Topic A: Rigid Motions & Congruence
Lessons 1–6 · Defining congruence through transformations.
01–04 — Lessons 1–4
Congruence via Rigid Motions
- Two figures are congruent if one can be mapped to the other by a rigid motion (or composition of rigid motions).
- Congruent figures have the same shape AND size.
- Corresponding parts of congruent figures are congruent.
- Write congruence statements in order of corresponding parts: △ABC ≅ △DEF means A↔D, B↔E, C↔F.
- Use rigid motions to identify which transformation maps one to the other.
△ABC ≅ △DEF means: ∠A ≅ ∠D, ∠B ≅ ∠E, ∠C ≅ ∠F, AB ≅ DE, BC ≅ EF, AC ≅ DF.
Practice — Lessons 1–4
Q1 · Multiple Choice
If △ABC ≅ △DEF, which pair of corresponding angles are congruent?
Q2 · Multiple Choice
Two figures are congruent if and only if:
05–06 — Lessons 5–6
Properties of Parallelograms
- Parallelogram: opposite sides parallel and congruent; opposite angles congruent; diagonals bisect each other.
- Rectangle: parallelogram with 4 right angles; diagonals are congruent.
- Rhombus: parallelogram with 4 congruent sides; diagonals are perpendicular.
- Square: both a rectangle and a rhombus.
- Prove it's a parallelogram by showing: 2 pairs of parallel sides, OR 2 pairs of congruent opposite sides, OR diagonals bisect each other.
In parallelogram ABCD: AB ∥ CD, AB ≅ CD, ∠A ≅ ∠C, diagonals AC and BD bisect each other.
Practice — Lessons 5–6
Q3 · Multiple Choice
In a parallelogram, the diagonals:
Q4 · Multiple Choice
Which quadrilateral has both perpendicular AND congruent diagonals?
Topic B: Triangle Congruence Criteria
Lessons 7–12 · SSS, SAS, ASA, AAS, and HL.
07–09 — Lessons 7–9
Triangle Congruence Shortcuts
- SSS: three pairs of congruent sides → triangles congruent.
- SAS: two sides and the included angle → congruent.
- ASA: two angles and the included side → congruent.
- AAS: two angles and a non-included side → congruent.
- HL (Hypotenuse-Leg): right triangles only — hypotenuse + leg → congruent.
- AAA and SSA do NOT prove congruence.
Given two sides AB ≅ DE and BC ≅ EF, and included ∠B ≅ ∠E → △ABC ≅ △DEF by SAS.
Practice — Lessons 7–9
Q5 · Multiple Choice
Two triangles have all three pairs of angles congruent (AAA). Are they congruent?
Q6 · Multiple Choice
You know AB ≅ DE, BC ≅ EF, and ∠B ≅ ∠E. What congruence criterion applies?
10–12 — Lessons 10–12
CPCTC Proofs
Once triangles are proven congruent, CPCTC allows you to conclude that corresponding parts are congruent.
- CPCTC: Corresponding Parts of Congruent Triangles are Congruent.
- Used as a reason AFTER you establish triangle congruence.
- Common proof structure: Given → prove triangles congruent (SSS/SAS/etc.) → apply CPCTC.
- Can prove segments, angles, or even parallel/perpendicular lines using CPCTC.
Prove △ABD ≅ △CBD by SAS. Then AB ≅ CB by CPCTC.
Practice — Lessons 10–12
Q7 · Multiple Choice
CPCTC can be used as a reason in a proof:
Q8 · Multiple Choice
CPCTC stands for:
Topic C: Quadrilateral Proofs
Lessons 13–18 · Proving quadrilaterals are parallelograms, rectangles, rhombi, or squares.
13–18 — Lessons 13–18
Proving Special Quadrilaterals
- Prove a parallelogram by: opposite sides ∥, opposite sides ≅, diagonals bisect each other, or one pair of sides both ∥ and ≅.
- Prove a rectangle: parallelogram + one right angle, OR parallelogram + congruent diagonals.
- Prove a rhombus: parallelogram + two consecutive sides ≅, OR parallelogram + perpendicular diagonals.
- Prove a square: show it is both a rectangle AND a rhombus.
- Use coordinates: slope for parallel/perpendicular, distance formula for congruent sides.
Given that both pairs of opposite sides of ABCD are congruent → ABCD is a parallelogram.
Practice — Lessons 13–18
Q9 · Multiple Choice
To prove a quadrilateral is a rectangle, you need to show it is a parallelogram and:
Q10 · Multiple Choice
In a rhombus, the diagonals are always: