles and Reuleaux triangle.

**Lessons in this chapter are organized into the following sections:**

**Tangent lines**– Students must use the properties of tangent lines to solve problems and complete a proof.**Arcs and chords**– Students must use properties of arcs and chords to solve problems.**Angle relationships in circles**– Students must use the properties of inscribed angles, central angles, intercepted arcs and circles to solve problems.**Segment relationships in circles**– Students must apply properties involving segment relationships in circles in order to solve problems.**Equations of circles**– Students must understand and solve the equation of a circle.**Reuleaux triangle**– Students must understand the Reuleaux triangle in order to use its properties to solve a set of problems.

### Chapter 9 – Transformations

**The materials in this chapter introduce and cover transformations. **It is organized into sections that teach, reinforce and test students on the concepts of transformations and translations with matrices.

**Lessons in this chapter are organized into the following sections:**

**Transformations**– Students are taught how to apply the concepts of geometric transformation including translation, reflection, rotation, symmetry, dilation and tessellation in order to solve problems.**Translations with matrices**– Students learn how to use matrices to translate and rotate geometric figures on a coordinate plane.

### Chapter 10 – Surface Area and Volume

**The materials in this chapter introduce and cover surface area and volume. **It is organized into sections that teach, reinforce and test students on the concepts of surface area, volume, spheres and volume of similar solids.

**Lessons in this chapter are organized into the following sections:**

**Surface area**– Students are required to use formulas of surface area for prisms, cylinders, pyramids and cones in order to solve problems.**Volume**– Students are required to use formulas of volume for prisms, cylinders, pyramids and cones in order to solve problems.**Spheres**– Students are required to use formulas of surface area and volume for various spheres in order to solve problems.**Volume of similar solids**– Students change the dimensions of a geometric solid in order to see what effect it has on the volume of the solid.

### Chapter 11 – Special Geometric Relations

**The materials in this chapter introduce and cover special geometric relations. **It is organized into sections that teach, reinforce and test students on the concepts of geometric mean, The Pythagorean Theorem, special triangle relations, angles of elevation and depression and vectors.

**Lessons in this chapter are organized into the following sections:**

**Geometric mean**– Students are required to use the concept of geometric mean in order to solve problems.**The Pythagorean Theorem**– Students are taught how to use the Pythagorean Theorem to find the missing lengths of right triangles. Students will also recognize and apply the trigonometric ratios.**Special triangle relations**– Students must apply properties of special right triangles and basic triangle trigonometry in order to solve problems.**Angles of elevation and depression**– Students must use the concept of angels of elevation and depression in order to solve problems.**Vectors**– Students will learn what vectors are and how to add two vectors together.

### Chapter 12 – Extensions

**The materials in this chapter introduce and cover extensions. **It is organized into sections that teach, reinforce and test students on the concepts of geometric constructions, non-Euclidean geometry, the architect and geometric art.

**Lessons in this chapter are organized into the following sections:**

**Geometric constructions**– Students are required to construct various geometric figures, such as the bisector of an angle.**Non-Euclidean geometry**– Students will gain an understanding of a non-Euclidean geometry called Taxicab Geometry. Students will use that geometry in order to solve problems.**The architect**– Students solve architectural blueprints by utilizing area and perimeter formulas.**Geometric art**– Students delve into geometric art, such as tessellations and tiling patterns.