Quadrilaterals · Lesson 7 of 13
Rhombuses and Their Angles
“Understand rhombuses through equal sides, parallelogram structure and angle relationships.”
• Construct and define a rhombus. • Prove that it is a parallelogram. • Find its opposite and adjacent angles. • Explain the overlap between rectangles, rhombuses and squares.
Cut four cardboard strips to exactly the same length and join them into a closed frame. Can you make only a square? Keep the lengths fixed and tilt the joints. The frame can become slanted while all four sides stay equal. This gives a family defined by side equality, just as rectangles were defined by right angles. The important question is what those equal sides force us to know about the opposite sides and angles.
We will construct one such frame with a 50° angle. Congruent and isosceles triangles will reveal its other angles and connect it with the parallelograms you already know.
4.4 Quadrilaterals with Equal Sidelengths
Draw AB and AD with the same length and ∠DAB = 50°. Set your compass to that common length. Draw arcs from B and D, and choose their intersection C on the other side of BD from A. Join BC and CD.
All four boundary sides now match: AB = BC = CD = DA. Choosing C distinct from A is essential; returning to A would not give four vertices. A different starting angle between 0° and 180° gives another member of the same family.
A quadrilateral in which all four sides have the same length.
Deduction 9 — What can we say about the angles in a rhombus?
Consider rhombus GAME and draw diagonal AE. In triangle GAE, GA = GE, so the angles at A and E are equal. In triangle MAE, MA = ME, so its angles at A and E are equal too.
The two triangles GAE and MAE are congruent by SSS: GA = MA, GE = ME, and AE is common. Their matching angles are equal. Combining this with the isosceles facts shows that all four small angles made by AE at A and E have the same size.
Equal alternate angles along AE imply EM ∥ GA and GE ∥ AM. Thus a rhombus is a parallelogram. Now the established parallelogram rules apply: opposite angles are equal and adjacent angles add to 180°.
Problem
Find all four angles of rhombus ABCD if ∠A = 50°.
Problem
Adjacent angles of a rhombus are x and 3x. Find them.
A rectangle and a rhombus are both parallelograms, but each adds a different condition. A rectangle requires right angles; a rhombus requires equal sides. A square satisfies both, so it belongs to the overlap.
Every square is a rhombus. A non-square rhombus, such as the 50°–130° example, proves that not every rhombus is a square. Likewise, a 6 cm by 3 cm rectangle proves that not every rectangle is a rhombus.
Problem
A rhombus has one right angle. Must it be a square?
Quiz
A rhombus is defined by:
Every rhombus is a parallelogram.
If one rhombus angle is 40°, an adjacent angle is:
Opposite angles of a rhombus are:
Is every rhombus a square?
Practice Problems
- A rhombus has one angle 72°. Find the other three.
- Adjacent rhombus angles are x and 3x. Find all four.
- Explain why a square belongs to both rectangles and rhombuses.
- Give a rhombus that is not a square and explain why.
- Why is every rhombus a parallelogram? Explain using a diagonal.
1. Adjacent angles are 180° − 72° = 108°. 2. The opposite angle is 72°, so the remaining angles are 108°, 72°, 108°.
1. 4x = 180°, so x = 45°. 2. The angles are 45°, 135°, 45°, 135°.
1. Its four right angles meet the rectangle condition. 2. Its four equal sides meet the rhombus condition.
1. Construct four equal sides with angles 60°, 120°, 60°, 120°. 2. It is a rhombus because its sides are equal. 3. Its angles are not 90°, so it is not a square.
1. A diagonal splits it into two triangles congruent by SSS. 2. Matching angles across the diagonal give equal alternate angles. 3. The converse alternate-angle test gives both pairs of opposite sides parallel, which defines a parallelogram.
Key Takeaways
• Four equal sides define a rhombus. • Congruence proves that its opposite sides are parallel. • A rhombus is therefore a parallelogram and inherits its angle rules. • Rectangles and rhombuses overlap in the squares. • One right angle in a rhombus forces a square.