Standardized Tests

SAT/ACT Standardized Test Prep

Intro To Geometry

Points, Lines, Rays, Segments, and Planes

Segment Addition Postulate

Arc Addition Postulate

Line Symmetry and Rotational Symmetry

Angles

Angle Addition Postulate

Adjacent Angles

Complementary and Supplementary Angles

Classifying Angles by Size

Vertical Angles

Classifying Pairs of Angles

Area and Perimeter

Area and Perimeter- Rectangles

Compound Figures- Area and Perimeter

Area and Perimeter- Triangles

The number pi

Area and Circumference- Circle

Area of Trapezoids

Trapezoid Midsegment (or Median)

Polygons

Diagonals of Polygons

Angles of Polygons

Area of Regular Polygons (Apothem)

Density

Population Density

Midpoint Formula

Distance Formula

Perimeter and Area in the Coordinate Plane

Logic

Conditional Statements
(Inverse, Converse, Contrapositive)

Intersections and Unions of Sets

Law of Syllogism

Law of Detachment

Lines and Graphs

Slope Formula

Horizontal Lines

Vertical Lines

Parallel and Perpendicular Slope

Equation of Perpendicular Bisector

Distance between a point and a line

Collinear Points

Proofs- Parallel Lines and Transversals

Transversals

Corresponding Angles

Alternate Interior Angles

Same-Side Interior Angles

Alternate Exterior Angles

Same Side Exterior Angles

Proving Parallel Lines Given Angles

Proving Angles Given Parallel Lines

Transformations

Transformations- Translations

Transformations- Reflections

Transformations- Rotations

Proofs- Triangle Congruence

Intro to Proofs

Proving Triangle Congruence SAS

Proving Triangle Congruence SSS

Proving Triangle Congruence ASA

Proving Triangle Congruence AAS

Proving Triangle Congruence CPCTC

Triangles

Triangle Classification

Equilateral Triangles

Pythagorean Theorem

Pythagorean Triples

Pythagorean Inequalities

Triangle Angle Sum Theorem

Side-Splitter Theorem

Parallel Lines Proportionality Theorem

Triangle Proportionality Theorem

Triangle Midsegment Theorem

Exterior Angle Theorem

Isosceles Triangles

Special Right Triangles 30-60-90 45-45-90

Triangle Inequality Theorem

Hinge Theorem

Triangle Longer Side Theorem

Triangle Larger Angle Theorem

Circumcenter, Incenter, Centroid, and Orthocenter

Circumcenter

Incenter

Centroid

Orthocenter

Quadrilaterals

Properties of Quadrilaterals

Properties of a Rhombus

Kites- Area and Properties

Proving Parallelograms

Trigonometry

Right Triangle Trigonometry

Word Problems- Angle of Depression and Elevation (Trigonometry)

Law of Sines

Area of triangle SAS formula

Law of Cosines

Heron’s Formula- Area of Triangles

Area of Oblique Triangles

Geometric Mean

Similar Right Triangles

Circles

Concentric Circles

Area of a Sector

Arc Length

Notes- Segments and Angles in Circles

Angles in Circles

Volume and Surface Area

Intro to Volume

Nets of Polyhedra

Volume Rectangular Prisms

Surface Area Rectangular Prisms

Volume and Surface Area of Cubes

Triangular Prisms- Surface Area

Cylinders Volume and Surface Area

Cones Volume and Surface Area

Pyramids Volume and Surface Area

Spheres Volume and Surface Area

Scale Factor

Perimeters of Similar Polygons

Areas of Similar Polygons

Similar Figures
(Simularity ratio, Area ratio, Volume ratio)

Distance Formula 3D

Diagonals of Rectangular Prisms and Cubes

Polyhedrons- Faces, Edges, and Vertices (Euler’s Formula)

Challenge Problems

Geometry Challenge Problems

In Summary

Geometry is a fundamental subject that is studied in high school and is a building block for many other mathematical concepts. Students learn about the different types of geometric shapes, such as circles, triangles, quadrilaterals, and their properties, and more advanced topics, such as congruence and similarity. These concepts are used to solve problems involving geometric shapes and are crucial for understanding more advanced mathematical concepts.

One important aspect of geometry is the use of deductive reasoning, which involves making logical conclusions based on given information. Students learn how to use this method of problem-solving to prove theorems and solve geometric problems.

Specific topics learned

Angles: Angles are formed when two lines intersect at a point. An angle is measured in degrees and is used to describe the amount of rotation around a point. Individual angles are either acute, obtuse, right, straight or reflex. Pairs of angles can be congruent (same size), complementary (sum to 90 degrees), supplementary (sum to 180 degrees), or none of these. More specific relationships of pairs of angles are vertical angles, linear pair, corresponding, alternate interior, alternate exterior, same-side interior, and many more.

Area and Perimeter: Area is a measure of the size of a two-dimensional shape and is typically expressed in square units, such as square inches or square feet. To calculate the area of a shape, students use formulas that depend on the type of shape being measured. Perimeter is a measure of the distance around the outside of a two-dimensional shape and is also typically expressed in linear units, such as inches or feet. To calculate the perimeter of a shape, students add up the lengths of all the sides of the shape.

Logic: Students learn about logic, which is the study of reasoning and argument. Logic is an important part of geometry because it is used to prove theorems and solve problems involving geometric shapes. Logic is introduced with conditional (if-then) statements and then law of syllogism and law of detachment.

Graphs of Linear Equations: Students learn how to create and interpret linear graphs.

Proofs: A large part of geometry courses is proofs. Students go through proofs with parallel lines and transversals, perpendicular lines, triangle congruence, triangle similarity, and more.

Transformations: 4 types of transformations are gone over in a geometry course. Translations, Reflections, Rotations, and dilations. Translations are movements of a figure that involve sliding the figure without rotating or flipping it. Reflections are movements of a figure that involve flipping the figure across a line. Rotations are movements of a figure that involve turning the figure around a fixed point. Dilations are changes in size of a figure using a scale factor.

Triangles: Triangles are studied in depth. Triangle Classification, Pythagorean Theorem, Triangle Angle Sum Theorem, Side-Splitter Theorem,
Triangle Midsegment Theorem, Exterior Angle Theorem, base angles theorem, special right triangles, triangle inequality theorem, hinge theorem, Circumcenter, Incenter, Centroid, and Orthocenter are just a few of the many in depth topics studied about triangles.

Quadrilaterals: Just like triangles, quadrilaterals are studied in depth as well. Properties of parallelograms, kites, trapezoids and all other quadrilaterals are gone over.

Trigonometry: Trigonometry is another detailed analysis of triangles and the relationships between their sides and angles. The three main trigonometric functions are sine (sin), cosine (cos), and tangent (tan), which are defined in terms of the ratios of the lengths of the sides of a right triangle. Students also learn about the inverse trigonometric functions, which are used to solve problems involving angles in right triangles. The inverse trigonometric functions are denoted with the prefix “arc,” such as arc sine (arcsin), arc cosine (arccos), and arc tangent (arctan). In addition to these concepts, students also learn how to use trigonometry to solve real-world problems involving lengths, distances, and angles, such as finding the height of a building or the distance across a river.

Circles: Circles are also studied in depth. Central, inscribed and exterior angles in triangles are analyzed in depth. Also relationships with chords inside of circles and secant and tangent lines outside of circles.

Volume and Surface Area: Students learn about volume and surface area, which are used to measure and calculate the size of three-dimensional objects such as prisms, pyramids, cylinders, cones, spheres and more. Volume is a measure of the amount of space occupied by an object and is typically expressed in cubic units, such as cubic inches or cubic feet. To calculate the volume of a three-dimensional object, students use formulas that depend on the shape of the object. Surface area is a measure of the total area of the surface of a three-dimensional object and is also typically expressed in square units. To calculate the surface area of a three-dimensional object, students add up the areas of all the faces of the object.

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