A diagonal of a rectangle is inclined to one side of the rectangle at 35°.
A diagonal of a rectangle is inclined to one side of the rectangle at 35°. The acute angle between the diagonals is
(a) 55°
(b) 70°
(c) 45°
(d) 50°
Let the diagonals $AC$ and $BD$ of rectangle $ABCD$ intersect at $O$. Given that the diagonal $AC$ is inclined to side $AD$ at $35^\circ$. Therefore,
$$\angle OAD = 35^\circ$$
Since $ABCD$ is a rectangle,
$$\angle DAB = 90^\circ$$
Hence,
$$\angle OAB = 90^\circ - 35^\circ = 55^\circ$$
Also, the diagonals of a rectangle are equal and bisect each other. Therefore,
$$AC = BD$$
and
$$OA = \frac{AC}{2}, \qquad OB = \frac{BD}{2}$$
Since $AC = BD$, we get
$$OA = OB$$
Thus, $\triangle AOB$ is an isosceles triangle, so its base angles are equal:
$$\angle OAB = \angle OBA = 55^\circ$$
Using the angle sum property of a triangle,
$$\angle AOB + \angle OAB + \angle OBA = 180^\circ$$
$$\angle AOB + 55^\circ + 55^\circ = 180^\circ$$
$$\angle AOB = 180^\circ - 110^\circ = 70^\circ$$
Therefore, the acute angle between the diagonals is
$$\boxed{70^\circ}$$
Key Properties of Rectangle Diagonals You Must Know
Before solving, make sure these four properties are committed to memory. They are tested directly in NCERT exercises and are foundational for geometry problems at every level.
| Property | Statement | Used in This Problem? |
|---|---|---|
| Equal Diagonals | $$AC = BD$$ | ✅ Yes |
| Diagonals Bisect Each Other | $$AO = OC = BO = OD$$ | ✅ Yes |
| Each Angle of a Rectangle is $90^\circ$ | $$\angle A = \angle B = \angle C = \angle D = 90^\circ$$ | ✅ Yes |
| Opposite Sides are Equal and Parallel | $$AB = CD,\quad AD = BC$$ | Indirectly |
For a deeper understanding of quadrilateral properties and how they connect to NCERT topics, explore our NCERT Solutions for Class 9 Maths — worked solutions aligned to the latest CBSE curriculum.
Common Mistakes Students Make in Rectangle Angle Problems
Rectangle-diagonal questions are usually straightforward, but many students lose marks because they identify the wrong angle or forget key properties of diagonals. Review these common mistakes before attempting similar problems.
| Mistake | Why It Happens | How to Avoid It |
|---|---|---|
| Choosing $55^\circ$ as the answer | Confusing $\angle OAB$ with $\angle AOB$. | Always identify where the required angle is located before solving. |
| Forgetting that diagonals bisect each other | Treating $\triangle OAB$ as a scalene triangle. | Write down $$OA = OB$$ as one of your first steps. |
| Using $35^\circ$ directly in the angle sum | Not finding $\angle OAB$ first. | Use the rectangle corner: $$\angle DAB = 90^\circ$$ Therefore, $$\angle OAB = 90^\circ - 35^\circ = 55^\circ$$ |
| Choosing $110^\circ$ | Finding the obtuse angle formed by the diagonals at $O$. | Always check whether the question asks for the acute angle or the obtuse angle. |
| Not drawing a diagram | Trying to solve the question mentally. | Sketch the rectangle and mark the intersection point $O$ before starting any calculations. |
Similar Practice Problems
Once you understand the method, test your skills on these similar rectangle-diagonal problems. They follow the same geometric principles used in the solved example above.
1. Rectangle Diagonal at $25^\circ$
A diagonal of a rectangle is inclined to one side at $25^\circ$. Find the acute angle between the diagonals.
Answer: $50^\circ$
2. Acute Angle Between Diagonals is $80^\circ$
The acute angle between the diagonals of a rectangle is $80^\circ$. Find the angle the diagonal makes with the longer side.
Answer: $40^\circ$
3. Rectangle Diagonal at $40^\circ$
In a rectangle, one diagonal makes an angle of $40^\circ$ with a side. Find both angles formed at the intersection of the diagonals.
Answer: $80^\circ$ and $100^\circ$
For fully worked NCERT-style solutions to quadrilateral problems like these, visit NCERT Solutions for Class 11 Maths for advanced geometry connections, or explore NCERT Solutions for Class 12 Maths to see how these angle concepts extend into vectors and coordinate geometry.
Frequently Asked Questions
Find answers to common questions.
Do the diagonals of a rectangle bisect each other at right angles?
No. The diagonals of a rectangle bisect each other (meet at their midpoints), but they do NOT bisect each other at right angles unless the rectangle is also a square. In a square, both equal-length and perpendicular-bisecting properties hold. In a general rectangle, the angle between diagonals depends on the ratio of length to breadth.
Why are the diagonals of a rectangle equal in length?
Rectangle diagonals are equal because a rectangle is a parallelogram with all angles equal to 90°. Using the Pythagorean theorem, both diagonals span the same two sides (length and breadth), giving them identical lengths. This property is unique to rectangles among parallelograms — rhombuses and general parallelograms do not have equal diagonals.
What is the acute angle between the diagonals of a rectangle if one diagonal is inclined at 35° to a side?
The acute angle between the diagonals is 70°. Since the diagonals of a rectangle are equal and bisect each other, triangle OAB (formed at the intersection O) is isosceles with OA = OB. With ∠OAB = ∠OBA = 55°, the angle at O works out to 180° − 55° − 55° = 70° by the angle sum property of a triangle
How is this type of problem relevant to JEE preparation?
At the JEE foundation level (Class 9–10), rectangle and quadrilateral angle problems build the geometric intuition needed for JEE Main coordinate geometry and JEE Advanced geometry problems. Understanding why triangles formed inside a rectangle are isosceles directly helps with circle-chord problems, vector problems, and transformation geometry — all of which appear
Which NCERT chapter covers rectangle diagonal properties?
Rectangle diagonal properties are covered in NCERT Class 9 Maths, Chapter 8 — Quadrilaterals. The theorems on parallelogram diagonals apply directly to rectangles as a special case. These are standard topics for CBSE board exams and form the base for coordinate geometry studied in Class 10 and beyond.
What is the formula for the acute angle between the diagonals of a rectangle?
If a diagonal makes an angle θ with one side of the rectangle, the acute angle between the two diagonals is 2θ. This comes directly from the isosceles triangle formed at the centre: both base angles equal θ, so the apex angle = 180° − 2θ — but the other angle at O (vertically opposite) = 2θ, which is the acute angle when θ < 45°.