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Advanced Geometry
Basics of Geometry
Patterns and Inductive Reasoning – 8.A.4b, 9.C.4a, 9.C.4b, 9.C.4c
Find and describe pattern
Use inductive reasoning to make real-life conjectures
Points, Lines, and Planes – 9.B.4
Understand and use the basic undefined terms and defined terms of geometry
Sketch the intersections of lines and planes
Segment and their Measures – 7.A.4b, 9.B.4
Use segment postulates
Use the distance formula to measure distances
Angles and their Measures – 7.A.4b, 7.B.4, 9.B.4, 9.C.4b
Use angle postulates
Classify angles as acute, right, obtuse, or straight
Segment and Angle Bisectors – 7.A.4b, 9.B.4
Bisect a segment and an angle
Angle Pair Relationships – 7.A.4b, 9.B.4
Identify vertical angles and linear pairs
Identify complementary and supplementary angles
Introduction to Perimeter, Circumference, and Area – 7.A.4b, 9.B.4
Find the perimeter and area of common plane figures
Use a general problem-solving plan
Reasoning and Proof
Conditional Statements – 9.B.4, 9.C.4b, 9.C.4c
Recognize and analyze a conditional statement
Write postulates about points, lines, and planes using conditional statements
Definitions and Biconditional Statements – 9.B.4, 9.C.4c
Recognize and use definitions and biconditional statements
Deductive Reasoning – 8.A.4b, 9.C.4a, 9.C.4b
Use symbolic notation to represent logical statements
Form conclusions by applying the laws of logic of true statements
Reasoning with Properties from Algebra – 6.A.4, 8.A.4b, 9.C.4c
Use properties from algebra
Use properties of length and measure to justify segment and angle relationships
Proving Statements about Segments – 9.B.4, 9.C.4c
Justify statements about congruent segments
Write reasons for steps in a proof
Proving Statements about Angles –9.B.4, 9.C.4c
Use angle congruence properties
Prove properties about special pairs of angles
Perpendicular and Parallel Lines
Lines and Angles – 9.B.4, 9.C.4b
Identify relationships between lines
Identify angles formed by transversals
Proof and Perpendicular Lines – 9.B.4, 9.C.4c
Write different types of proofs
Prove results about perpendicular lines
Parallel Lines and Transversals – 9.B.4, 9.C.4c
Prove and use results about parallel lines and transversals
Use properties about parallel lines to solve real-life problems
Proving lines are parallel – 9.B.4, 9.C.4c
Prove that two lines are parallel
Use properties of parallel lines to solve real-life problems
Using Properties of Parallel Lines – 9.B.4, 9.C.4c
Use properties of parallel lines in real-life situations
Construct Parallel lines using straightedge and compass
Parallel Lines in the Coordinate Plane – 7.A.4b, 8.B.4a, 9.B.4
Find slopes of lines and use slope to identify parallel lines in a coordinate plane
Write equations of parallel lines in a coordinate plane
Perpendicular Lines in the Coordinate Plane – 7.A.4b, 8.B.4a, 9.B.4
Use slope to identify perpendicular lines in a coordinate plane
Write equations of perpendicular lines
Congruent Triangles
Triangles and Angles – 9.B.4, 9.C.4c
Classify triangles by their sides and angles
Find angle measures in triangles
Congruence and Triangles – 9.B.4, 9.C.4c
Identify congruent figures and corresponding parts
Prove that two triangles are congruent
Proving Triangles are Congruent: SSS and SAS – 7.A.4b, 9.B.4, 9.C.4c
Prove that triangles are congruent using the SSS and SAS Congruence Postulates
Use congruence postulates in real-life problems, such as bracing a structure
Proving Triangles are Congruent: ASA and AAS – 9.B.4, 9.C.4c
Prove that triangles are congruent using the ASA Congruence Postulate, and the
AAS Congruence Theorem
Use congruence postulates and theorems in real-life problems
Using Congruent Triangles –9.B.4, 9.C.4c
Use congruent triangles to plan and write proofs
Use congruent triangles to prove constructions are valid
Isosceles, Equilateral, and Right Triangles – 9.B.4, 9.C.4c
Use properties of isosceles and equilateral triangles
Use properties of right triangles
Triangles and Coordinate Proof – 7.A.4b, 9.B.4, 9.C.4c
Place geometric figures in a coordinate plane
Write a coordinate proof
Properties of Triangles
Perpendicular and Bisectors – 9.B.4, 9.C.4c
Use properties of perpendicular bisector
Use properties of angle bisectors to identify equal distances
Bisectors of a Triangle – 9.B.4, 9.C.4c
Use properties of perpendicular bisectors of a triangle
Use properties of angles bisectors of a triangle
Medians and Altitudes of a Triangle – 7.A.4b, 9.B.4
Use properties of medians of a triangle
Use properties of altitudes of a triangle
Midsegment Theorem – 7.A.4b, 9.B.4, 9.C.4c
Identify the midsegments of a triangle
Use properties of midsegments of a triangle
Inequalities in One Triangle – 9.B.4, 9.C.4c
Use triangle measurements to decide which side is longest or which angle is
largest
Use the Triangle Inequality
Indirect Proof and Inequalities in Two Triangles – 9.B.4, 9.C.4c
Read and write an indirect proof
Use the Hinge Theorem and its converse to compare side lengths and angle
measures
Quadrilaterals
Polygons – 9.B.4, 9.C.4b
Identify, name, and describe polygons
Use the sum of the measures of the interior angles of a quadrilateral
Properties of Parallelograms – 9.B.4, 9.C.4c
Use some properties of parallelograms
Use properties of parallelograms in real-life situations
Proving Quadrilaterals are Parallelograms – 7.A.4b, 9.B.4, 9.C.4c
Prove that a quadrilateral is a parallelogram
Use coordinate geometry with parallelograms
Rhombuses, Rectangles, and Squares – 7.A.4b, 9.B.4, 9.C.4c
Use properties of sides and angles of rhombuses, rectangles, and squares
Use properties of diagonals of rhombuses, rectangles, and squares
Trapezoids and Kites – 7.A.4b, 9.B.4, 9.C.4c
Use properties of trapezoids
Use properties of kites
Special Quadrilaterals – 7.A.4b, 9.B.4, 9.C.4c
Identify special quadrilaterals based on limited information
Prove that a quadrilateral is a special type of quadrilateral, such as a rhombus or
trapezoid
Areas of Triangles and Quadrilaterals – 7.A.4b, 9.B.4, 9.C.4c
Find the areas of squares, rectangles, parallelograms, trapezoids, kites,
rhombuses, and triangles
Transformations
Rigid Motion in a Plane – 7.A.4b, 9.B.4
Identify the three basic rigid transformations
Use transformations in real-life situations
Reflections – 9.A.4b, 9.B.4, 9.C.4c
Identify and use reflections in a plane
Identify relationships between reflections and line symmetry
Rotations – 7.B.4, 9.A.4b, 9.B.4, 9.C.4c
Identify rotations in a plane
Use rotational symmetry in real-life situations
Translations and Vectors – 7.A.4b, 8.C.4b, 9.A.4b, 9.B.4
Identify and use translations in a plane
Use vectors in real-life situations
Glide Reflections and Compositions – 9.A.4b, 9.B.4
Identify glide reflections in a plane
Represent transformations as compositions of simpler transformations
Frieze Patterns – 9.A.4b, 9.B.4, 9.C.4b
Use transformations to classify frieze patterns
Use frieze patterns to design border patterns in real-life situations
Similarity
Ratio and Proportion – 6.D.4, 7.C.4a, 9.B.4
Find and simplify the ratio of two numbers
Use proportions to solve real-life problems
Problem Solving in Geometry with Proportions – 6.D.4, 7.C.4a, 7.C.4b, 9.B.4
Use properties of proportions
Similar Polygons – 6.D.4, 7.C.4a, 9.B.4
Identify similar polygons
Use similar polygons to solve real-life problems
Similar Triangles – 6.D.4, 7.C.4a, 9.B.4, 9.C.4c
Identify similar triangles
Use similar triangles in real-life situations
Proving Triangles are Similar – 6.D.4, 7.C.4a, 9.B.4, 9.C.4c
Use similarity theorems to prove that two triangles are similar
Use similar triangles to solve real-life problems
Proportions and Similar Triangles – 6.D.4, 7.C.4a, 9.B.4, 9.C.4c
Use proportionality theorems to calculate segment lengths
Dilations – 6.D.4, 7.A.4a, 7.C.4a, 9.A.4b, 9.B.4
Identify dilations and use properties of dilations to create a real-life perspective
drawing
Right Triangles and Trigonometry
Similar Right Triangles – 6.D.4, 7.C.4a, 9.B.4, 9.C.4c
Solve problems involving similar triangles formed by the altitude drawn to the
hypotenuse of a right triangle
Use geometric mean to solve problems
The Pythagorean Theorem – 7.A.4b, 7.C.4a, 9.B.4, 9.C.4c
Prove the Pythagorean Theorem
Use the Pythagorean Theorem to solve real-life problems
The Converse of the Pythagorean Theorem – 9.B.4, 9.C.4c
Use the converse of the Pythagorean Theorem to solve problems
Use side lengths to classify triangles by their angle measure
Special Right Triangles – 7.A.4b, 9.B.4
Find the side lengths of special right triangles
Trigonometric Ratios – 6.C.4, 7.A.4b, 7.C.4a, 9.B.4, 9.D.4
Find the sine, cosine, and the tangent of an acute angle
Use trigonometric ratios to solve real-life problems
Solving Right Triangles – 6.C.4, 9.B.4, 9.D.4
Solve a right triangle
Use right triangles to solve real-life problems
Vectors – 6.C.4, 7.A.4b, 8.C.4b, 9.D.4
Find the magnitude and the direction of a vector
Add vectors
Circles
Tangents to Circles – 9.B.4, 9.C.4c
Identify segments and lines related to circles
Use properties of a tangent circle
Arcs and Cords – 9.B.4, 9.C.4c
Use properties of arcs of circles
Use properties of chords of circles
Inscribed Angles – 9.B.4, 9.C.4c
Use inscribed angles to solve problems
Use properties of inscribed polygons
Other Angle Relationships in Circles – 9.B.4, 9.C.4c
Use angles formed by tangents and chords to solve problems in geometry
Use angles formed by lines that intersect a circle to solve problems
Segment Lengths in Circles – 9.B.4, 9.C.4c
Find the lengths of segments of chords, tangents, and secants
Equations of Circles – 9.B.4
Write the equation of a circle
Use the equation of a circle and its graph to solve problems
Locus – 9.B.4, 9.C.4b
Draw the locus of points that satisfy given conditions
Area of Polygons and Circles
Angle Measures in Polygons – 9.B.4, 9.C.4c
Find the measures of interior and exterior angles of polygons
Use measures of angles of polygons
Areas of Regular Polygons – 7.A.4b, 9.A.4b, 9.B.4, 9.C.4c
Find the area of an equilateral triangle
Find the area of a regular polygon
Perimeters and Area of Similar Figures – 6.D.4, 7.A.4b, 9.B.4
Compare perimeters and areas of similar figures
Use perimeters and areas of similar figures to solve real-life problems
Circumference and Arc Length – 7.A.4b, 9.B.4
Find the circumference of a circle and the length of a circular arc
Use circumference and arc length to solve real-life problems
Areas of Circles and Sectors – 7.A.4b, 9.B.4
Find the area of a circle and a sector of a circle
Geometric Probability – 9.B.4, 10.C.4a
Find a geometric probability
Use geometric probability to solve real-life problems
Surface Area and Volume (If Time Allows)
Exploring Solids – 9.B.4
Use properties of polyhedra
Use Euler’s Theorem in real-life situations
Surface Area of Prisms and Cylinders – 7.A.4b, 9.A.4a
Find the surface area of a prism
Find the surface area of a cylinder
Surface Area of Pyramids and Cones – 7.A.4b, 9.A.4a
Find the surface area of a pyramid
Find the surface area of a cone
Volume of Prisms and Cylinders –7.A.4b, 9.B.4
Use volume postulates
Find the volume of prisms and cylinders in real-life situations
Volume of Pyramids and Cones – 7.A.4b, 9.A.4a, 9.B.4
Find the volume of pyramids and cones
Surface Area and Volume of Spheres –7.A.4b, 9.B.4
Find the surface area of a sphere
Similar Solids – 6.D.4, 7.A.4b, 9.B.4
Find and use the scale factor of similar solids