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Geometry Syllabus

TERM 1

Lesson # Topics and Objectives

1 -2 Unit I: LOGIC

Introduction to Logic

a. Define statement, compound statement, and logical connectives.

b. Determine the truth values for a negation, conjunction, and disjunction.

c. Differentiate between exclusive and inclusive “or.”

NOTE: De Morgan’s Laws have been removed from the syllabus

3. The Conditional Statement

a. Identify the hypothesis and conclusion of a conditional statement, including those of hidden

conditionals.

b. Determine the truth value for a conditional statement.

c. Write the converse, inverse and contrapositive of a conditional.

d. Show that a conditional statement and its contrapositive are logically equivalent

e. Show that the converse and the inverse are not logically equivalent to the original

conditional statement.

4 The Biconditional

a. Define biconditional.

b. Determine the truth value for a biconditional.

NOTE: Defining a tautology and a contradiction has been moved to

the introduction to indirect proof.

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5-6

Unit II: INTRODUCTION TO GEOMETRY

Definitions Involving Segments and Angles

a. Write the symbols and diagrams (if applicable) for undefined terms point, line, plane, half-

plane, between, distance, and angle measure.

b. Write the symbols and diagrams (if applicable) and the definitions of collinear, segment,

endpoint, midpoint, ray, opposite rays, segment bisector, angle, vertex, interior of an angle,

adjacent angles, vertical angles, angle bisector, acute angle, right angle, obtuse angle,

perpendicular lines, perpendicular bisector, complementary angles, supplementary angles,

linear pair.

c. Apply definitions to problems which require the solution to a linear equation, system of

linear equations, or factorable quadratic equation.

7-9 Basic Postulates

a. Define postulate and theorem.

b. State:

1. Two points determine a line.

2. Two lines intersect in at most one point.

3. Given a line, there exists a point not on it.

4. If B is between A and C, then A, B, and C are distinct collinear points

and AB + BC =AC.

5. Every distance is a non-negative real number.

6. The distance between two points is zero if and only if the two points are equal.

7. Given a positive real number d and a ray AB , there exists AC such that AC = d

and point C is on AB (Segment Existence Postulate).

8. Every angle measure is a real number between 0 and 180.

9. If P lies in the interior of ABC , then m ABP + m PBC = m ABC

10. Given a real number r between 0 and 180 and a ray AB , there exist two angles,

CAB and DAB such that m CAB = m DAB =r, where C and D lie in

opposite half-planes of AB (Angle Existence Postulate).

11. the reflexive, symmetric, and transitive properties of equality.

12. the addition, subtraction, multiplication, and division properties of equality.

13. the substitution property of equality.

10-12 Writing Basic Proofs

a. Apply the definitions, postulates, and properties to write proofs involving segments and

angles, including the following theorems involving angles:

1. If two angles are right angles then they are congruent.

2. If two angles form a linear pair then they are supplementary.

3. If two angles are supplementary to the same or congruent angles then they are

congruent.

4. If two angles are complementary to the same or congruent angles then they are

congruent.

5. If two angles are vertical then they are congruent.

b. Use theorems in proofs.

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13 Unit III: CONGRUENT TRIANGLES

Introduction to Congruent Triangles

a. Define congruent triangles.

b. Name the corresponding parts of congruent triangles.

c. Symbolize congruent triangles using the correct order of correspondence.

d. Solve problems involving congruent triangles which require the solution to a linear

equation, system of linear equations, or factorable quadratic equation.

14 -15 Methods For Proving Triangle Congruence

a. State and apply the SAS Postulate.

b. State, without proof, and apply the ASA and SSS Theorems.

c. Use counterexamples to show why AAA and SSA are not valid methods for proving triangles

congruent.

16-18 Proving Overlapping Triangles Congruent, Proving Corresponding Parts of Congruent

Triangles Congruent, and Double Congruence Proofs

19-21 Isosceles Triangles

a. Define scalene, isosceles, and equilateral triangles; median of a triangle, altitude of a

triangle.

b. Define legs, base, vertex angle, and base angles of an isosceles triangle.

c. Prove and apply: If two sides of a triangle are congruent, then the angles opposite those sides

are congruent.

d. Prove an equilateral triangle is equiangular.

e. Prove the bisector of the vertex angle of an isosceles triangle is also the median and the

altitude to the base.

f. Solve problems involving isosceles triangles which require algebraic solutions.

g. Use isosceles triangles to prove triangles congruent.

22-23 Proving Lines Perpendicular

a. Prove: At a given point on a given line, one and only one perpendicular can be drawn to that

line.

b. Prove: From a given point not on a given line, one and only one perpendicular can be drawn

to that line.

c. Prove that any point on the perpendicular bisector of a line segment is equidistant from the

endpoints of the line segment, and conversely.

d. Prove that if two points are each equidistant from the endpoints of a line segment, then the

points determine the perpendicular bisector of the line segment.

e. Prove that if two lines intersect to form congruent adjacent angles, then the lines are

perpendicular.

f. Use the previous two theorems to prove lines are perpendicular.

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24 -25 Unit IV: INEQUALITIES

Indirect Proof

(Include definition of tautology and contradiction.)

26 Inequality Postulates

a. State and apply the transitive and the substitution postulates of inequality.

b. State and apply “a whole is greater than any of its parts.”

NOTE: The following postulates have been removed from the syllabus:

addition, subtraction, multiplication and division postulates.

27-29 Inequalities Involving Sides and Angles of a Triangle

State, prove, and apply:

a. The Exterior Angle Inequality: The measure of an exterior angle of a triangle is greater than

the measure of either remote interior angle.

b. In a triangle, the greater angle is opposite the longer side.

c. In a triangle, the longer side is opposite the greater angle.

d. The Triangle Inequality

30-32 UNIT V: PARALLEL LINES AND ANGLE SUMS

Introduction to Parallel Lines

a. Define parallel lines, transversal, alternate interior angles, consecutive interior angles, and

corresponding angles.

b. Prove that two lines cut by a transversal are parallel if:

1. a pair of alternate interior angles are congruent.

2. a pair of corresponding angles are congruent.

3. a pair of consecutive interior angles are supplementary.

c. Prove that in a plane, two lines to the same line are parallel.

d. Use the theorems in proofs.

33-35 Properties of Parallel Lines

a. State the Parallel Postulate.

b. Prove that if two parallel lines are cut by a transversal then:

1. alternate interior angles formed are congruent.

2. corresponding angles formed are congruent.

3. same-side interior angles formed are supplementary.

c. Prove that two lines parallel to a third line are parallel to each other.

d. Use the theorems in proofs and in problems which require solutions to linear equations,

systems of linear equations, and quadratic equations.

36 The Triangle Angle-Sum Theorem

a. Prove the sum of the measures of the angles of a triangle is 180 degrees.

b. Apply the theorem to problems requiring numeric and algebraic solutions.

c. Prove the No Choice Theorem: If two angles of a triangle are congruent, respectively,

to two angles of another triangle, then the third angles are congruent.

d. Prove the acute angles of a right triangle are complementary.

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37 Polygon Angle-Sum Theorems

a. Define convex polygon: a polygon is convex if and only if a half- plane of the line joining any

two consecutive vertices of the polygon contains all the other vertices of the polygon.

b. Prove the sum of the measures of the interior angles of a polygon with n sides is 180(n -2)

degrees.

c. Prove the sum of the measures of the exterior angles of a polygon is 360 degrees.

d. Solve problems which apply the theorems to polygons and regular polygons.

38 Prove and apply: The AAS Theorem

39 Prove and apply: The Converse of the Base Angles Theorem

40 Prove and apply: The Exterior Angle Theorem

41 Prove and apply: The Hy-Leg Theorem

42 -43 UNIT VI: QUADRILATERALS

The Properties of a Parallelogram

a. Prove both pairs of opposite sides are congruent, both pairs of opposite angles are

congruent, and the diagonals bisect each other.

b. Use properties in proofs.

c. Solve problems which require algebraic solutions.

44 -45 The Properties of a Rectangle, Rhombus, and Square

46-47 Proving a Quadrilateral is a Parallelogram

48-49 Proving a Quadrilateral is a Rectangle, Rhombus, or Square

50 The Properties of a Trapezoid and Isosceles Trapezoid

Use inclusive definition for trapezoid:

“A trapezoid is a quadrilateral with at least one pair of parallel sides.”

51 Proving a Quadrilateral is a Trapezoid or Isosceles Trapezoid

52-53 UNIT VII: SIMILARITY

Solving Proportions

a. State and apply properties of proportions.

a/b = c/d if and only if:

1. ad = bc

2. b/a = d/c

3. a/c = b/d

4. (a+b)/b = (c+d)/d

5. (a-b)/b=(c-d)/d

b. Solve proportions which lead to quadratic equations or linear equations.

c. Solve problems involving the mean proportional between two numbers.

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54-55 The Line Segment Joining the Midpoints of Two Sides of a Triangle

a. Prove the theorem: If a line segment joins the midpoints of two sides of a triangle, the

segment is parallel to the third side and its length is one-half the length of the third

side.

b. Prove that the median of a trapezoid is parallel to its bases and its length is one-half

the sum of the lengths of its bases.

c. Apply the theorems to proofs and problems which lead to linear or quadratic

equations.

56 Proportionality Theorems

a. If a line is parallel to one side of a triangle and intersects the other two sides, the line

divides those sides proportionally.

b. Parallel lines intercept proportional segments on any two transversals.

57 Similar Polygons

a. Define similar polygons and ratio of similitude (constant of proportionality/scale factor).

b. Solve problems leading to algebraic solutions.

c. Prove and apply: The ratio of the perimeters of two similar triangles equals their ratio

of similitude.

58-59 AA, SSS, and SAS Similarity Theorems

a. Apply the theorems to proofs and problems.

b. Prove that two medians of a triangle intersect at a point that is a trisection point for

both of them.

60 Using Similar Triangles to Prove Segments Proportional

61 Proving Products of Segments Equal

62-63 Proportions in a Right Triangle

a. Prove the Altitude-on-Hypotenuse Theorems

b. Solve problems leading to linear and quadratic equations.

64 Simplifying Square Roots (Review)

a. Simplify square roots and their products and quotients.

b. Rationalize monomial denominators containing square roots.

65-66 The Pythagorean Theorem

a. Prove the Pythagorean Theorem.

b. Solve problems involving rhombuses, trapezoids, isosceles triangles.

c. Solve problems leading to quadratic equations.

d. Use Pythagorean triples to solve problems.

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Geometry Syllabus

TERM 2 – NOTE: solving systems of equations have been removed from the syllabus - this

is a topic is covered in algebra I and can be reviewed via numerical geometry problems.

Lesson # Topics and Objectives

1 UNIT I: RIGHT TRIANGLE TRIGONOMETRY

Ratios of sides for the Special Right Triangles:

300-600-900, 450-450-900

2-4 Trigonometric Ratios

a. Define the trigonometric ratios in a right triangle.

b. Use the relationship between the sine and cosine of complementary angles.

c. Find side lengths and angle measures in right triangles.

d. Solve angle of elevation/depression problems using trigonometry.

5 Evaluating Expressions Using Graphing Calculator

a. On the home screen, evaluate general expressions containing parentheses, powers,

radicals, and . Use Ans and 2nd ENTER (ENTRY).

b. Evaluate sine/cosine/tangent/sin-1/cos-1/tan-1

c. Define a function using Y = and use the table to evaluate the function for different

values of x. Use the auto and ask settings in the table setup.

6-7 UNIT II: AREA

Find the area of a triangle, parallelogram, rectangle, rhombus, square, and trapezoid, including

those containing a 30o, 450, or 600 angle. (partially review)

NOTE: “The ratio of the areas of two similar triangles is equal to the square of their ratio of

similitude “ – moved to dilation unit (must include all polygons!)

8-9 Special Area Formulas

a. Derive and apply the formula 2 3

4

sA for the area of an equilateral triangle

with side s.

b. Derive and apply the formula 1 2

1

2A d d for the area of a rhombus

with diagonals d1 and d2.

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10 UNIT III: COORDINATE GEOMETRY

The Coordinate Plane (Review)

a. Label axes and quadrants.

b. Define origin, ordered pair, x-coordinate (abscissa), y-coordinate (ordinate).

c. Use absolute value to find the lengths of segments parallel to the axes.

NOTE: graphing review have been moved to the lesson before equations for lines

11 Area in the Coordinate Plane

Find the area of a polygon in the coordinate plane, enclosing it in a rectangle if necessary.

12 The Distance Formula

a. Derive the distance formula.

b. Use the distance formula to find lengths of segments, show three points are collinear, solve

problems, and write informal proofs (e.g. prove a triangle is scalene, isosceles, equilateral,

or right; prove a quadrilateral is a parallelogram, rhombus, rectangle, or square).

13 The Midpoint Formula

a. Derive the midpoint formula.

b. Solve problems and write proofs (e.g. find the length of a median of a triangle, prove a

quadrilateral is a parallelogram).

14 The Slope Formula

a. Define slope.

b. Identify lines with positive, negative, zero, or undefined slopes.

c. Find slopes of parallel and perpendicular lines.

d. Solve problems and write proofs (e.g. show three points are collinear, prove a triangle is a

right triangle, prove a quadrilateral is a parallelogram, rhombus, rectangle, square, or

trapezoid).

15 Proving Theorems Using Variable Coordinates

16 Find the point on a directed line segment that partitions the given segment into a given ratio.

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17 Graphing Equations of Lines (Review)

a. Graph equations of lines by creating a table or by identifying x- and y-intercepts.

b. Graph an equation of a line using the slope and y-intercept.

18-19 The Slope-Intercept Form of the Equation of a Line (Review) and the Point-Slope Form of the

Equation of a Line

a. Given its equation, identify the slope and y-intercept of a line.

b. Given conditions, write an equation of the line satisfying those conditions.

c. Write an equation of the perpendicular bisector of a given segment, and of the medians and

altitudes of a given triangle.

20 The Center-Radius Form of the Equation of a Circle

a. Define circle and use the distance formula to derive the center-radius form of the equation

of a circle.

b. Given conditions, write an equation of the circle satisfying those conditions.

c. Find the equation of the circle by completing the square.

d. Graph a circle given its center and radius.

21 UNIT IV: CIRCLE GEOMETRY

Definitions Involving Circles

a. Define circle, radius, chord, diameter, arc, semi-circle, major arc, minor arc, midpoint of an

arc, central angle, degree measure of an arc, congruent circles, congruent arcs.

b. Prove arcs of a circle are congruent if and only if central angles which intercept them are

congruent if and only if chords which intercept them are congruent.

22 Theorems Involving Chords

a. A diameter perpendicular to a chord bisects the chord and its arcs.

b. The perpendicular bisector of a chord passes through the center of a circle.

c. In a circle, congruent chords are equidistant from the center.

d. Chords equidistant from the center of a circle are congruent.

23-24 Tangents and Tangent Circles

a. Define tangent line to a circle.

b. Through each point on a circle, there exists one and only one tangent line.

c. A line is perpendicular to a radius of a circle at its outer endpoint if and only if the line is

tangent to the circle.

d. Two tangent segments to a circle from the same external point are congruent.

e. Define line of centers, common internal/external tangent, internally /externally tangent

circles.

f. If two circles are tangent, their line of centers passes through their point of tangency and is

perpendicular to their common tangent.

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25-26 Inscribed Angles

Prove and apply:

a. The measure of an inscribed angle of a circle is one-half the measure of its intercepted arc.

b. In a given circle, inscribed angles that intercept the same or congruent arcs are congruent.

c. An angle inscribed in a semi-circle is a right angle.

d. If two chords are parallel, then the arcs between them are congruent.

e. If a quadrilateral is inscribed in a circle (ie: is cyclic), then each pair of opposite angles is

supplementary.

27 Tangent- Chord Angles

Prove and apply: The measure of an angle formed by a tangent and a chord is one-half the

measure of its intercepted arc.

28 Angles with Vertices Inside or Outside a Circle

Prove and apply:

a. The measure of an angle formed by two chords intersecting inside a circle is one-half the

sum of its intercepted arcs.

b. The measure of an angle formed by two secants, two tangents, or a secant and a tangent

intersecting outside a circle is one-half the difference of its intercepted arcs.

29 Solve Problems Involving Angles Related to a Circle

NOTE: Cyclic Quadrilateral Theorem moved to Inscribed Angle lesson

30 Two Chords Intersecting Inside a Circle

Prove and apply:

If two chords intersect inside a circle, the product of the lengths of the segments of one chord is

equal to the product of the lengths of the segments of the other.

31 Secants and Tangents Drawn to a Circle from an External Point

Prove and apply:

a. If two secant segments are drawn from an external point to a circle, then the product of the

lengths of one secant segment and its external part is equal to the product of the other

secant segment and its external part.

b. If a tangent segment and a secant segment are drawn from an external point to a circle,

then the square of the length of the tangent segment is equal to the product of the lengths

of the secant segment and its external part.

32 Solving Problems Involving Segments Related to a Circle

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33-34 Circumference and Area of a Circle

a. Justify circumference and area formulas using limit concepts.

b. Find arc length and area of a sector.

c. Define radian measure of an angle as the constant of proportionality between the length of

the arc intercepted by angle (drawn as a central angle in a circle) to the radius of the circle.

35-36 Concurrency Theorems

a. Prove:

1. The perpendicular bisectors of the sides of a triangle are concurrent at a point, called

the circumcenter of the triangle, that is equidistant from the vertices of the triangle.

2. The bisectors of the angles of a triangle are concurrent at a point, called the incenter of

the triangle, that is equidistant from the sides of the triangle.

3. The lines containing the altitudes of a triangle are concurrent at a point called the

orthocenter of the triangle.

4. The medians of a triangle are concurrent at a point, called the centroid of the triangle,

that is two-thirds of the way from any vertex of the triangle to the midpoint of the

opposite side. (Prove concurrency of the medians using coordinate geometry. The other

part of the theorem was already proved in lessons 58-59 of Term 1)

b. Apply theorems to problems requiring numeric or algebraic solutions.

c. Find the coordinates of the circumcenter, orthocenter, and centroid of a triangle if the

coordinates of its vertices are given.

NOTE: Construction unit now appears after solid geometry unit.

37 UNIT VI: TRANSFORMATIONS

Line Reflections

a. Define transformation, reflection in line , fixed point, image, preimage.

b. Use the notation ( ) 'r A A and : 'r A A

c. State coordinate rules for reflecting in the x-axis, y-axis, the line y = x and the line y = − x.

NOTE: Symmetry has been to a separate lesson towards the end of this unit.

38 Point Reflection

a. Define reflection in point P

b. Use the notation ( ) 'PR A A and : 'PR A A

c. State coordinate rule for reflecting through the origin.

39 Translations

a. Define translation.

b. Use the notation and rule , ( , ) ( , )a bT x y x a y b

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40 Rotations

a. Define rotation.

b. Use the notation , ( ) 'PR A A and , : 'PR A A

c. State coordinate rules for rotating 90o, 180o, and 270o counterclockwise about the origin.

41-42 Dilations

a. Define dilation.

b. Use the notation and rule for dilating with center the origin ( , ) ( , )kD x y kx ky

c. Prove and apply: The ratio of the areas of two similar polygons is equal to the square of

their ratio of similitude.

43 Compositions of Transformations

a. Use the notation for composition of functions.

b. Name a single transformation equivalent to a given composition of transformations.

c. Define and identify glide reflections.

44 Identify line, point, and rotational symmetries in figures, especially regular polygons.

45 Properties of Transformations

a. State the properties which remain invariant under reflections, translations, and rotations:

distance, area, angle measure, parallelism, collinearity, and midpoint.

b. Define isometry (rigid motion), direct isometry, and opposite isometry.

c. State that two polygons are congruent if and only if there exists an isometry that maps one

polygon to the other.

d. State that two polygons are similar if and only if there exists a similarity transformation

(i.e., a composition of an isometry and a dilation) that maps one polygon to the other.

46 Writing proofs using transformations

a. Identify a sequence of transformations that maps on polygon onto another.

b. Use transformations to prove two polygons congruent or similar.

c. Use transformations to prove two circle similar.

47 Transformations of Graphs

Graph the image of a line, circle, and polygon in the coordinate plane after a transformation.

NOTE’s:

Parabolas have been removed from this lesson. They are no longer studied in this geometry course.

Constructions of transformations has been moved to the construction unit.

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48 UNIT VII: SOLID GEOMETRY

Surface Area of Prisms and Regular Pyramids

a. Define prism, base, lateral edge, lateral face, lateral area, surface area, regular pyramid.

b. State that the lateral edges of a prism are congruent and parallel, the lateral edges of a

regular pyramid are congruent, and the lateral faces of a regular pyramid are isosceles

triangles.

c. Solve problems involving the surface areas of prisms and regular pyramids.

d. Use the Pythagorean Theorem to solve problems involving surface and space diagonals

49 Surface Area of Cylinders, Cones, and Spheres

a. Define cylinder, cone, sphere, great circle.

b. Apply the formula for the lateral surface area of a cylinder

50 Volumes of Prisms and Cylinders

a. Find volume of solids formed from prisms and cylinders.

b. Use cross sections of solids, dissection arguments,

Cavelieri’s principle, and informal limit arguments

51 Volumes of Pyramids and Cones

a. Find volume of solids formed from pyramids and cones.

b. Find the volume of a truncated pyramid and a truncated cone.

c. Use cross sections of solids, dissection arguments,

Cavelieri’s principle, and informal limit arguments

52 Volume of a Sphere

a. Find volume of a sphere (and a hemisphere!)

b. Identify that the intersection of a plane and a sphere is a circle.

c. Note that if two planes which intersect a sphere are equidistant from the center of the

sphere, then the intersection is two congruent circles.

53-54 Applications of Surface Area and Volume

a. Identify the shapes of planar cross-sections of three-dimensional objects.

b. Identify three-dimensional objects generated by rotations of two dimensional objects.

c. Express surface area or volume of a solid whose dimensions are given algebraic

expressions.

d. Solve problems leading to quadratic equations.

e. Apply concepts of density based on area and volume in modeling situations (persons

per square mile, BTUs per cubic foot).

f. Apply geometric methods to designing an object or a structure to satisfy constraints

such as area, volume, mass and cost.

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55-56 UNIT V: CONSTRUCTIONS

Basic Constructions

Construct and justify each construction:

a. a segment congruent to a given segment

b. an angle congruent to a given angle

c. the bisector of a given angle

d. a bisector of a given segment

e. a line perpendicular to a given line at a given point on the line

f. a line perpendicular to a given line from a given point not on the line

57-58 Applications of Basic Constructions

Construct and justify each construction:

a. Construct a line parallel to a given line through a given point not on the line.

b. Construct a triangle with given side lengths.

c. Construct an equilateral triangle with given side length.

d. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

e. Construct points of concurrency (centroid, circumcenter, incenter, and orthocenter).

f. Construct a circle circumscribed about a given triangle.

g. Construct a circle inscribed in a given triangle.

h. Construct a triangle that is congruent to a given triangle (via SSS or SAS or ASA).

i. Construct a triangle that is similar to a given triangle. (via AA or SAS Theorem for similarity).

Suggested:

j. Construct a line tangent to a given circle through a given point on or outside the circle.

k. Locate the center of a given circle.

59-60 Applications of Constructions to Transformations

a. Construct the line of reflection given a preimage and its image.

b. Construct the center of a point reflection or a rotation given a preimage and its image.

c. Construct the image of a point or line segment under the following transformations:

1. a line reflection

2. a translation

3. a point reflection/rotation of 180°

4. a rotation of 60°

5. a dilation