Congruence

Basic Geometry Definitions- HSG-CO.A.1

• Parallel and Perpendicular lines - Remember your lines by thinking of Railroad tracks.
• Right Angles - A right angle forms when two perpendicular lines meet.



Geometric Transformations within a Plane- HSG-CO.A.2

• Rotation - Think of a Ferris wheel. It rotates around a fixed point in the center. The cars change positions as it rotates.
• Translations - The object stays the same, it just moves to a different location.



Graphing Complex Transformations - HSG-CO.A.4



• Complex Transformations - Complex Transformations are nothing more than addition of positive and negative numbers.
• Rotating a Figure - Negate the x value by switching the x value signs (make the positive numbers negative and the negative numbers positive).



Rotations, Reflections, and Translations of Geometric Shapes- HSG-CO.A.4



• Glide Reflection - A Glide reflection is when a Translation and a Reflection both occur.
• Reflection - A reflection is a mirror image of the object.



Drawing Transformed Figures- HSG-CO.A.5



• Rotating a Geometric Object - Common formulas you can use to find where to place the rotated object.
• Transformations - The shape moves around the Center of Rotation.



Rigid Motions to Transform Figures- HSG-CO.B.6



• Rigid Motions - You can rotate the shape. To find how many degrees to rotate the object follow these steps.
• Rigid Motions and Symmetry - If the shape is symmetrical you can reflect the shape onto itself across the line of symmetry.



Rigid Motions and Congruent Triangles- HSG-CO.B.7



• Coordinate Notation - The Mathematical way to describe Transformations.
• Congruence - What makes something congruent?



Proving Triangle Congruence- HSG-CO.B.8



• Proving Congruence - These triangles are congruent because they have equal side, angle, side.
• Congruent Triangles - SSS- Side, Side, Side and SAS- Side, Angle, Side and ASA- Angle, Side, Angle.



Geometric Proofs On Lines and Angles- HSG-CO.C.9



• Coordinate Geometry Proofs - Remember to draw and label a graph.
• Angles - Two angles that share the same vortex or corner point.

Triangle Proofs - HSG-CO.C.10



• Proofs - Lines that are perpendicular form right angles (90°) to each other.
• Proofs and Properties - If two sides of a triangle are congruent, the angles opposite these sides are congruent.



Proving Theorems of Parallelograms- HSG-CO.C.11



• Parallelograms - Opposite Angles of a Parallelogram are Congruent.
• Proofs for Parallelograms - Opposite angles of a parallelogram are congruent. Consecutive angles are supplementary.



Making Bisectors of Angles and Lines - HSG-CO.D.12



• Making Bisectors - Place compass on the vertex (V) and draw a line across the each leg of the angle at any point.
• Making Perpendicular Lines - Place your compass on point B and using the same distance make an arc above and below the line.



Making Perpendicular and Parallel Lines- SG-CO.D.12



• Draw Perpendicular Lines - Place your compass on point K and then make marks a little way down (P and Q).
• Draw Parallel Lines - Adjust your compass so it's the same length as your lower arc.



Inscribing Shapes in Circles- HSG-CO.D.13



• Inscribe a Triangle - Find the incenter of the triangle: formed by taking the intersection of the angle bisectors of the three vertices of the triangle.
• Construct a Hexagon - Move your compass to the arc you just drew and draw another. Repeat until you have 6 vertices.

Similarity, Right Triangles, & Trigonometry

Dilations and Parallel Lines - HSG-SRT.A.1a

• Denominators Can't Be Zero - The point on the line's graph that the line intersects the Y‐axis.
• Common Integer Factors - Making Parallel lines using slope intercept form.



Dilations and Scale Factors- HSG-SRT.A.1b



• Dilations - Write down all your coordinates for the original triangle.
• Dilation - The scale factor tells you how much to enlarge the image.



Similarity Transformations- HSG-SRT.A.2



• Similarity Transformations - Transform circle A to circle B by translating it and performing a dilation.
• Translate it! - Find the Scale factor for this dilation by calculating the ratio of radii.

Angle Sum and Difference, Double Angle and Half Angle Formulas - HSG-SRT.A.2

• Double and Half Angle Formulas - Use these formulas to find the values on unknown trig functions.
• Angle sum and difference - Follow this example to find the sum and difference of angles.



Corresponding Angles of Similar of Triangles- HSG-SRT.A.3



• What Does Correspond Mean? - Find the length of the unknown side.
• Triangle Ratios - Find Corresponding Lengths in Similar Triangles



Proving Triangle Theorems- HSG-SRT.B.4



• Triangle Theorems - The measure of an exterior angle of a triangle is equal to the sum of the measures of the two non‐adjacent interior angles of the triangle.
• Types of Triangles - Equilateral, Isosceles, Scalene



Congruent Triangles: SSS and SAS Theorems- HSG-SRT.B.5



• SSS Theorem - Side‐Side‐Side
• SAS Theorem - Side‐Angle‐Side



Congruent Triangles: ASA and AAS Theorems - HSG-SRT.B.5



• AAS Theorem - Angle‐ Angle‐ Side
• ASA Theorem - Angle‐ Side‐ Angle



Triangles (Similarity and Congruence)- HSG-SRT.C.6



• Similar Triangles - Find the missing side in similar triangles.
• Similarity Statements - AA similarity theorem states that two triangles are similar is two angles of one triangle are congruent to two angles of the other triangle.



Tangent Ratios- HSG-SRT.C.6



• Sine, Cosine and Tangent Ratios - Opposite and adjacent sides are in reference to the angle you are finding the ratio for.
• Tangent Ratios - Find the Tangent ratio for C.



Cos and Sin Trigonometric Ratios - HSG-SRT.C.7



• Cosine Ratio - A visual.
• Sine Ratio - Same.



Trigonometric Ratios and the Pythagorean Theorem- HSG-SRT.C.8



• Trigonometric Ratios - Find how high the rocket is shot into the sky.
• Find Cosine and Sine - The math of the rocket.



Working with Right Triangles - HSG-SRT.C.8



• Pythagorean Theorem - A practical application.
• With Triangles - Use it on a right triangle. Use to find the missing side.



Proving the Formula A = 1/2 ab sin(C)- HSG-SRT.D.9



• Proving Triangle Formula - A = 1/2 ac Sin A
• Using the Formula - ...when you don’t have the height.



Area of Triangle Using Trigonometry - HSG-SRT.D.9



• Area of a Triangle - Area = 1/2 base (height)
• Prove the Area Formula - See it in action.



Using Sine and Cosine - HSG-SRT.D.10



• Cosine Graphs - Cosine curves usually start high on the y axis then dip down.
• Sine Graphs - Sine curves usually start near or close to 0 on the y axis.



Law of Cosines - HSG-SRT.D.10



• Law of Sines - When you know 2 sides and one angle and you want to find the measure opposite the known angle.
• Another look at Law of Sines - When you know 2 angles and 1 side and want to know the side opposite the known angle.



Graphs Dealing with Sine and Cosine Problems - HSG-SRT.D.10



• Law of Cosines - Use to find missing sides.
• When to use law of Cosines - Use to find the missing side of the triangle when we know two sides and the angle between the two known sides.



Application of the Standard Law of Sines - HSG-SRT.D.10



• How many Triangles? - When given these measurements how many triangles can you make?
• Inverse Sine - When you need to find an unknown angle, you will use the inverse sin.



Law of Sines and the Ambiguous Case - HSG-SRT.D.10



• When to use Law of Sines - When you know 2 sides and one angle and you want to find the measure opposite the known angle.
• Law of Sines - Use the law of sines to find the missing sides and angles in a triangle.



Find the Missing Angle Using Trigonometry - HSG-SRT.D.11



• Inverse Cosine or Tangent - Use when you know the length of the sides but you don't know any angles.
• Law of Cosines - Use to find the angles of a triangle when we know all three sides.

Circles




Similarity of Circles - HSG-C.A.1



• Dilating Circles - Find the scale factor of dilation by finding the ratio of the radii.
• Translating Circles - If h is positive, the shape moves to the right; if it is negative, it moves to the left.



Circles: Inscribed Angles, Arcs and Chords- HSG-C.A.2



• Diameter, Radius, Chords - The length of the line through the center of a circle and touching two points on its edge.
• Inscribed Angle, Minor and Major Arcs - An angle made from two chords of a circle that meet on the circumference.



Graphs Dealing with Tangent, Cotangent, Secant, Cosecant Problems - HSG-C.A.2



• Tangent, Cotangent - Tangent graphs go through negative to positive infinity, crossing through 0.
• Secant, Cosecant - Secant is the reciprocal of cosine.



Perimeter of Polygons with Inscribed Circles- HSG-C.A.3



• Inscribed Quadrilateral - The opposite angles in the quadrilateral or supplementary or added together they equal 180°.
• Inscribed Right Triangle - Find the measure of the missing angle if < B = 70°



Angles in Inscribed Right Triangles and Quadrilaterals- HSG-C.A.3



• Inscribed Circles - If is 5 and is 3 what is ?
• Contained Circles - A circle contained in a triangle. It touches all three sides of the triangle.



Constructing and Using Tangent Lines- HSG-C.A.4



• Tangent Lines - A line that touches the circle in exactly one point, never going into the circle.
• Tangent Points - When you have a right triangle, use the Pythagorean theorem to find the measure of the missing side.



Measurements of Arcs- HSG-C.B.5



• Length of an Arc - X is the degrees of the arc.
• Arc and Circumference - The distance around the outside of a circle.



Area of Sectors of A Circle- HSG-C.B.5



• Slice of Circles - A slice of a circle is called a sector.
• Area of a Sector - M is the measure of degrees of the arc around the sector.



Arc Length and Radian Measure - HSG-C.B.5



• Degrees and Radians - To change from degrees to radians and back again.
• Radian Measure - A measure based on the radius of a circle. A radian is created when the radius is wrapped around the edge of a circle.

Expressing Geometric Properties with Equations




Finding the Equation of Circles - HSG-GPE.A.1



• Find the equation of a circle... - When you know the center point and the length of the radius.
• Using End Points - When you are given the end points of the diameter of the circle.



Finding the Equation of a Parabola- HSG-GPE.A.2



• Equation of a Parabola - Given the focus of a parabola and the directrix point, find the equation.
• Parabola, Directrix, Vertex - A curve where any point is at an equal distance from the focus.



Equations of Ellipses- HSG-GPE.A.3



• Equation of an Ellipse - |a| = the distance from the center to the vertex
• Ellipse, Focus, Vertex - Two points inside an ellipse that lie on the longest (major) axis.



Equations of Hyperbolas- HSG-GPE.A.3



• Equation of a Hyperbola - Semi‐ conjugate axis.
• Hyperbola - The distances of any point on a hyperbola for the focus or the directrix are always the same ratio.



Using Coordinates To Prove Theorems- HSG-GPE.B.4



• Prove Theorems using Coordinates - Use the unknown point on the circle and the circle center in the first equation.
• Using Points - To find if two points are on the same circle, you must calculate the distance from the center for both points. If they are the same distance, they are on the same circle.



Slopes of Parallel and Perpendicular Lines- HSG-GPE.B.5



• Slopes of Parallel and Perpendicular Lines - Two Parallel lines have the same slope.
• Slope- Intercept Form - Find the equation for perpendicular lines when given the equation of one line.



Finding Midpoints of Line Segments- SG-GPE.B.6



• Find Midpoints - To find the midpoint of coordinates AB, you use the formula.
• Coordinate Midpoints - When given two coordinates, find the midpoint using the formula.



Area and Perimeter in the Coordinate Plane- HSG-GPE.B.7



• Area in a Coordinate Plane - Take the difference between the x‐ coordinates to find the base and the height of the triangle.
• Perimeter in a Coordinate Plane - Use the distance formula to find the length of the length and the width.



Similar Polygons: Ratio of Perimeters & Areas - HSG-GPE.B.7



• Perimeter of Similar Triangles - Use the numbers in the perimeter formula.
• Ratio of Similar Polygons - Ratio of Perimeters = (similarity ratio)²



Finding the Length of Line Segments - HSG-GPE.B.7



• Find the Length of a Line Segment - Using the distance formula, you can find the length of a line segment whose endpoints are (5, 4) and (3, 6).
• Distance Formula - Your points will be written: (x₁, y₁); (x₂, y₂)

Geometric Measurement & Dimension




Cavalieri's Principle - HSG-GMD.A.1



• Cavalieri's Principle - When you are given the area of a part of a solid, and are asked to calculate the volume, use this formula.
• Penny Cross Sections - The principle is illustrated by a stack of coins. There is the same number of coins in each stack and they have the same volume no matter how they are rearranged.



Volume of Cylinders and Trigular Prisms- HSG-GMD.A.3



• Volume of Cylinders - The root formula to remember.
• Volume of Triangular Prisms - Use the formula to find the volume.



Volume of Cones and Spheres- HSG-GMD.A.3



• Volume of a Cone - You just need radies and height.
• Volume of a Sphere - Surface area of a sphere too.



Cross-sections of Three-dimensional Objects - HSG-GMD.B.4



• Three Dimensional Objects - Identify the 2‐dimensional shapes that make up 3 dimensional objects.
• Discover 2D and 3D Shapes - Discover what 2‐dimensional shapes make up 3‐dimensional shapes.

Modeling with Geometry




Using Shapes and Measures to Describe Objects- HSG-MG.A.1



• Pizza Objects - Everyday objects are made of shapes. Can you find all the shapes in the picture below?
• Finding Shapes - Look around the room you're in, what shapes do you see?



Using Density in Real-life Situations - HSG-MG.A.2



• Population Density - A measurement of population per unit.
• Find Volume and Mass - Mass = volume x density



Using Geometry in Design Problems - HSG-MG.A.3



• Find the Area of an Unusual Shape - Find the area of the shapes within the shape. In the shape below you have two triangles and one rectangle.
• Geometry in Real Situations - How many boxes can you fit in your trunk that is 20 inches long, 10 inches wide and 40 inches high?



Complete the Truth Table - HSG-MG.A.3



• Truth Tables - Below is a chart explaining some of the symbols that you may find in a truth table.
• Truth Statements - A truth table helps find all possible truth values of a statement.



Conditionals Using Logic Tables - HSG-MG.A.3



• Conditionals - The symbol for a conditional statement is: ↄ
• Biconditionals - The symbol for biconditional.



Conjunctions Using Logic Tables - HSG-MG.A.3



• Conjunction - Examples of conjunctions using words.
• Recognize Conjunction - The statement is only true if both A and B are true. All other statements are false.



Disjunction, Conditionals and Biconditionals - HSG-MG.A.3



• Can you tell the difference? - Is this statement a conjunction, disjunction or biconditional?
• Conjunction, Disjunction, Biconditional - The only time the statement is false is when both parts of the statement are false.



Disjunction Based Logic Tables - HSG-MG.A.3



• Disjunction Tables - The only time the statement is false is when both P and Q are false, all other times it is true.
• The Symbol V - It is easy to see the logic of a disjunction when you see the logic in sentences.



Negation and Conjunction In Logic Statements - HSG-MG.A.3



• Negation - A negation is formed by placing the word "not" in the original statement.
• Table Form Negation - With a negation, true will become false and false will become true.



Related Conditional: converse, inverse, contra- positive- - HSG-MG.A.3



• Converse - The converse of a statement is formed by interchanging the hypothesis and the conclusion of the original statement.
• Inverse and Contra-positive - The inverse is formed when you negate the hypothesis and the conclusion of the original statement.



Truth Values of Compound Sentences - HSG-MG.A.3



• Compound Sentences - A compound sentence is when two or more thoughts are connected in one sentence.
• Determine Truth Value In Compound Sentences - Determine the validity of the entire statement.



Logic : Conjunctions, Disjunctions, and Biconditionals - HSG-MG.A.3



• Conditionals - An example of a conditional statement (note it uses the words if...then).
• Recognizing Conditionals - The Conditional statement is only false then A is True and B is False.



Truth Value of Open Sentences - HSG-MG.A.3



• Open Sentences - To find the variable, simply solve for x.
• What's A Domain? - With more information you can determine the truth value of open sentences. A domain is given to provide more information.



Using Venn Diagrams Problems - High School Modeling



• Venn Diagrams - Venn Diagrams help to compare and contrast two different things.
• Venn Diagrams Word Problems - There are 13 students in a class, 5 of those students play basketball and baseball; 3 play just basketball and 5 play just baseball.