Topic A: Compass & Straightedge Constructions
Lessons 1–6 · Basic constructions using only compass and straightedge.
01–03 — Lessons 1–3
Basic Constructions
Geometric constructions use only a compass (for circles/arcs) and a straightedge (for lines — no measurements).
- Copy a segment: use compass to transfer length.
- Copy an angle: reproduce the angle's arc and chord at a new vertex.
- Perpendicular bisector: find midpoint of segment; line perpendicular to it.
- Angle bisector: divide an angle into two equal halves.
- Constructions are exact — no measuring with rulers or protractors.
Perpendicular bisector of AB: set compass > AB/2, draw arcs from A and B, connect the intersections.
Practice — Lessons 1–3
Q1 · Multiple Choice
What is the purpose of the perpendicular bisector of a segment?
Q2 · Multiple Choice
In a construction, a 'straightedge' is used to:
04–06 — Lessons 4–6
Circle Constructions & Special Figures
- Inscribed circle: tangent to all sides of a triangle (incircle) — constructed using angle bisectors.
- Circumscribed circle: passes through all vertices (circumcircle) — constructed using perpendicular bisectors.
- Equilateral triangle: set compass = side length, draw arcs from endpoints.
- Regular hexagon: radius = side length, mark 6 points around center circle.
- Square: use perpendicular bisectors and circle.
To construct an equilateral triangle: draw segment AB, arc from A (radius AB), arc from B — their intersection is C.
Practice — Lessons 4–6
Q3 · Multiple Choice
The circumscribed circle of a triangle passes through:
Q4 · Multiple Choice
Which center is found using angle bisectors?
Topic B: Formal Proof
Lessons 7–12 · Two-column proofs, paragraph proofs, and geometric reasoning.
07–09 — Lessons 7–9
Introduction to Geometric Proof
Geometric proof uses definitions, postulates, and previously proven theorems to establish that a statement is always true.
- Given: information you are told is true.
- Prove: the statement you must show is true.
- Two-column proof: statements on left, reasons on right.
- Reasons: definition, postulate, theorem, or given.
- Each step follows logically from what came before.
Given: AB ≅ CD, CD ≅ EF. Prove: AB ≅ EF. Reason: Transitive Property of Congruence.
Practice — Lessons 7–9
Q5 · Multiple Choice
In a two-column proof, the right column lists:
Q6 · Multiple Choice
If AB = BC and BC = CD, then AB = CD by the:
10–12 — Lessons 10–12
Angle Relationships & Proofs
- Vertical angles: congruent. (Formed by two intersecting lines, across from each other.)
- Linear pair: supplementary (add to 180°).
- Complementary: two angles adding to 90°.
- Transversal + parallel lines: alternate interior angles ≅, corresponding angles ≅, co-interior angles supplementary.
- Use these theorems as reasons in proofs.
Given: ∠1 and ∠2 are vertical angles. Prove: ∠1 ≅ ∠2. Reason: Vertical Angles Theorem.
Practice — Lessons 10–12
Q7 · Multiple Choice
Two parallel lines are cut by a transversal. Alternate interior angles are:
Q8 · Short Answer
If ∠A and ∠B are supplementary and ∠A = 65°, what is ∠B?
Topic C: Rigid Motions
Lessons 13–19 · Translations, reflections, rotations, and composites.
13–15 — Lessons 13–15
Translations & Reflections
- Rigid motion (isometry): a transformation that preserves distance and angle measure.
- Translation: slides a figure. Vector notation (a, b): move right a, up b.
- Reflection: flips a figure over a line of reflection.
- Reflect over x-axis: (x, y) → (x, −y).
- Reflect over y-axis: (x, y) → (−x, y).
- Reflect over y = x: (x, y) → (y, x).
Point A(3, 5) reflected over x-axis → A'(3, −5). Reflected over y-axis → A'(−3, 5).
Practice — Lessons 13–15
Q9 · Multiple Choice
A translation moves a figure 3 right and 2 down. Point (5, 7) maps to:
Q10 · Short Answer
Reflect (4, −3) over the y-axis. Give the x-coordinate of the image.
16–19 — Lessons 16–19
Rotations & Composition
- Rotation: turns a figure around a center point by an angle.
- Rotate 90° CCW about origin: (x, y) → (−y, x).
- Rotate 180° about origin: (x, y) → (−x, −y).
- Rotate 270° CCW (= 90° CW): (x, y) → (y, −x).
- Composition: apply two or more transformations in sequence.
- A glide reflection = translation + reflection.
Rotate (2, 3) by 90° CCW about origin: (2, 3) → (−3, 2).
Practice — Lessons 16–19
Q11 · Multiple Choice
Point (5, −2) is rotated 180° about the origin. The image is:
Q12 · Multiple Choice
A composition of two reflections over parallel lines is equivalent to a: