Rhombus diagonals cross at 90°
Which shapes cross their diagonals at a right angle?
Crossing at a right angle?
In a square the diagonals cross at 90°. In a leaning parallelogram they cross at some other angle. What about a rhombus, a leaning shape with four equal sides?
At what angle do the diagonals of a rhombus cross?
What this lesson covers
The idea
The diagonals of a rhombus intersect each other at an angle of 90°.
Crossing at a right angle?
In a square the diagonals cross at 90°. In a leaning parallelogram they cross at some other angle. What about a rhombus, a leaning shape with four equal sides?
At what angle do the diagonals of a rhombus cross?
- Always 90°
- Always 60°
- It depends on the rhombus
Measure the crossing
Both diagonals are drawn, and the angle where they cross is measured. Drag B and D. For which shapes does the angle read 90°? What do those shapes have in common?
A right angle in a rhombus
The diagonals of a rhombus intersect each other at an angle of 90°.
Every shape that gave 90° had four equal sides: a rhombus. The right angle came with the equal sides.
Why? Compare ∆AOB and ∆COB: AB = CB (sides of a rhombus), OA = OC (the diagonals bisect each other), and OB is common. So they are congruent by SSS, and ∠AOB = ∠COB.
These two angles lie on a straight line, so ∠AOB + ∠COB = 180°. Equal angles that add up to 180° are 90° each.
Notes
The diagonals of a rhombus intersect each other at an angle of 90°.
Check yourself
The diagonals of rhombus ABCD meet at O. How large is ∠AOB?
Answer: 90 °
The diagonals of a rhombus intersect at 90°, so ∠AOB = 90°.
In rhombus ABCD the diagonals meet at O, and ∠OAB = 35°. How large is ∠OBA?
Answer: 55 °
The diagonals cross at 90°, so ∠AOB = 90°. In triangle AOB, 35° + 90° + ∠OBA = 180°, so ∠OBA = 55°.
Which quadrilateral always has diagonals that cross at 90°?
In rhombus ABCD with diagonals meeting at O, why is ∆AOB ≅ ∆COB?
- A rhombus — correct. Yes! In every rhombus the diagonals intersect at a right angle.
- A rectangle. A 6 cm by 4 cm rectangle has diagonals that cross at an angle that is not 90°. Only a square, a special rectangle, does.
- Any parallelogram. In a leaning parallelogram with unequal sides, the diagonals do not cross at 90°.
- AB = CB, OA = OC and OB is common: SSS — correct. Yes! The sides AB and CB of a rhombus are equal, the diagonals bisect each other so OA = OC, and OB is shared.
- Because ∠AOB = 90°. That is what we want to show. It cannot be used to prove the triangles congruent.
- Because the triangles look alike. Looking alike is not a proof. We need matching sides: SSS.