Focus Areas for Geometry
- Establish criteria for congruence of triangles based on rigid motions
- Establish criteria for similarity of triangles based on dilations and proportional reasoning
- Informally develop explanations of circumference, area, and volume formulas
- Apply the Pythagorean Theorem to the coordinate plane
- Prove basic geometric theorems
- Extend work with probability
Math Standards by Domain
Congruence and Transformations
What Students Learn:
Students learn precise definitions of geometric terms and understand transformations in the plane. They explore rigid motions (translations, rotations, reflections) that preserve distance and angle, and learn how these transformations determine congruence.
Key Skills:
- Define angle, circle, perpendicular line, parallel line, and line segment precisely
- Represent transformations using transparencies and geometry software
- Compare transformations that preserve distance/angle vs. those that don't
- Describe symmetries of shapes (rotations and reflections that carry a shape onto itself)
- Draw transformed figures given a rotation, reflection, or translation
- Use rigid motions to decide if two figures are congruent
A rectangle has two lines of symmetry (vertical and horizontal through its center) and 180° rotational symmetry about its center.
To determine if two triangles are congruent, find a sequence of rigid motions that maps one onto the other.
What Students Learn:
Students understand that congruence means corresponding sides and angles are equal, and learn the criteria for triangle congruence (SSS, SAS, ASA). They see how these criteria follow from the definition of congruence in terms of rigid motions.
Key Skills:
- Use rigid motions to show triangles are congruent if and only if corresponding parts are congruent
- Apply SSS (Side-Side-Side) congruence criterion
- Apply SAS (Side-Angle-Side) congruence criterion
- Apply ASA (Angle-Side-Angle) congruence criterion
- Explain how each criterion follows from rigid motion definitions
- Use congruence criteria to solve problems and prove relationships
Given two triangles with sides 3, 4, 5 and sides 3, 4, 5, the triangles are congruent by SSS.
If two triangles have two sides of length 5 and 7 with an included angle of 60°, they are congruent by SAS.
What Students Learn:
Students prove fundamental geometric theorems about lines, angles, triangles, and parallelograms. This develops logical reasoning and proof-writing skills essential to geometric thinking.
Key Skills:
- Prove vertical angles are congruent
- Prove properties of parallel lines cut by a transversal (alternate interior angles, corresponding angles)
- Prove the triangle angle sum theorem (angles sum to 180°)
- Prove base angles of isosceles triangles are congruent
- Prove the triangle midsegment theorem
- Prove properties of parallelograms (opposite sides congruent, diagonals bisect each other)
- Prove rectangles are parallelograms with congruent diagonals
Prove the triangle angle sum: Draw a line through one vertex parallel to the opposite side. Use alternate interior angles to show the three angles form a straight line (180°).
What Students Learn:
Students learn to make formal geometric constructions using compass and straightedge or dynamic geometry software. They construct basic elements and inscribed polygons in circles.
Key Skills:
- Copy a segment and copy an angle
- Bisect a segment and bisect an angle
- Construct perpendicular lines and perpendicular bisectors
- Construct a line parallel to a given line through a point
- Construct an equilateral triangle inscribed in a circle
- Construct a square inscribed in a circle
- Construct a regular hexagon inscribed in a circle
To construct a perpendicular bisector of segment AB: Open compass to more than half AB's length, draw arcs from A and B above and below AB. Connect the intersection points to create the perpendicular bisector.
Similarity, Right Triangles, and Trigonometry
What Students Learn:
Students understand similarity as a transformation involving dilations (enlargements or reductions). They learn that similar figures have equal corresponding angles and proportional corresponding sides, and establish the AA criterion for triangle similarity.
Key Skills:
- Verify that dilations preserve angle measures but scale distances by the scale factor
- Use similarity transformations to determine if figures are similar
- Apply the AA (Angle-Angle) criterion for triangle similarity
- Prove the Pythagorean Theorem using triangle similarity
- Prove theorems: a line parallel to one side of a triangle divides the other sides proportionally
- Use similarity to solve problems involving geometric figures
Two triangles with angles 50°, 60°, 70° and 50°, 60°, 70° are similar by AA (two pairs of equal angles).
If a triangle has sides 3, 4, 5 and a similar triangle has shortest side 6, the other sides are 8 and 10 (scale factor of 2).
What Students Learn:
Students understand that in right triangles, the ratios of sides depend only on the angles, leading to definitions of sine, cosine, and tangent. They learn special right triangle ratios and the relationship between complementary angles.
Key Skills:
- Understand sine, cosine, and tangent as ratios in right triangles
- Use trigonometric ratios for acute angles: sin(θ) = opposite/hypotenuse, cos(θ) = adjacent/hypotenuse, tan(θ) = opposite/adjacent
- Recognize that sin(θ) = cos(90° - θ) for complementary angles
- Derive and use ratios for 30-60-90 triangles (1 : √3 : 2)
- Derive and use ratios for 45-45-90 triangles (1 : 1 : √2)
- Use the Pythagorean Theorem with trigonometry
In a right triangle with angle 30° and hypotenuse 10, the opposite side is 10 · sin(30°) = 10 · 0.5 = 5.
In a 45-45-90 triangle with legs of length 5, the hypotenuse is 5√2.
What Students Learn:
Students apply trigonometric ratios and the Pythagorean Theorem to solve real-world problems involving right triangles, such as finding heights, distances, and angles in practical contexts.
Key Skills:
- Use sine, cosine, and tangent to find missing sides in right triangles
- Use inverse trig functions to find missing angles
- Apply the Pythagorean Theorem to find missing sides
- Solve applied problems: finding heights of buildings, distances across rivers, angles of elevation
- Combine multiple right triangles to solve complex problems
- Choose appropriate trigonometric ratios based on given information
A ladder leans against a wall at a 70° angle. If the ladder is 20 feet long, how high up the wall does it reach?
Solution: height = 20 · sin(70°) ≈ 20 · 0.94 ≈ 18.8 feet
Circles
What Students Learn:
Students prove all circles are similar and explore relationships among inscribed angles, central angles, radii, chords, and tangents. They learn to construct inscribed and circumscribed circles of triangles.
Key Skills:
- Prove that all circles are similar (through dilations)
- Understand that a central angle has twice the measure of an inscribed angle subtending the same arc
- Prove that an inscribed angle subtending a diameter is a right angle
- Understand that a radius is perpendicular to a tangent at the point of tangency
- Construct the inscribed circle (incircle) of a triangle
- Construct the circumscribed circle (circumcircle) of a triangle
- Prove properties of cyclic quadrilaterals (opposite angles are supplementary)
If a central angle measures 80°, an inscribed angle subtending the same arc measures 40°.
Any angle inscribed in a semicircle is a right angle (90°).
What Students Learn:
Students derive formulas for arc length and sector area using similarity and proportional reasoning. They understand radian measure and learn to convert between degrees and radians.
Key Skills:
- Understand that arc length is proportional to radius for a given angle
- Define radian measure as the constant of proportionality
- Calculate arc length: s = rθ (where θ is in radians)
- Derive the formula for sector area: A = (1/2)r²θ
- Convert between degrees and radians: 180° = π radians
- Solve problems involving arc length and sector area
In a circle with radius 6, a central angle of π/3 radians (60°) creates an arc of length s = 6 · π/3 = 2π.
The sector area is A = (1/2) · 36 · π/3 = 6π square units.
Coordinate Geometry and Measurement
What Students Learn:
Students derive equations of circles and parabolas using distance formulas and geometric definitions. They learn to complete the square to identify centers and radii of circles.
Key Skills:
- Derive the equation of a circle: (x - h)² + (y - k)² = r²
- Use the Pythagorean Theorem to justify the circle equation
- Complete the square to find the center and radius from a general equation
- Derive the equation of a parabola from focus and directrix
- Graph circles and parabolas from their equations
Circle with center (3, -2) and radius 5: (x - 3)² + (y + 2)² = 25
Given x² + y² - 6x + 4y - 12 = 0, complete the square: (x - 3)² + (y + 2)² = 25, center (3, -2), radius 5.
What Students Learn:
Students use coordinates and algebraic methods to prove geometric theorems. They prove slope criteria for parallel and perpendicular lines and use coordinates to verify properties of polygons.
Key Skills:
- Prove parallel lines have equal slopes
- Prove perpendicular lines have slopes that are negative reciprocals (m₁ · m₂ = -1)
- Find equations of parallel or perpendicular lines through given points
- Prove a quadrilateral is a rectangle, parallelogram, etc., using coordinates
- Find points that partition segments in given ratios
- Use the distance formula to compute perimeters and areas
Prove ABCD with A(0,0), B(4,0), C(4,3), D(0,3) is a rectangle: Show opposite sides are parallel (equal slopes) and adjacent sides are perpendicular (slopes multiply to -1).
What Students Learn:
Students develop informal arguments for volume formulas and apply them to solve problems. They explore cross-sections of 3D objects and understand how scale factors affect length, area, and volume.
Key Skills:
- Give informal arguments for formulas: circumference (C = 2πr), circle area (A = πr²)
- Argue for volume formulas of cylinders (V = πr²h), cones (V = (1/3)πr²h), spheres (V = (4/3)πr³)
- Use volume formulas to solve problems
- Identify 2D cross-sections of 3D objects
- Understand how scale factor k affects: length (×k), area (×k²), volume (×k³)
- Verify triangle inequalities experimentally
A cylinder with radius 3 and height 10 has volume V = π · 9 · 10 = 90π cubic units.
If you scale a cube by factor 2, its volume increases by factor 2³ = 8.
What Students Learn:
Students apply geometric concepts to model real-world objects and situations. They use geometric shapes, density concepts, and design constraints to solve practical problems.
Key Skills:
- Model real objects using geometric shapes (cylinder for tree trunk, sphere for ball)
- Apply density concepts: people per square mile, BTUs per cubic foot
- Solve design problems with physical constraints
- Minimize cost or maximize efficiency using geometric reasoning
- Work with design systems based on geometric ratios
Design a cylindrical can holding 355 mL that minimizes surface area (material cost). Use the volume constraint to express radius in terms of height, then minimize surface area.
Probability
What Students Learn:
Students understand conditional probability, independence, and how to use two-way tables to calculate and interpret probabilities. They learn to recognize independence in everyday situations.
Key Skills:
- Describe events using set notation (unions, intersections, complements)
- Understand independence: P(A and B) = P(A) · P(B)
- Calculate conditional probability: P(A|B) = P(A and B) / P(B)
- Use two-way frequency tables to find conditional probabilities
- Recognize independence in everyday contexts
- Determine if events are independent from two-way tables
In a class, 60% are girls and 40% boys. 30% of girls play sports and 50% of boys play sports. What's P(plays sports | boy)? Answer: 0.50 or 50%.
Are gender and playing sports independent? No, because P(sports|boy) ≠ P(sports|girl).
What Students Learn:
Students apply probability rules including the addition rule and multiplication rule. They learn to use permutations and combinations to calculate probabilities and apply probability concepts to make fair decisions.
Key Skills:
- Apply the Addition Rule: P(A or B) = P(A) + P(B) - P(A and B)
- Apply the Multiplication Rule: P(A and B) = P(A) · P(B|A)
- Use permutations to count ordered arrangements
- Use combinations to count unordered selections
- Calculate probabilities of compound events
- Use probability to make fair decisions (random number generators, lotteries)
- Analyze decisions using probability (medical testing, product quality control)
How many ways can you choose 3 students from 10? C(10,3) = 10!/(3!·7!) = 120
If you randomly select 3 students from 10 (6 girls, 4 boys), what's the probability all 3 are girls? P = C(6,3)/C(10,3) = 20/120 = 1/6