Geometry

COURSE OVERVIEW

Geometry focuses on the development of abstract thinking, critical thinking and reasoning of geometric concepts.  The course will emphasize skill development, practical applications to the real world, and contain the major Geometric concepts that students need to be successful in Geometry. Areas of concentration include, but are not limited to the following: basics of geometry, reasoning and proof, parallel and perpendicular lines, transformations, congruency, similarity, quadrilaterals and other polygons, circles, surface area and volume, and right triangle trigonometry.

• Identify types of points, lines, planes, rays, and segments.

• Recognize parallel and perpendicular lines and their properties.

• Find lengths and measurements of angles and segments based on properties of midpoints and bisectors.

• Apply and use the distance and midpoint formulas.

• Find perimeter and areas of rectangles, squares, circles, and irregular shapes.

• Classify quadrilaterals by their properties.

• Use and apply the properties of parallelograms.

• Place quadrilaterals in the coordinate plane and use standard position to name coordinates.

• Solve a system of linear equations in applications to quadrilaterals.

• Use coordinate geometry to prove properties of quadrilaterals.

• Use the Pythagorean Theorem, its converse, and determine a set of Pythagorean triples.
• Simplify square roots and rationalize denominators.

• Use and apply Pythagorean Theorem to real-world scenarios.

• Evaluate Pythagorean Theorem answers in leave answers simplest radical form.

• Solve and apply special right triangles.

• Identify congruent and similar figures, and apply properties of each.

• Prove triangles congruent in accordance to the congruence theorems.

• Find the midpoint of a line segment in coordinate geometry.

• Find the slope of a line and use geometric properties to find parallel lines and perpendicular lines in the coordinate plane.

• Find the equation of a line based on geometry properties that are given.

• Identify angles formed by lines and a transversal.

• Apply angle properties to solve when given angle pairs and a transversal.

• Prove two lines are parallel using a two column proof method and special angle pairs.

• Identify triangles by their angle measures and their side lengths.

• Identify the properties of isosceles and right triangles.

• Find the surface area and volume of 3D shapes.

• Utilize nets to identify 3D shapes.

• Identify, name, and classify polygons.

• Find the measurements of the angles of polygons.

• Measure the interior and exterior angles of a polygon.

• Recognize and implement the properties of all quadrilaterals.

• Write and simplify the ratio of two numbers.

• Solve problems using a proportion.

• Identify and use the properties of similar polygons.

• Prove that two triangles are similar.

• Use proportionality theorems to set up and solve problems.
• Prove that two triangles are congruent.

• Apply and use the Pythagorean Theorem and its converse.

• Find the lengths of the sides of 45 deg.-45 deg.-90 deg. and 30 deg.-60 deg.-90 deg. triangles.

• Find a side or angle of a right triangle by using the sine, cosine, or tangent ratio.

• Solve real world problems using angles of elevation and depression.

• Use the relationship between the radius and a tangent line, and the relationship between two tangents and one point.

• Apply and use properties of congruent chords, arcs, and central angles, inscribed angles, and inscribed quadrilaterals.

• Find arc length, area of a sector, and arc measure of a circle.

Geometry

CC.2.3.HS.A.1: Use geometric figures and their properties to represent transformations in the plane.

CC.2.3.HS.A.2: Apply rigid transformations to determine and explain congruence.

CC.2.3.HS.A.3: Verify and apply geometric theorems as they relate to geometric figures.

CC.2.3.HS.A.5: Create justifications based on transformations to establish similarity of plane figures.

CC.2.3.HS.A.6: Verify and apply theorems involving similarity as they relate to plane figures.

CC.2.3.HS.A.7: Apply trigonometric ratios to solve problems involving right triangles.

CC.2.3.HS.A.8: Apply geometric theorems to verify properties of circles.

CC.2.3.HS.A.9: Extend the concept of similarity to determine arc lengths and areas of sectors of circles.

CC.2.3.HS.A.10: Translate between the geometric description and the equation for a conic section.

CC.2.3.HS.A.11: Apply coordinate geometry to prove simple geometric theorems algebraically.

CC.2.3.HS.A.12: Explain volume formulas and use them to solve problems.

CC.2.3.HS.A.13: Analyze relationships between two-dimensional and three-dimensional objects.

CC.2.3.HS.A.14: Apply geometric concepts to model and solve real world problems.

Numbers and Operations

CC.2.1.HS.F.3: Apply quantitative reasoning to choose and interpret units and scales in formulas, graphs, and data displays.

CC.2.1.HS.F.4: Use units as a way to understand problems and to guide the solution of multi-step problems.

CC.2.1.HS.F.5: Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

Measurement, Data, and Probability

CC.2.4.HS.B.5: Make inferences and justify conclusions based on sample surveys, experiments, and observational studies.

Congruency

CCSS.Math.Content.HSG.CO.A.1: Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

CCSS.Math.Content.HSG.CO.A.4: Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

CCSS.Math.Content.HSG.CO.B.7: Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

CCSS.Math.Content.HSG.CO.B.8: Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

CCSS.Math.Content.HSG.CO.C.9: Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.

CCSS.Math.Content.HSG.CO.C.10: Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

CCSS.Math.Content.HSG.CO.C.11: Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

Similarity, Right Triangles, & Trigonometry

CCSS.Math.Content.HSG.SRT.A.2: Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

CCSS.Math.Content.HSG.SRT.A.3: Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

CCSS.Math.Content.HSG.SRT.B.4: Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.

CCSS.Math.Content.HSG.SRT.B.5: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

CCSS.Math.Content.HSG.SRT.C.6: Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

CCSS.Math.Content.HSG.SRT.C.8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

Circles

CCSS.Math.Content.HSG.C.A.1: Prove that all circles are similar.

CCSS.Math.Content.HSG.C.A.2: Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

CCSS.Math.Content.HSG.C.A.3: Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

CCSS.Math.Content.HSG.C.A.4: Construct a tangent line from a point outside a given circle to the circle.

Expressing Geometric Principles with Equations

CCSS.Math.Content.HSG.GPE.B.6: Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

CCSS.Math.Content.HSG.GPE.B.7: Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

Geometric Measurement & Dimension

CCSS.Math.Content.HSG.GMD.A.1: Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone.

CCSS.Math.Content.HSG.GMD.A.3: Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

Modeling with Geometry

CCSS.Math.Content.HSG.MG.A.1: Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).

Unit:  Tools of Geometry

Lessons:

• Introduction to points, lines and planes
• Measuring and constructing segments
• Midpoint and distance formulas
• Using angle bisectors and segment bisectors
• Proving statements about segments and angles
• Finding perimeter and area of regular and irregular polygons

Unit:  Quadrilaterals and Other Polygons        

Lessons:

• Angles of Polygons and the polygon sum theorem
• Quadrilateral Mapping
• Properties of parallelograms
• Proving that a quadrilateral is a parallelogram
• Properties of special parallelograms
• Properties of Trapezoids and Kites
• Placing quadrilaterals in the coordinate plane based on given properties

Unit:  Relationships within Triangles

Lessons:

• Perpendicular and Angle bisectors
• Bisectors of Triangles
• Medians and Altitudes of Triangles
• Inequalities in two triangles

Unit:  Surface Area and Volume

Lessons:

• Three dimensional figures and nets
• Surface area of prisms and cylinders
• Surface area of cones and pyramids
• Volumes of prisms and cylinders
• Volumes of cones and pyramids
• Surface area and volume of spheres

Unit:  Circles and Their Properties

Lessons:

• Lines and segments that intersect circles
• Finding arc lengths and measures
• Finding area of sectors of circles
• Using chords and their properties
• Inscribed angles and polygons in circles
• Angle relationships in circles
• Segment relationships in circles
• Circles in the coordinate plane

Unit:  Angle Pairs, Transversals and Parallel Lines

Lessons:

• Pairs of lines and angles
• Parallel lines and their transversals
• Proofs with parallel lines
• Proofs with perpendicular lines
• Equations of parallel and perpendicular lines

Unit:  Congruence

Lessons:

• Angles in Triangles
• Congruent Polygons
• Proving Triangle Congruence by SAS and SSS
• Equilateral and Isosceles Triangles
• Proving Triangle Congruence by ASA and AAS
• Using Congruent Triangles
• Coordinate Proofs

Unit:  Similarity

Lessons:

• Similar Polygons
• Proving Triangle Similarity by AA
• Proportionality and ratios
• Scale Models and applications of similarity

Unit:  Right Triangle Trig and The Pythagorean Theorem

Lessons:

• The Pythagorean Theorem
• Converse of Pythagorean Theorem
• Special Right Triangles
• Introduction to right triangle trig
• Use the tangent ratios to solve for missing sides
• Use the sine and cosine ratios to solve for missing sides
• Use the inverse of sine, cosine, and tangent to solve for angles
• Solve applications of right triangle trig to find angles of elevation and depression