Geometry

PA Common Core Mathematics Standards:
CC.2.3.HS.A.3 Verify and apply geometric theorems as they relate to geometric figures.
CC.2.3.HS.A.11 Apply coordinate geometry to prove simple geometric theorems algebraically.
 

• Name points, lines, angles, and planes with proper symbolic notation.
• Feature algebra concepts to segment and angle measurements.
• Identification of angle relationships. 
• Steps to midpoint and distance formula given a coordinate plane.

• Proper symbolic notation identifies different geometric figures.
• Properties of angles and lines allow us to solve for unknown values.
   

• Recognize and name points, lines, and planes.
• Use properties of segments to write and solve algebraic equations to find segment measures.
• Calculate distance and midpoints of segments from the coordinate plane. 
• Classify and name parts of angles.
• Use angle relationships to find angle measures. 

• Why is it important to use proper notation and vocabulary in geometry?
• How do we use postulates and definitions to explain relationships between geometric figures?
• How can we use the relationships between parts and wholes of angles and segments to solve geometric problems and justify reasoning?
   

• How can we represent and name points, lines, and planes accurately?
• How do points, lines, and planes relate to one another in space?
• How does the midpoint formula help in dividing a line segment into equal parts?
• How does the distance between two points help us understand their relationship on a graph?
• What properties exist between angles, and how can we use them to find unknown measurements?
• How do we accurately find the measure of whole line segments given algebraic expressions for each part of the segment?
   

• Desmos: https://www.desmos.com/scientific
• IXL: https://www.ixl.com/math/ 
• Scientific Calculators
• All Things Algebra: Geometry Curriculum
• Collaborative Geometry Google Drive:
• https://drive.google.com/drive/folders/1y3VVUGFBuFh7Zx8521Ez62HwDU0FAoG4?usp=drive_link

• Point
• Line (collinear)
• Plane (coplanar)
• Segment
• Segment Addition
• Midpoint (segments)
• Distance Formula
• Midpoint Formula
• Coordinates
• Angle
• Congruence
• Bisect
• Perpendicular
• Vertical Angles
• Adjacent Angles
• Linear Pair
• Complementary Angles
• Supplementary Angles

• Quizzes 
• Tests
• IXL
• Check for Understanding Mini-Quizzes
• All Things Algebra Activities
   

PA Common Core Mathematics Standards:

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

• Conditional statements to determine their truth values.
• Steps to write two column algebraic proofs.
• Steps to write two column angle and segment proofs.
   

• Postulates, definitions, and properties of congruence allow us to use logical thinking to write two column proofs involving algebra, segments, and angles. 
• Truth value can be determined by a given conditional statement and its related statements
   

• Identify the hypothesis and conclusion of an if-then statement.
• Convert statements into if-then statements. 
• Changing conditional statements to determine the truth value of related statements.
• Use properties of equality to complete multi-step algebraic proofs (two column format).
• Use definitions, postulates, and properties of congruence to write segment and angle proofs (two column format). 

• How can conditional statements be written in different forms (converse, inverse, contrapositive), and how do their truths compare?
• How do conditional statements support the structure of geometric proofs?
   

• What is a conditional statement, and how is it structured?
• How do the converse, inverse, and contrapositive of a conditional statement differ from the original?
• How are conditional statements used in geometric proofs and theorems?
• How can we use properties of equality to justify each step in solving an equation in a two column proof?
• How do properties of angle pairs (like complementary, supplementary, vertical, and adjacent angles) help us construct valid proofs?
• How can we use definitions, postulates, and theorems to prove relationships between line segments?

• Desmos: https://www.desmos.com/scientific
• IXL: https://www.ixl.com/math/ 
• Scientific Calculators 
• All Things Algebra: Geometry Curriculum
• Collaborative Geometry Google Drive:
• https://drive.google.com/drive/folders/1y3VVUGFBuFh7Zx8521Ez62HwDU0FAoG4?usp=drive_link

• Conditional Statements
• Inverse
• Converse
• Contrapositive
• Biconditional
• Hypothesis
• Conclusion
• Distributive Property
• Substitution Property
• Reflexive Property
• Symmetric Property
• Transitive Property
• Two-Column Proof
• Congruence Properties
• Definition of Congruence
• Definition of Midpoint
• Segment Addition Postulate
• Angle Addition Postulate

• Test
• IXL
• All Things Algebra Activities
   

PA Common Core Mathematics Standards:

• CC.2.3.HS.A.3 Verify and apply geometric theorems as they relate to geometric figures.
• CC.2.3.HS.A.11 Apply coordinate geometry to prove simple geometric theorems algebraically.

• Identification of types of lines by their slopes.
• Slope of a line given ordered pairs. 
• Steps to convert and solve standard form equations to slope-intercept.
• Identification of angle relationships involving parallel lines.

• Slope can determine the different types of lines. 
• Slope-intercept form identifies the slope and y-intercept of a line.
• Angle relationships of parallel lines can determine unknown measurements of angles.

• Determine whether lines are parallel, perpendicular, or neither given slope.
• Calculate slope given ordered pairs of lines and determine if they are parallel, perpendicular, or neither. 
• Convert equations of lines from standard form to slope-intercept and determine if they are parallel, perpendicular, or neither. 
• Write the equation of a line in slope-intercept form using the point-slope equation.
• Determine the type of angle relationships given parallel lines cut by a transversal.
• Determine if angle relationships are congruent or supplementary based on their position on parallel lines cut by a transversal.  

• How does the relationship between slopes help us prove whether two lines are parallel or perpendicular without graphing?
• How do different forms of linear equations (slope-intercept, point-slope, standard form) show slope?
• Why are some angle pairs congruent and others supplementary when two parallel lines are cut by a transversal?

• What do different types of slopes (positive, negative, zero, and undefined) tell us when comparing two or more lines?
• How can we determine if two lines are parallel or perpendicular by looking at their slopes?
• How do lines and points define planes, and what relationships exist between lines when they intersect or lie within the same plane?
• How can we identify corresponding, alternate interior, alternate exterior, and same-side interior angles?
• How do we use algebra to find missing angle measures in parallel line diagrams?
   

• Desmos: https://www.desmos.com/scientific
• IXL: https://www.ixl.com/math/ 
• Scientific Calculators
• All Things Algebra: Geometry Curriculum
• Collaborative Geometry Google Drive:
• https://drive.google.com/drive/folders/1y3VVUGFBuFh7Zx8521Ez62HwDU0FAoG4?usp=drive_link
   

• Slope
• Parallel Lines
• Perpendicular Lines
• Negative Reciprocals
• Slope-Intercept Form
• Standard Form
• Point-Slope Form
• Perpendicular Bisector
• Parallel Planes
• Skew Lines
• Transversal
• Corresponding Angles
• Alternate Interior Angles
• Alternate Exterior Angles
• Consecutive Interior Angles

• Quiz 
• Test
• IXL
• Check for Understanding Mini-Quizzes
• All Things Algebra Activities

PA Common Core Mathematics Standards:

• CC.2.3.HS.A.3 Verify and apply geometric theorems as they relate to geometric figures.
• CC.2.3.HS.A.2 Apply rigid transformations to determine and explain congruence.
• CC.2.3.HS.A.6 Verify and apply theorems involving similarity as they relate to plane figures. 

• Side and angle measures determine the triangle classification. 
• Feature algebra concepts to find triangle measurements. 
• Steps to complete two column proofs proving triangles congruent.

• Properties of scalene, isosceles, and equilateral triangles relating to their parts.
• Proofs develop critical thinking and logical reasoning skills.

• Classify triangles by their side lengths and angle measures. 
• Calculate unknown angle measures using the triangle sum and exterior angle sum theorems. 
• Determine side and angle measurements using the properties of isosceles and equilateral triangles. 
• Write two column proofs proving triangles are congruent by the SSS, SAS, ASA, AAS, or HL postulates.
• Write two column proofs proving parts (sides and angles) are congruent using the theorem Corresponding Parts of Congruent Triangles are Congruent (CPCTC).

• How can the properties of sides and angles help us classify and understand different types of triangles?
• How do angle measurements help us understand and solve problems involving triangles?
• How can we use known properties and relationships to prove that two triangles are congruent?

• How can we classify triangles based on their side lengths and angle measures?
• How can we determine the measure of an unknown angle in a triangle if we know the other two angles?
• Given two equal sides in a triangle and the included angle, how can you determine if the triangle is isosceles?
• Why can we say all sides and angles are equal in an equilateral triangle?
• How can we use definitions, postulates, and properties of congruence to prove that two triangles are congruent (SSS, SAS, ASA, AAS, HL)?
• How does CPCTC help us prove that parts (angles and sides) of congruent triangles are congruent? 
   

• Desmos: https://www.desmos.com/scientific
• IXL: https://www.ixl.com/math/ 
• Scientific Calculators
• All Things Algebra: Geometry Curriculum
• Collaborative Geometry Google Drive:
• https://drive.google.com/drive/folders/1y3VVUGFBuFh7Zx8521Ez62HwDU0FAoG4?usp=drive_link
   

• Acute 
• Obtuse
• Right
• Equiangular
• Equilateral
• Isosceles
• Scalene
• Bisect
• Triangle Sum Theorem
• Exterior Angle Theorem
• Vertex
• Base Angles
• Congruence Statement
• Reflexive Property
• SSS Congruence
• SAS Congruence
• ASA Congruence
• AAS Congruence
• HL Congruence
• CPCTC

• Quizzes 
• Tests
• IXL
• Check for Understanding Mini-Quizzes
• All Things Algebra Activities

PA Common Core Mathematics Standards:

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

• Properties of similar triangles.
• Steps to solve proportions.
• Features of scale factors related to similar figures.

• Similar figures are related to each other with a common ratio.

• Use extended ratios to find angle and side measurements of triangles.
• Apply scale factor to similar triangles to find side measurements and perimeters.
• Calculate side measures of similar triangles using proportions.
• Determine if triangles are similar by SSS~, SAS~, or AA~.

• What is the difference between triangle similarity and triangle congruence, and when do we use each?
• What properties of similar triangles help us solve real-world problems?
   

• What does it mean for two triangles to be similar?
• How can you use angle and side relationships to determine if triangles are similar?
• What are the different methods to prove triangles are similar (AA, SSS, SAS)?
• Why does knowing two angles are congruent guarantee triangle similarity?
• How can you use proportions to find missing side lengths in similar triangles?
• How can similar triangles be used to find distances or heights indirectly?
• How does the scale factor between similar triangles affect side lengths and perimeters?
   

• Desmos: https://www.desmos.com/scientific
• IXL: https://www.ixl.com/math/ 
• Scientific Calculators 
• All Things Algebra: Geometry Curriculum
• Collaborative Geometry Google Drive:
• https://drive.google.com/drive/folders/1y3VVUGFBuFh7Zx8521Ez62HwDU0FAoG4?usp=drive_link
   

• Ratio
• Extended Ratio
• Proportion
• Scale Factor
• Similarity Statement
• AA ~
• SSS~
• SAS~

• Test
• IXL
• All Things Algebra Activities

PA Common Core Mathematics Standards:

• CC.2.3.HS.A.7 Apply trigonometric ratios to solve problems involving right triangles.
• CC.2.3.HS.A.3 Verify and apply geometric theorems as they relate to geometric figures.
• CC.2.3.HS.A.14 Apply geometric concepts to model and solve real world problems.

• Steps to using Pythagorean Theorem.
• Feature trigonometric functions in right triangles. 
• Identification of special right triangles.

• In real-world jobs (ex architecture or construction), right triangle trigonometry is used to find needed sides and angles. 
• The Pythagorean Theorem shows the relationship between the lengths of the sides in a right triangle and can be used in real-world jobs. 

• Use Pythagorean Theorem to find missing side lengths of right triangles.
• Determine if three side lengths can form a triangle, then classify the type of triangle, using the triangle inequality theorem.
• Calculate side lengths of special right triangles using their specific properties.
• Use trigonometric ratios to find side lengths and angle measurements of right triangles.
• Determine side or angle measurements using angles of elevation or depression. 
   

• How can we use the Pythagorean Theorem to solve real-world and mathematical problems involving right triangles?
• How do trigonometric ratios (sine, cosine, and tangent) help us find unknown sides or angles in right triangles?
• How do inverse trigonometric functions help us solve right triangles?
   

• How do you determine if a triangle is a right, acute, or obtuse triangle using side lengths?
• What are some real-life scenarios where the Pythagorean Theorem is useful?
• Why are special right triangles helpful shortcuts in problem-solving?
• How do you determine which trigonometric ratio to use (sine, cosine, or tangent)?
• How can you use trigonometric ratios to find a missing side or angle?
• How can you use a diagram to set up and solve a right triangle problem?
   

• Desmos: https://www.desmos.com/scientific
• IXL: https://www.ixl.com/math/ 
• Scientific Calculators 
• All Things Algebra: Geometry Curriculum
• Collaborative Geometry Google Drive:
• https://drive.google.com/drive/folders/1y3VVUGFBuFh7Zx8521Ez62HwDU0FAoG4?usp=drive_link

• Pythagorean Theorem
• Legs
• Hypotenuse
• Triangle Inequality Theorem 
• 45-45-90 Triangle
• 30-60-90 Triangle
• Trigonometry
• Sine
• Cosine
• Tangent
• sin-1
• cos-1
• tan-1
• Angle of Elevation
• Angle of Depression

• Quizzes 
• Tests
• IXL
• Check for Understanding Mini-Quizzes
• All Things Algebra Activities
   

PA Common Core Mathematics Standards:

• 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.7 Apply trigonometric ratios to solve problems involving right triangles.
• CC.2.3.HS.A.13 Analyze relationships between two-dimensional and three-dimensional objects.

• Interior and exterior angle sum formula.
• Features of polygon properties. 
• Classification of quadrilaterals.
   

• Polygon number of sides and how it relates to the measure of each angle in the polygon.
• Properties, classifications, and angle relationships of quadrilaterals, allow us to solve problems involving side lengths and angles.

• Apply the interior angle sum formula to any polygon to find angle and side measurements.
• Use the property of exterior angles of polygons to a sum of 360º.
• Apply the properties of parallelograms to find unknown side and angle measurements.
• Apply the properties of parallelograms to the remainder of the quadrilateral family consisting of rectangles, rhombi, and squares.
• Explore the properties of trapezoids and kites.

• What is the relationship between the number of sides of a polygon and the sum of its interior angles?
• Why do the exterior angles of any polygon always add up to 360 degrees?
• How do the properties of quadrilaterals compare and contrast with each other?

• How does the interior angle sum formula for a polygon relate to triangle sum?
• What methods can be used to find the interior or exterior angle of a regular polygon?
• How do interior and exterior angles relate to each other in polygons?
• What are the properties of parallelograms?
• How are parallelograms related to rectangles, rhombi, and squares?
• What type of symmetry do the quadrilaterals have?
• How do the properties of a trapezoid differ from an isosceles trapezoid?
• How do the sides and diagonals in a kite differ from the other quadrilaterals?

• Desmos: https://www.desmos.com/scientific
• IXL: https://www.ixl.com/math/ 
• Scientific Calculators 
• All Things Algebra: Geometry Curriculum
• Collaborative Geometry Google Drive:
• https://drive.google.com/drive/folders/1y3VVUGFBuFh7Zx8521Ez62HwDU0FAoG4?usp=drive_link
   

• Polygon
• Quadrilateral
• Pentagon
• Hexagon
• Heptagon
• Octagon
• Nonagon
• Decagon
• Regular Polygon
• Interior Angle
• Exterior Angle
• Parallelogram
• Supplementary
• Diagonals
• Rectangle
• Bisect
• Consecutive Angles
• Rhombus
• Square
• Trapezoid
• Isosceles Trapezoid
• Midsegment of Trapezoid
• Kite

• Quizzes 
• Tests
• IXL
• Check for Understanding Mini-Quizzes
• All Things Algebra Activities

PA Common Core Mathematics Standards:

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

• Identification of circle parts. 
• Features of central and inscribed angles and their corresponding arcs.
• Arc length and its relation to circumference.
• Identification of segments in circles.
• Identification of center and radius of a circle from standard equation.

• Understanding the relationship between a circle's radius and its key measurements—diameter, circumference, and area—is fundamental to real-world problems.
• The position of the vertex of an angle and the arc(s) it intercepts on a circle will determine how we calculate measurements of unknown angles and segment lengths. 
• Understanding the standard form equation for a circle and the geometric features each part represents. 

• Identify and name parts of circles with symbolic notation.
• Calculate the area and circumference of a circle.
• Use the properties of central angles to find unknown angles and arcs of circles.
• Calculate the arc length which is a portion of the circle's circumference. 
• Using congruent chord and arc properties of a circle, find segment and angle measures using trigonometry functions.
• Use the properties of inscribed angles to find the measurement of the angle and its corresponding arc.
• Calculate the tangent line of a circle using the Pythagorean Theorem.
• Determine the measure of angles and arcs formed by intersecting lines in circles.
• Calculate the segment length(s) in circles formed by intersecting lines.
• Write the equation of a circle in standard form given the center and radius, two endpoints, or graph.

• How can we use formulas to find the area and circumference of circles, and why are these measurements important in real-world contexts?
• How do the relationships between angles and arcs in a circle help us understand and solve geometric problems?
• How can we use the properties of segments in circles to solve problems involving lengths and relationships within and around circles?
• How can we use the standard equation of a circle to identify and describe its center and radius, and solve problems involving points on or inside the circle?
   

• How are the radius, diameter, and circumference of a circle related?
• How do we find the area and circumference of a circle, and what do these measurements represent?
• How do central angles, arcs, and chords help describe parts of a circle?
• How does arc length relate to the circumference of a circle?
• How is the measure of an inscribed angle related to the arc it intercepts?
• What happens when multiple inscribed angles intercept the same arc?
• How do inscribed angles help us understand the properties of cyclic (inscribed) quadrilaterals?
• What is the relationship between a tangent and a radius at the point of contact?
• How are angles formed by chords, tangents, and secants related to arcs?
• How can you identify the center and radius of a circle by looking at its equation?
   

• Desmos: https://www.desmos.com/scientific
• IXL: https://www.ixl.com/math/ 
• Scientific Calculators 
• All Things Algebra: Geometry Curriculum
• Collaborative Geometry Google Drive:
• https://drive.google.com/drive/folders/1y3VVUGFBuFh7Zx8521Ez62HwDU0FAoG4?usp=drive_link

• Circle
• Radius
• Chord
• Diameter
• Secant
• Tangent
• Point of Tangency
• Area of a Circle
• Circumference
• Central Angles
• Arc
• Arc Measures
• Arc Length
• Semicircle
• Minor Arc
• Major Arc
• Inscribed Angles
• Intercepted Arc
• Inscribed Quadrilaterals
• Pythagorean Theorem
• Standard Form of Circle

• Quizzes 
• Tests
• IXL
• Check for Understanding Mini-Quizzes
• All Things Algebra Activities
   

PA Common Core Mathematics Standards:

• 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.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.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.

• Identification of plane figures.
• Steps to calculate area of plane or composite figures.
• Identification of 3-dimensional figures. 
• Steps to calculate surface area, volume, and density of 3-dimensional shapes.

• Understand that area represents the amount of space within a plane figure, and that complex shapes (composite figures) can be broken down into simpler geometric components (like rectangles, triangles, or circles).
• All 3-dimensional objects have a surface area; however, lateral surface area must be removed for overlapping faces. 
• All 3-dimensional objects have a volume which is used in real-world applications such as filling a pool with water or saving space when shipping a package.

• Calculate the area of plane figures given the formula.
• Determine the area of a sector (portion of the whole circle area). 
• Given a diagram, calculate the area of composite figures and area of shaded regions of composite figures. 
• Calculate the surface area of prisms, cylinders, pyramids, spheres, hemispheres, and cones.
• Calculate the volume of prisms, cylinders, pyramids, spheres, hemispheres, and cones.
• Determine the density of a 3-dimensional object.

• How can we use formulas and reasoning to find the area of plane and composite figures in real-world and mathematical contexts?
• How does understanding surface area help us design and build objects in the real world, like packaging, buildings, or art?
• What strategies can we use to find the volume of prisms and pyramids, and why is this important?

• How does the formula for area relate to the properties of a shape (like base, height, radius)?
• How can we calculate and apply the area of sectors?
• What is a composite figure, and how can we break it down into simpler shapes?
• How can we use area formulas of basic shapes to find the area of a composite figure?
• What strategies can we use to subtract areas when there are missing or hollow parts?
• Why do we use different formulas for different 3D figures?
• How can calculating surface area be useful in real-world contexts (e.g., packaging, painting, wrapping)?
• How do we calculate the volume of prisms, cylinders, pyramids, cones, and spheres?
• What is density, and how does it relate mass and volume in 3D objects?
   

• Desmos: https://www.desmos.com/scientific
• IXL: https://www.ixl.com/math/ 
• Scientific Calculators 
• All Things Algebra: Geometry Curriculum
• Collaborative Geometry Google Drive:
• https://drive.google.com/drive/folders/1y3VVUGFBuFh7Zx8521Ez62HwDU0FAoG4?usp=drive_link

• Area
• Base 
• Height
• Sector
• Area of a Sector
• Composite Figures
• Edge
• Prism
• Cylinder
• Pyramid 
• Cone
• Faces
• Lateral Area
• Surface Area
• Volume
• Slant Height
• Perimeter of Base
• Radius
• Sphere
• Hemisphere
• Great Circle 

• Quizzes 
• Tests
• IXL
• Check for Understanding Mini-Quizzes
• All Things Algebra Activities