Geometry is one of the most visual and logical branches of mathematics. In this course, you will learn to PROVE things — not just calculate them. Every theorem you master here was first proven by mathematicians thousands of years ago, and now YOU will prove them too. From the elegance of triangle congruence to the power of trigonometry, this course will sharpen your reasoning skills for life. I have seen students who "hated math" fall in love with geometry — and I am confident you will too!
— Mr. Augustine Anyimadu-Ahenkae
10
Interactive Lessons
300
Practice Questions
100+
Printable Worksheets
53
Standards Covered
4
Standards Frameworks
5 full tests — NY Regents, Ohio, Texas STAAR, Virginia SOL, Pennsylvania Keystone · 150 questions with solutions
Master Geometry with our comprehensive worksheet collection — proofs, constructions, transformations, and more
Geometric reasoning, proofs, and rigid motions
Geometry is built on a foundation of undefined terms (point, line, plane), definitions, postulates, and theorems. Every geometric fact we use must be justified by one of these. **Undefined Terms:** • Point — no size, just location • Line — no width, extends forever in both directions • Plane — no thickness, extends forever in all directions **Key Postulates:** • Through any two points, there is exactly one line • Through any three non-collinear points, there is exactly one plane • If two points lie in a plane, the line through them lies in that plane • If two planes intersect, they intersect in exactly one line **Types of Proofs:** • Two-column proof: Statements | Reasons • Paragraph proof: Written in sentence form • Flow proof: Uses arrows to show logical flow **Why This Matters:** Proof-writing develops logical thinking used in law, medicine, computer science, and engineering. Every theorem in geometry — and every formula you use — was proven from these basic foundations.
Write a two-column proof: Given that ∠1 and ∠2 are supplementary and m∠1 = 110°, prove that m∠2 = 70°.
Given: Points A, B, and C are collinear with B between A and C. AB = 3x + 1, BC = 2x + 4, AC = 26. Find x and all segment lengths.
Using deductive reasoning, prove: If two angles are both supplementary to the same angle, then they are congruent.
Students often struggle with the difference between a postulate and a theorem. I tell them: a postulate is something we AGREE to accept (like rules of a game), while a theorem is something we PROVE. Every proof must have a reason for every statement — "it looks right" is never a valid reason in geometry!
Transformations move or resize figures in the coordinate plane. Understanding transformations is essential for geometry, computer graphics, and physics. **Rigid Motions (Isometries) — preserve shape AND size:** • Translation: (x, y) → (x + a, y + b) — slides the figure • Reflection over x-axis: (x, y) → (x, −y) • Reflection over y-axis: (x, y) → (−x, y) • Reflection over y = x: (x, y) → (y, x) • Rotation 90° CCW about origin: (x, y) → (−y, x) • Rotation 180° about origin: (x, y) → (−x, −y) • Rotation 270° CCW (= 90° CW): (x, y) → (y, −x) **Dilation — preserves shape but NOT size:** • Dilation with center at origin, scale factor k: (x, y) → (kx, ky) • If k > 1: enlargement; if 0 < k < 1: reduction • Dilation preserves angle measures and creates similar figures **Composition of Transformations:** Applying two or more transformations in sequence.
Triangle ABC has vertices A(1, 2), B(3, 5), C(4, 1). Translate the triangle by (x + 3, y − 2). Find the image vertices.
Reflect point P(3, −4) over the y-axis, then rotate the result 90° counterclockwise about the origin.
Dilate triangle with vertices A(2, 4), B(6, 4), C(4, 8) using a scale factor of 1/2 centered at the origin.
I have students use tracing paper to physically perform transformations before working algebraically. For rotations, I teach the "hand trick": for 90° CCW, switch x and y then negate the new x. Always label pre-image with original letters and image with prime notation (A, B, C → A', B', C').
Two triangles are congruent if and only if all three pairs of corresponding sides and all three pairs of corresponding angles are congruent. However, we don't need to check all six — just the right combination! **The Five Congruence Shortcuts:** • SSS: All three sides match • SAS: Two sides and the INCLUDED angle (between those sides) • ASA: Two angles and the INCLUDED side (between those angles) • AAS: Two angles and a NON-included side • HL: Right triangles only — hypotenuse and one leg **NOT Valid Shortcuts:** • AAA — proves similarity, NOT congruence • SSA — does NOT guarantee congruence (the "ambiguous case") **CPCTC Strategy:** First prove triangles congruent using SSS/SAS/ASA/AAS/HL, then use CPCTC to prove individual parts congruent.
Given: AB ≅ DE, BC ≅ EF, AC ≅ DF. Prove: △ABC ≅ △DEF.
Given: ∠A ≅ ∠D, AB ≅ DE, ∠B ≅ ∠E. Prove: △ABC ≅ △DEF, then find BC if EF = 8.
In right triangles △PQR and △XYZ, ∠Q = ∠Y = 90°, PR ≅ XZ (hypotenuses), and PQ ≅ XY (legs). Are the triangles congruent?
Students confuse SAS with SSA. I use this memory trick: SAS is valid because the angle is BETWEEN the two sides (like a sandwich — the angle is the filling). SSA is NOT valid because the angle is on the outside. Also remind students: AAA only proves similarity, not congruence — two triangles can have the same angles but be different sizes!
Proportional reasoning, trigonometry, and quadrilaterals
Similar triangles have the same shape but different sizes. The three similarity shortcuts are: **AA (Angle-Angle):** If two pairs of angles are equal, the triangles are similar. (Since angles sum to 180°, two equal angles guarantee the third is also equal.) **SAS~ (Side-Angle-Side Similarity):** Two sides in proportion AND the included angle equal. **SSS~ (Side-Side-Side Similarity):** All three pairs of sides in proportion. **Key Proportionality Theorems:** • Triangle Proportionality Theorem: If a line is parallel to one side of a triangle and intersects the other two sides, it divides those sides proportionally. • Midsegment Theorem: The midsegment is parallel to the third side and equals half its length. • Geometric Mean: In a right triangle, the altitude to the hypotenuse creates two triangles similar to the original and to each other. **Applications:** Similar triangles are used in surveying, architecture, map-making, and indirect measurement.
In △ABC and △DEF, ∠A = ∠D = 50° and ∠B = ∠E = 70°. Are the triangles similar? Find the scale factor if AB = 6 and DE = 9.
In △PQR, a line parallel to QR intersects PQ at S and PR at T. If PS = 4, SQ = 6, and PT = 5, find TR.
The midsegment of a triangle connects the midpoints of two sides. If the third side has length 18, what is the length of the midsegment?
Students often confuse similarity with congruence. I use this analogy: congruence is like identical twins (same size AND shape), while similarity is like a photo and its enlargement (same shape, different size). For AA similarity, remind students that if two angles match, the third MUST match too — so you only need to check two!
Trigonometry connects angle measures to side length ratios in right triangles. The three primary ratios are: **SOH-CAH-TOA:** • sin θ = Opposite / Hypotenuse • cos θ = Adjacent / Hypotenuse • tan θ = Opposite / Adjacent **Special Right Triangles:** • 45-45-90: legs = x, x; hypotenuse = x√2 • 30-60-90: short leg = x; long leg = x√3; hypotenuse = 2x **Finding Missing Sides:** Set up a trig ratio and solve. **Finding Missing Angles:** Use inverse trig (sin⁻¹, cos⁻¹, tan⁻¹). **Pythagorean Theorem:** a² + b² = c² (for any right triangle) **Applications:** Architecture, navigation, physics, engineering — trigonometry is everywhere!
In right triangle ABC with right angle at C, if ∠A = 35° and hypotenuse AB = 20, find BC (opposite side) and AC (adjacent side).
A 30-60-90 triangle has a short leg of 5. Find the other two sides.
A ladder 15 feet long leans against a wall. The base is 6 feet from the wall. What angle does the ladder make with the ground?
I teach SOH-CAH-TOA with the phrase "Some Old Hippos Can Always Hear Their Old Age." For special triangles, I have students draw and memorize the two triangles — they appear on every standardized test. Remind students: inverse trig finds ANGLES, regular trig finds SIDES.
Quadrilaterals are four-sided polygons. Understanding their properties is essential for proofs, coordinate geometry, and real-world applications. **Parallelogram Properties:** • Opposite sides parallel and congruent • Opposite angles congruent • Consecutive angles supplementary • Diagonals bisect each other **Rectangle:** All parallelogram properties + right angles + congruent diagonals **Rhombus:** All parallelogram properties + 4 equal sides + perpendicular diagonals **Square:** All rectangle AND rhombus properties **Trapezoid:** • Exactly one pair of parallel sides • Midsegment = (base₁ + base₂)/2 • Isosceles trapezoid: equal legs, equal base angles, equal diagonals **Hierarchy:** Square ⊂ Rectangle ⊂ Parallelogram ⊂ Quadrilateral Square ⊂ Rhombus ⊂ Parallelogram ⊂ Quadrilateral
In parallelogram ABCD, ∠A = 3x + 10 and ∠C = 5x − 30. Find x and all angles.
The diagonals of rhombus ABCD are AC = 16 and BD = 12. Find the side length of the rhombus.
Trapezoid ABCD has bases AB = 14 and CD = 8. Find the length of the midsegment.
I draw the "family tree" of quadrilaterals on the board — students love seeing how a square is the most special (it qualifies as a rectangle, rhombus, AND parallelogram). For parallelogram proofs, I teach students to always check: opposite sides, opposite angles, and diagonals. These three properties cover almost every parallelogram proof.
Circle theorems, coordinate geometry, and volume
Circles have rich properties connecting angles, arcs, and segments. Mastering these is essential for geometry proofs and standardized tests. **Arc and Angle Relationships:** • Central angle = intercepted arc • Inscribed angle = (1/2) × intercepted arc • Angle formed by two chords inside circle = (1/2)(arc₁ + arc₂) • Angle formed by two secants/tangents outside circle = (1/2)|arc₁ − arc₂| • Tangent-chord angle = (1/2) × intercepted arc **Chord, Secant & Tangent Segment Relationships:** • Two chords intersecting inside: AE × EC = BE × ED • Two secants from external point: whole × external = whole × external • Tangent-secant from external point: tangent² = whole secant × external secant **Inscribed Polygon Theorems:** • Opposite angles of an inscribed quadrilateral are supplementary • An angle inscribed in a semicircle is a right angle (Thales' Theorem)
An inscribed angle intercepts an arc of 110°. What is the measure of the inscribed angle?
Two chords AB and CD intersect inside a circle at point E. AE = 4, EB = 9, CE = 6. Find ED.
A tangent and a secant are drawn from external point P. The tangent touches at T (PT = 8). The secant passes through the circle with external segment = 4 and whole length = 16. Verify the relationship.
I group circle angle theorems by WHERE the vertex is: (1) At the CENTER = central angle = arc. (2) ON the circle = inscribed angle = half the arc. (3) INSIDE the circle = half the SUM of arcs. (4) OUTSIDE the circle = half the DIFFERENCE of arcs. This "location determines the formula" approach helps students remember all four cases.
**2D Area Formulas:** • Rectangle: A = lw • Triangle: A = (1/2)bh • Parallelogram: A = bh • Trapezoid: A = (1/2)(b₁ + b₂)h • Circle: A = πr² • Regular polygon: A = (1/2) × perimeter × apothem **3D Volume Formulas:** • Prism/Cylinder: V = Bh (B = base area) • Pyramid/Cone: V = (1/3)Bh • Sphere: V = (4/3)πr³ **3D Surface Area Formulas:** • Rectangular prism: SA = 2(lw + lh + wh) • Cylinder: SA = 2πr² + 2πrh • Cone: SA = πr² + πrl (l = slant height) • Sphere: SA = 4πr² **Cavalieri's Principle:** Two solids with equal heights and equal cross-sections at every level have equal volumes. This explains why a cylinder and an oblique cylinder with the same base and height have the same volume.
Find the volume and surface area of a cylinder with radius 4 cm and height 10 cm.
A square pyramid has a base side of 6 m and a height of 8 m. Find its volume and lateral surface area.
Find the volume of a sphere with radius 6 inches.
I group formulas by the rule: "Prisms and cylinders = Bh; Pyramids and cones = (1/3)Bh; Sphere = (4/3)πr³." The factor of 1/3 for pyramids/cones can be demonstrated by filling a cone-shaped cup three times to fill a cylinder of the same base and height. Students remember what they see!
Coordinate geometry connects algebra and geometry by placing figures on the coordinate plane. **Key Formulas:** • Distance: d = √[(x₂−x₁)² + (y₂−y₁)²] • Midpoint: M = ((x₁+x₂)/2, (y₁+y₂)/2) • Slope: m = (y₂−y₁)/(x₂−x₁) **Line Relationships:** • Parallel: same slope, different y-intercept • Perpendicular: slopes are negative reciprocals (m₁ × m₂ = −1) **Equation of a Circle:** • (x − h)² + (y − k)² = r² • Center (h, k), radius r • To find center/radius from general form, complete the square **Proving Geometric Properties with Coordinates:** • Prove a quadrilateral is a parallelogram: show opposite sides have equal slopes • Prove a triangle is a right triangle: show two sides have perpendicular slopes • Prove a triangle is isosceles: show two sides have equal lengths
Find the distance and midpoint between A(1, 3) and B(7, 11).
Write the equation of a circle with center (3, −2) and radius 5.
Prove that the quadrilateral with vertices A(0,0), B(4,0), C(5,3), D(1,3) is a parallelogram.
I remind students that the distance formula is just the Pythagorean Theorem in disguise — the horizontal and vertical distances are the legs, and the distance is the hypotenuse. For circle equations, I emphasize that the signs inside the parentheses are OPPOSITE to the center coordinates: center (3, −2) gives (x − 3)² + (y + 2)².
**Parallel Lines Cut by a Transversal:** When two parallel lines are cut by a transversal: • Corresponding angles are congruent (F-pattern) • Alternate interior angles are congruent (Z-pattern) • Alternate exterior angles are congruent • Co-interior (same-side interior) angles are supplementary (C-pattern) **Triangle Theorems:** • Triangle Sum: ∠A + ∠B + ∠C = 180° • Exterior Angle: exterior angle = sum of two non-adjacent interior angles • Isosceles Triangle: base angles are congruent • Triangle Inequality: sum of any two sides > third side **Polygon Angle Sums:** • Interior angle sum = (n − 2) × 180° • Each interior angle of regular polygon = (n − 2) × 180° / n • Exterior angle sum of any convex polygon = 360° • Each exterior angle of regular polygon = 360° / n
Two parallel lines are cut by a transversal. One angle measures 3x + 15 and its alternate interior angle measures 5x − 25. Find x and both angles.
Find the sum of interior angles of a decagon (10 sides) and each interior angle of a regular decagon.
In triangle ABC, ∠A = 50° and ∠B = 70°. An exterior angle at C is formed. Find the exterior angle.
I teach parallel line angle pairs with letter patterns: F = corresponding, Z = alternate interior, C = co-interior. Students draw these letters on the diagram and immediately identify the angle relationship. For polygon angle sums, I show students the "triangle fan" — any polygon can be divided into (n−2) triangles, each contributing 180°.