Overview
- Topic: Theorems and properties of a rhombus.
- Focus: Basic properties, diagonal properties, angle relationships, worked examples, and true/false statements.
- Purpose: Summarize key facts and demonstrate example problems.
Properties of a Rhombus
- A rhombus is a parallelogram with all sides congruent.
- Opposite angles are congruent.
- Consecutive angles are supplementary (sum to 180°).
- Diagonals bisect each other (they meet at their midpoints).
- Diagonals are perpendicular (they meet at right angles).
- Each diagonal bisects the vertex angles it connects.
Diagonal Properties (Theorems)
- The diagonals of a rhombus are perpendicular, forming four right angles.
- Each diagonal bisects a pair of opposite angles, dividing them into two equal angles.
- Because diagonals bisect each other, corresponding half-segments are congruent.
Key Terms and Definitions
- Rhombus: Parallelogram with four equal sides.
- Opposite angles: Angles across the figure from each other, congruent in a rhombus.
- Consecutive angles: Adjacent angles along a side, supplementary in a rhombus.
- Diagonals bisect: A diagonal divides an angle or segment into two equal parts.
- Perpendicular: Two lines that intersect at 90°.
Worked Example 1 — Rhombus CARE (Given ∠CAR = 96°)
- Given: CARE is a rhombus; measure ∠CAR = 96°.
- Find: ∠CEA, ∠ACE, ∠ECR, ∠CRE.
Table: Angle Measures for Rhombus CARE
| Angle | Reason | Measure (°) |
|---|
| ∠CAR | Given | 96 |
| ∠CER (opposite ∠CAR) | Opposite angles congruent | 96 |
| ∠CEA (half of ∠CER by diagonal bisect) | Diagonal bisects vertex angle | 48 |
| ∠ACE (supplementary to ∠CAR) | Consecutive angles supplementary | 84 |
| ∠ECR (half of ∠ACE by diagonal bisect) | Diagonal bisects vertex angle | 42 |
| ∠CRE (opposite ∠ECR) | Opposite angles congruent | 42 |
- Verification: 96° + 84° = 180° (consecutive angles supplementary).
Worked Example 2 — Numbered Angles in a Rhombus
- Given: One vertex angle (∠D) = 72°; diagonals bisect vertex angles.
- Find: Angles labeled 1–5 (as in diagram logic).
Table: Numbered Angle Results
| Label | Reason | Measure (°) |
|---|
| ∠D | Given | 72 |
| ∠1 + ∠5 (same vertex sum) | Consecutive angles supplementary to ∠D | 108 |
| ∠1 | Diagonal bisects vertex angle (∠C) so ∠1 = ∠5 | 54 |
| ∠5 | Diagonal bisects vertex angle | 54 |
| ∠2 = ∠3 | Opposite/paired by symmetry from diagonals | 54 |
- Note: If ∠1 + ∠5 = 108°, equal halves give 54° each.
True/False Statements (Examples)
- Statement: Diagonals of a rhombus are congruent. — False.
- Statement: A half-diagonal segment is perpendicular to the other half-diagonal. — True (diagonals are perpendicular).
- Statement: Diagonals bisect vertex angles, so related angle pairs are congruent. — True.
- Statement: Opposite consecutive angles sum to 180°. — True.
- Statement: Some specific internal angle equals 90° without information on square property. — False (only diagonals intersect at 90°, not necessarily vertex angles).
Example Problems and Quick Methods
- To find angle bisected by a diagonal: divide the vertex angle by 2.
- To find a consecutive angle: subtract the given angle from 180°.
- To find angles formed by diagonals: use perpendicular property (right angles) and bisected-vertex property.
- To find congruent segments from diagonals: use diagonal bisection (half segments are equal).
Action Items / Practice Steps
- Identify given elements: side congruence, given angle measures, diagonal intersections.
- Apply properties in order: opposite angles, consecutive supplementary, diagonal bisects (angles and segments), perpendicular diagonals.
- Check with simple arithmetic: sums (180°), halves (÷2), and right angles (90°).
Summary
- Core rhombus facts: four equal sides, opposite angles equal, consecutive angles supplementary, diagonals perpendicular, diagonals bisect angles and each other.
- Use these properties to quickly compute unknown angles and segment relationships in problem figures.