Geometry worksheet with proofs for parallelograms and rectangles.

Let's solve each of the three proofs step by step, filling in the reasons for each statement. These are standard geometry proofs involving parallelograms, rectangles, and their properties.

---

🔷 Proof 1: Opposite Angles in a Parallelogram are Congruent

Given: ABCD is a parallelogram
Prove: ∠ADC ≅ ∠CBA

| Statements | Reasons |
|-----------|--------|
| 1. ABCD is a parallelogram | 1. Given |
| 2. AB ∥ CD and AD ∥ CB | 2. Definition of a parallelogram (opposite sides are parallel) |
| 3. ∠DAC ≅ ∠BCA and ∠BAC ≅ ∠DCA | 3. Alternate Interior Angles Theorem (since AB ∥ CD and AD ∥ CB, with diagonal AC as transversal) |
| 4. AC ≅ CA | 4. Reflexive Property of Congruence |
| 5. ΔADC ≅ ΔCBA | 5. ASA (Angle-Side-Angle) Congruence Postulate: ∠DAC ≅ ∠BCA, AC ≅ CA, ∠DCA ≅ ∠BAC |
| 6. ∠ADC ≅ ∠CBA | 6. CPCTC (Corresponding Parts of Congruent Triangles are Congruent) |

✔ Explanation: We used diagonal AC to create two triangles. By proving the triangles congruent via ASA, we conclude that opposite angles are congruent.

---

🔷 Proof 2: Diagonals of a Parallelogram Bisect Each Other

Given: JKLM is a parallelogram
Prove: JN ≅ NL and KN ≅ NM

| Statements | Reasons |
|-----------|--------|
| 1. JKLM is a parallelogram | 1. Given |
| 2. ∠JMK ≅ ∠LKM and ∠MJL ≅ ∠KLJ | 2. Alternate Interior Angles Theorem (JM ∥ LK and JL ∥ MK, diagonals as transversals) |
| 3. JM ≅ LK | 3. Opposite sides of a parallelogram are congruent |
| 4. ΔJMN ≅ ΔLKN | 4. AAS (Angle-Angle-Side) Congruence: ∠JMK ≅ ∠LKM, ∠MJL ≅ ∠KLJ, JM ≅ LK |
| 5. JN ≅ NL and KN ≅ NM | 5. CPCTC (Corresponding Parts of Congruent Triangles are Congruent) |

✔ Explanation: We use the diagonals JL and KM intersecting at point N. By showing that triangles JMN and LKN are congruent (using AAS), we deduce that the segments formed by the intersection (the halves of the diagonals) are congruent — meaning diagonals bisect each other.

---

🔷 Proof 3: Diagonals of a Rectangle are Congruent

Given: DEFG is a rectangle
Prove: DF ≅ EG

| Statements | Reasons |
|-----------|--------|
| 1. DEFG is a rectangle | 1. Given |
| 2. ∠DGF and ∠EFG are right angles | 2. Definition of a rectangle (all angles are right angles) |
| 3. ∠DGF ≅ ∠EFG | 3. All right angles are congruent |
| 4. DG ≅ EF | 4. Opposite sides of a rectangle are congruent |
| 5. FG ≅ GF | 5. Reflexive Property of Congruence |
| 6. ΔDFG ≅ ΔEGF | 6. SAS (Side-Angle-Side) Congruence: DG ≅ EF, ∠DGF ≅ ∠EFG, FG ≅ GF |
| 7. DF ≅ EG | 7. CPCTC (Corresponding Parts of Congruent Triangles are Congruent) |

✔ Explanation: Since a rectangle is a special type of parallelogram with four right angles, we use two triangles formed by the diagonals. Using SAS, we prove the triangles congruent and thus the diagonals are congruent.

---

✔ Summary of Key Concepts Used:

- Definition of a parallelogram: Opposite sides are parallel.
- Alternate Interior Angles Theorem: When two parallel lines are cut by a transversal, alternate interior angles are congruent.
- Reflexive Property: A segment or angle is congruent to itself.
- ASA & AAS: Methods to prove triangle congruence.
- CPCTC: Once triangles are proven congruent, their corresponding parts are congruent.
- Rectangle properties: All angles are 90°, opposite sides are congruent, and diagonals are congruent.

These proofs demonstrate fundamental geometric reasoning using triangle congruence and properties of quadrilaterals.

Let me know if you'd like this formatted as a printable PDF or need diagrams!