# High School Level—Geometry

Overview

#### About

#### 01: Fundamental Geometric Concepts

In this introductory lesson, we define point, line, and plane; use and understand the terms space, collinear, intersection, segment, and ray; learn terminology of various expressions relative to points, lines, and planes; and establish a system of linear measurement.

#### 02: Angles and Angle Measure

We explore the definition of an angle and learn its parts; establish a system of angle measurement; recognize and classify types of angles; and show angle relationships.

#### 03: Inductive Reasoning and Deductive Reasoning

We use inductive reasoning to discover mathematical relationships, recognize real-world applications of inductive reasoning, and understand conditional statements and deductive reasoning.

#### 04: Preparing Logical Reasons for a Two-Column Proof

We review properties of equality for real numbers; summarize and review postulates related to points, lines, planes, and angles; and introduce new theorems related to points, lines, planes, and angles.

#### 05: Planning Proofs in Geometry

We discuss the key elements of a two-column proof; learn how to draw and label a diagram for a proof; write a plan for the proof; use strategy to write a two-column proof; and write a two-column proof.

#### 06: Parallel Lines and Planes

We identify parallel lines, skew lines, parallel planes, transversals, and the angles formed by them; and we state and apply postulates and theorems about angles formed when parallel lines are intersected by a transversal.

#### 07: Triangles

We classify triangles according to their sides and angles, and use theorems about the angles of a triangle.

#### 08: Polygons and Their Angles

We distinguish between convex polygons and concave polygons; name convex and regular polygons; and find measures of interior and exterior angles of convex polygons.

#### 09: Congruence of Triangles

We identify congruent parts of congruent triangles; state and apply the SSS, SAS, and ASA postulates; and use those postulates to prove triangles congruent.

#### 10: Variations of Congruent Triangles

We deduce that segments or angles are congruent by first proving two triangles congruent; use two congruent triangles to prove other, related facts; and prove two triangles congruent by first proving two other triangles congruent.

#### 11: More Theorems Related to Congruent Triangles

We use the isosceles triangle theorem, its converse, and related theorems; and use the AAS theorem and right triangle theorems to prove triangles congruent.

#### 12: Median, Altitudes, Perpendicular Bisectors, and Angle Bisectors

We discuss definitions of median, altitude, perpendicular bisector, angle bisector, and related terms; state and apply theorems related to them; and learn their points of concurrence.

#### 13: Parallelograms

We state and apply the definition of a parallelogram, state and apply theorems related to the properties of a parallelogram, and prove that certain quadrilaterals are parallelograms.

#### 14: Rectangles, Rhombuses, and Squares

We identify rectangles, rhombuses, and squares; and state and apply properties and theorems related to their properties.

#### 15: Trapezoids, Isosceles Trapezoids, and Kites

We learn to identify trapezoids, isosceles trapezoids, and kites, and we state and apply properties and theorems related to them.

#### 16: Inequalities in Geometry

We review properties of inequality for real numbers and relate them to segments and angles; state and apply the inequality relations for one triangle and for two triangles.

#### 17: Ratio, Proportion, and Similarity

In this lesson we explain how to express a ratio in its simplest form; identify, write, and solve proportions; use ratios and proportions to solve problems; express a given proportion in other equivalent forms; and apply the properties of similar polygons using ratios and proportions.

#### 18: Similar Triangles

We state and apply the AA Similarity Postulate, the SAS Similarity Theorem, and the SSS Similarity Theorem. We learn to solve for unknown measurements using the new postulates and theorems related to similarity and to apply the Triangle Proportionality Theorem, the Triangle Angle-Bisector Theorem, and related theorems.

#### 19: Right Triangles and the Pythagorean Theorem

We apply proportions and the concepts of proportionality in right triangles, use and apply the geometric mean between two values, state and apply the relationships that exist when the altitude of a triangle is drawn to the hypotenuse, state and apply the Pythagorean Theorem and its converse, and relate the Pythagorean Theorem to inequalities.

#### 20: Special Right Triangles

We explore how to apply relationships in a 45°-45°-90° right triangle and in a 30°-60°-90° right triangle and use those relationships in the development of the unit circle.

#### 21: Right-Triangle Trigonometry

We define and apply the tangent, sine, and cosine ratios for an acute angle and solve right-triangle problems using those ratios.

#### 22: Applications of Trigonometry in Geometry

We address how to select the correct trigonometric ratio to use in problem solving, and how to use trigonometry to solve real-life problems.

#### 23: Tangents, Arcs, and Chords of a Circle

We apply basic definitions and concepts related to circles, and state and apply properties and theorems regarding circles and their tangents, chords, central angles, and arcs.

#### 24: Angles and Segments of a Circle

We apply basic definitions and theorems related to inscribed angles; state and apply theorems involving angles with vertices not on the circle formed by tangents, chords, and secants; and state and apply theorems involving lengths of chords, secant segments, and tangent segments.

#### 25: The Circle as a Whole and Its Parts

We state and apply the formulas for the circumference and area of a circle, and for the arc lengths and the areas of sectors of a circle.

#### 26: The Logic of Constructions through Applied Theorems (Part I)

In sample exercises, we review lessons and solve problems having to do with segments, angles, parallel and perpendicular lines, circles and arcs, and others.

#### 27: The Logic of Constructions through Applied Theorems (Part II)

Continuing sample exercises, we review lessons and solve problems having to do with triangles, isosceles triangles, proportions, hexagons, and others.

#### 28: Areas of Polygons

We address the derivation of the area formulas and apply those formulas to find the areas of a rectangle, square, parallelogram, triangle, trapezoid, and regular polygon.

#### 29: Prisms, Pyramids, and Polyhedra

We explore definitions of a polyhedron, prism, pyramid, and related terms; understand the logical derivation of area and volume formulas; and apply theorems to compute the lateral area, total area, and volume of prisms and pyramids.

#### 30: Cylinders, Cones, and Spheres

We explain the definitions of cylinder, cone, and sphere; explain the logical derivation of area and volume formulas; and apply theorems to compute the lateral areas, total areas, and volumes of cylinders, cones, and spheres.