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Geometry Learning Outcomes (Residue)
Unit 1: Introducing Geometry
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Understanding geometric definitions is fundamental to sharing a common language of Geometry.
Points, lines and planes are undefined terms in geometry and are used in defining most other geometric
figures.
An angle is formed by two rays with a common endpoint which is called a vertex. An angle divides a plane
in to two regions; the interior and exterior of the angle.
Polygons are closed planar figures formed by connecting line segments endpoint to endpoint and they are
classified by the number of sides.
A circle is a set of points that lie in a plane that are all equidistant from a given point.
Unit 2: Reasoning in Geometry
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Inductive reasoning involves making conjectures that generalize patterns observed in data. (General to
specific)
Deductive reasoning is the process of showing that certain statements follow logically from agreed upon
assumptions and proven facts. (Specific to general)
Special angles are formed when two or more lines are crossed by a transversal.
The definitions of supplementary angles and adjacent angles can be used to prove the Linear Pair theorem
and the Vertical Angle theorem.
When the lines are parallel and a transversal crosses them, the following can be said: corresponding angles
are congruent, alternate interior angles are congruent, alternate exterior angles are congruent, same side
interior angles are supplementary and same side exterior angles are supplementary.
Unit 3: Using tools of Geometry
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Formal geometric constructions can be completed with a variety of tools and methods including compasses,
rulers, straightedges, protractors, paper and pencil, patty paper, reflective devices and dynamic geometric
software.
The term perpendicular describes two lines, segments, rays, etc. that meet at a 90 degree angle. The term
bisect means to cut in half.
The shortest distance from a given point to a given line will be the perpendicular segment that connects the
point to the line.
The slopes of parallel lines are equal, the slope of perpendicular lines are negative reciprocals of each other.
The points of concurrency in a triangle relate the perpendicular bisectors to the circumcenter and the angel
bisectors to the incenter.
Unit 4: Discovering and Proving Triangle Properties
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The sum of the three angles of any triangle is 180 degrees.
To prove that two triangles are congruent, each set of corresponding sides and each set of corresponding
angles must be proven congruent.
Triangle congruence conjectures can be used as a shortcut to prove whether or not two triangles are
congruent. (SSS, SAS, ASA, AAS, HL)
Once two triangles have been proven congruent, you can then assume that all corresponding parts of those
two triangles are also congruent.
A flowchart proof is used to demonstrate relationships among the ideas in a proof. It can be written
vertically or horizontally.
Unit 5: Discovering and Proving Polygon Properties/ Transformations
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The polygon-sum conjecture (n – 2)180 can be used to find the sum of the angles of any polygon.
The sum of the exterior angles of any polygon is 360 degrees.
The diagonal properties of a quadrilateral can be used to identify the specific quadrilateral type.
Quadrilaterals can be organized into a hierarchy of properties.
A midsegment of a trapezoid is parallel to the bases and its length is the average of the lengths of the bases.
A midsegment of a triangle is parallel to its base and is half the length of the base.
A transformation is a movement of a geometric figure.
Rigid transformations (isometries) do not change the shape or size of a figure, only its location.
The line of reflection is the perpendicular bisector of all segments joining points on the original figure with
corresponding points on the image.
Translations occur when a figure is reflected over parallel lines.
Rotations occur when a figure is reflected over intersecting lines.
Unit 6: Discovering and Proving Circle Properties
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Every line tangent to a circle is perpendicular to the radius at the point of tangency.
Two tangent segments from the same point outside of a circle are congruent.
Congruent chords determine congruent central angles, intercept congruent arcs and are equidistant from the
center.
A central angle is equal to the measure of its intercepted arc. An inscribed angle is equal to half of its
intercepted arc.
Pi is the ratio of the circumference to the diameter.
Arc length is measured in distance units and is found by circumference by the central angle divided by 360
degrees.
Unit 7: Area and Surface Area
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Perimeter is distance around a polygon.
Area of a plane figure is a measure of the space contained within a given perimeter. It is measured in square
units.
The area of a regular polygon can be found using perimeter and the apothem.
The circumference of a circle is the measure of the distance around the circle and the area is the flat surface
the circle covers.
A sector of a circle is the region between two radii and an arc of the circle.
Surface area is the sum of all surfaces of a solid.
Unit 8: The Pythagorean Theorem and Trigonometry
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The Pythagorean Theorem is a claim about areas of squares built on the sides of a right triangle. The area of
the square on the hypotenuse is the sum of the areas of the squares on the legs.
The Pythagorean Theorem can be used to find the length of any one of the missing sides of a right triangle
given the lengths of the other two.
Special right triangle ratios can be used to find the exact trigonometric ratios for 30, 45 and 60 degree
angles.
The distance formula can be found using the Pythagorean Theorem. The distance between any two points
can be found using the distance formula.
Trigonometric ratios can be used to find the missing lengths or angles of right triangles.
Sine = opposite over hypotenuse, Cosine = adjacent over hypotenuse and Tangent = opposite over adjacent.
Unit 9: Volume
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Volume is the amount of space contained within a three dimensional shape. Volume is measured in cubic
units, even when that shape has rounded elements.
Polyhedrons are made up of flat polygonal faces.
The height of a prism or cylinder is the perpendicular distance between the two bases. The height of a
pyramid or cone is the perpendicular distance between the base and the vertex.
The volume of an irregularly shaped object is measured by the amount of liquid it displaces.
Volumes can be used to find the density of an object and thus determine which materials they are made of.
Unit 10: Similarity
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Similar polygons (noted using the ~ symbol) are dilations of one another. All corresponding angles are
congruent and the lengths of corresponding sides are proportional.
Similarity shortcuts, like congruence shortcuts, refer to the congruence of angles, but they refer to the
proportionality rather than the congruence of the sides. (AA, SSS, SAS)
Similar triangles can be used to measure heights indirectly by using proportions.
If two figures are similar, then the ratio of any corresponding two-dimensional parts (area) is the square of
the scale factor.
If two figures are similar, then the ratio of their volumes is the cube of the scale factor.
Unit 11: Geometry as a Mathematical System (Proofs)
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Deductive reasoning systems help organize mathematical ideas so that circular reasoning can be avoided.
There are different formats for formal proofs. (Two-column, paragraph and flow-chart proofs)
The mathematical system of geometry is based on the assumptions of Algebraic and Arithmetic properties
and the postulates of geometry.
When writing a proof you should follow a logical set of steps from beginning with a conditional statement
to drawing an labeling any relevant pictures to proving what has been asked.