Chapter

Competencies

Assessment

Points, Lines, Planes, and Angles

The basic geometric figures are introduced as basic building blocks of geometry. Lines, points and planes are first introduced and the student learns to determine relationships between points and lines (such as points being collinear and coplanar, lines intersecting, etc.) Rays and line segments are later introduced and the concept of length is defined. The introduction of formal deductive reasoning begins here. The basic postulates of Euclidean geometry are introduced. Theorems are also introduced here. The student understands how theorems are proven using postulates and other theorems.

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Reasoning

The conditional, or "if-then" statement, is introduced and examined. The student views the many ways that a conditional can be expressed and determines its hypothesis and conclusion. The student determines the validity of the converse of a conditional and learns to disprove a condition using counterexamples. The two-column proof, a formal style of proving theorems, is introduced and the student learns how to present a proof by first using basic properties from algebra. The student discovers the detailed process of writing a proof.

Numerical properties of angles and lines are later defined. The student determines if angles are complementary or supplementary and lines are perpendicular by using definitions and theorems. These definitions and theorems and also use to enhance skills of proof writing.

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Parallel Lines and Planes

The concepts of parallel (and skew) lines and parallel planes are introduced. The student observes the many relationships between angles formed by a line (defined as a transversal) intersecting two lines (e.g., corresponding angles, alternate interior angles, etc.) The student determines whether pairs of angles—which are formed by many lines and line segments—have these characteristics. The student can further deduce properties if parallel lines are involved. These properties are used to prove more theorems.

The student classifies triangles according to its sides (scalene, isosceles and equilateral) and angles (acute, obtuse, right and equiangular). The fundamental theorem of the angles of a triangle (i.e.: the sum of their angles is 180 degrees) is introduced and this theorem will be the basis of other theorems about triangles and other polygons. The student also discerns between inductive and deductive reasoning.

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Triangles

Congruent figures, especially triangles, are examined. The student learns and applies postulates and theorems that validate congruence (if possible) between two triangles. The theorems and postulates supporting the congruence of triangles are also the basis for developing more complex deductive reasoning skills. Isosceles triangles are also examined.

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Quadrilaterals

First the properties of the parallelogram are examined. The students examine and use properties concerning different parts of the parallelogram. Afterwards, the student determines whether quadrilaterals with given properties are parallelograms.

The classification of quadrilaterals (e.g.: the parallelogram, rhombus, rectangle, square, trapezoid, isosceles trapezoid) are examined. The same skills of determining data sufficiency used in showing quadrilaterals are parallelograms are also developed here. The properties of these quadrilaterals are used to calculate parts of quadrilaterals.

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Inequalities in Geometry

Inequalities are observed in an algebraic sense and then used to develop inequalities evident in geometry. These inequalities are mainly deduced from triangles.

Different types of reasoning are also introduced. Inverses and contrapositives of conditions are defined and observed (with converses already introduced). The concepts of indirect proofs and paragraph proofs are introduced and here the student develops reasoning skills without the use of the two-column proof format.

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Similar Polygons

The concept of similarity of polygons is introduced and especially the concept of similar triangles is examined. As with congruence, the student determines similarity between two triangles using postulates and theorems. Given similarity of two polygons, students learn to calculate lengths of sides and perimeters. Similarity theorems are also used to prove other theorems concerning parallel lines and angle bisectors of triangles.

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Right Triangles

Many topics are introduced concerning right triangles. Similar triangles are examined with the altitude of a right triangle is drawn to its hypotenuse. The Pythagorean Theorem is also reviewed (since it has been introduced in a first-year algebra course.) The theorem is used not only to find lengths of sides of a right triangle, but also used to classify triangles as acute, right or obtuse. Special right triangles (45-45-90 and 30-60-90 triangles) are examined and triangle trigonometry is introduced to find lengths of sides of other right triangles. The student also applies triangle trigonometry to calculate real-life geometric problems.

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Circles

Terminology concerning circles are introduced. Theorems and formulas are then proven and derived so that measurements of parts of a triangle can be calculated. These parts include angles (central, inscribed and other), arcs, lengths of cords or parts of chords, and tangents.

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Constructions and Loci

Basic construction techniques using a compass and straightedge are introduced. The student then utilizes these techniques to create constructions with given properties. These constructions are also used to review previous postulates, theorems and formulas—especially those concerning congruent and similar triangles, and circles.

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Areas of Plane Figures

Formulas for area of shapes are introduced and utilized. These shapes include rectangles, parallelograms, triangles, rhombuses, trapezoids, regular polygons, circles and parts of circles. The student calculates areas of these shapes, using formulas as well as Pythagorean Theorem, special triangles and triangle trigonometry. These formulas are also used to calculate areas of similar figures and are even applied in probability situations.

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Formulas for finding the surface area and volume of solids (three-dimensional figures) are derived and introduced. These shapes include prisms, pyramids and spheres. In finding solids, the student also reviews the use of the Pythagorean Theorem, special right triangles, triangle trigonometry and area of plane figures in the previous chapter. Ratio of areas and volumes of similar solids are examined and calculated.

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Coordinate Geometry

The concept of the Cartesian plane is reviewed (since its introduction in a first-year algebra or pre-algebra course.) Geometric concepts are then examined in a Cartesian context (such as find the distance between two points.) Lines in a Cartesian plane and their characteristics are also reviewed (such as slope, determining if lines are parallel or perpendicular, the many forms of equations of lines, etc.) These concepts are then used to supply further proof of theorems, especially theorems involving triangles and quadrilaterals.

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