Context: We can compute the four angles between the diagonals to be , , , and . Q. Can you find the other angles?

Understand the Question The diagonals of a rectangle are equal in length and bisect each other, dividing the rectangle into four isosceles triangles: , , , and . In any isosceles triangle, angles opposite to equal sides are equal. Using the triangle angle sum property (), we can determine all the angles formed between the diagonals and the sides of the rectangle. Step 1 · Identify Isosceles Triangles Formed by Diagonals Let be the intersection point of the diagonals and . In a rectangle, the diagonals are equal () and bisect each other: Therefore, the four triangles formed by the diagonals—, , , and —are all isosceles triangles. Step 2 · Find Angles in Triangles with Angle In , since , the base angles are equal. Let . Using the angle sum property of a triangle: Similarly, in , (vertically opposite to ) and : Step 3 · Find Angles in Triangles with Angle In , since , let the base angles be . Using the angle sum property: Similarly, in , and : Step 4 · Find the Interior Angles of the Rectangle Combine adjacent angles at each vertex of the rectangle: Each interior angle of the rectangle is . Answer > The remaining angles are: Common Mistakes Assuming Diagonals are Perpendicular: The diagonals of a rectangle are not necessarily perpendicular (unless the rectangle is a square); do not assume . Sign Errors in Subtraction: When solving , ensure signs are distributed properly: , not .

Quadrilaterals | IT

Question 10

Context: We can compute the four angles between the diagonals to be $x$ , $x$ , $180^\circ - x$ , and $180^\circ - x$ .

Q. Can you find the other angles?

Solution

Understand the Question

The diagonals of a rectangle are equal in length and bisect each other, dividing the rectangle into four isosceles triangles: $\triangle AOB$ , $\triangle BOC$ , $\triangle COD$ , and $\triangle DOA$ .
In any isosceles triangle, angles opposite to equal sides are equal.
Using the triangle angle sum property ( $180^\circ$ ), we can determine all the angles formed between the diagonals and the sides of the rectangle.

Step 1 · Identify Isosceles Triangles Formed by Diagonals

Let $O$ be the intersection point of the diagonals $AC$ and $BD$ . Diagram 1

In a rectangle, the diagonals are equal ( $AC = BD$ ) and bisect each other: $AO = OC = BO = OD$

Therefore, the four triangles formed by the diagonals— $\triangle AOB$ , $\triangle BOC$ , $\triangle COD$ , and $\triangle DOA$ —are all isosceles triangles.

Step 2 · Find Angles in Triangles with Angle $x$

In $\triangle AOB$ , since $AO = OB$ , the base angles are equal. Let $\angle OAB = \angle OBA = \alpha$ .

Using the angle sum property of a triangle: $\angle OAB + \angle OBA + \angle AOB = 180^\circ$

$2\alpha + x = 180^\circ$

\begin{aligned} 2\alpha &= 180^\circ - x \\[0.6em] \alpha &= \dfrac{180^\circ - x}{2} \\[0.6em] &= 90^\circ - \dfrac{x}{2} \end{aligned}

$\angle OAB = \angle OBA = 90^\circ - \dfrac{x}{2}$

Similarly, in $\triangle COD$ , $\angle COD = x$ (vertically opposite to $\angle AOB$ ) and $OC = OD$ : $\angle OCD = \angle ODC = 90^\circ - \dfrac{x}{2}$

Step 3 · Find Angles in Triangles with Angle $180^\circ - x$

In $\triangle BOC$ , since $OB = OC$ , let the base angles be $\angle OBC = \angle OCB = \beta$ .

Using the angle sum property: $\angle OBC + \angle OCB + \angle BOC = 180^\circ$

$2\beta + (180^\circ - x) = 180^\circ$

\begin{aligned} 2\beta &= 180^\circ - (180^\circ - x) \\[0.6em] 2\beta &= x \\[0.6em] \beta &= \dfrac{x}{2} \end{aligned}

$\angle OBC = \angle OCB = \dfrac{x}{2}$

Similarly, in $\triangle DOA$ , $\angle DOA = 180^\circ - x$ and $OD = OA$ : $\angle ODA = \angle OAD = \dfrac{x}{2}$

Step 4 · Find the Interior Angles of the Rectangle

Combine adjacent angles at each vertex of the rectangle:

\begin{aligned} \angle DAB &= \angle OAD + \angle OAB = \dfrac{x}{2} + \left(90^\circ - \dfrac{x}{2}\right) = 90^\circ \\[0.6em] \angle ABC &= \angle OBA + \angle OBC = \left(90^\circ - \dfrac{x}{2}\right) + \dfrac{x}{2} = 90^\circ \\[0.6em] \angle BCD &= \angle OCB + \angle OCD = \dfrac{x}{2} + \left(90^\circ - \dfrac{x}{2}\right) = 90^\circ \\[0.6em] \angle CDA &= \angle ODC + \angle ODA = \left(90^\circ - \dfrac{x}{2}\right) + \dfrac{x}{2} = 90^\circ \end{aligned}

Each interior angle of the rectangle is $90^\circ$ .

Answer

The remaining angles are:

$\angle OAD = \angle ODA = \angle OBC = \angle OCB = \dfrac{x}{2}$
$\angle OAB = \angle OBA = \angle OCD = \angle ODC = 90^\circ - \dfrac{x}{2}$
$\angle DAB = \angle ABC = \angle BCD = \angle CDA = 90^\circ$

Common Mistakes

Assuming Diagonals are Perpendicular: The diagonals of a rectangle are not necessarily perpendicular (unless the rectangle is a square); do not assume $x = 90^\circ$ .
Sign Errors in Subtraction: When solving $2\beta + (180^\circ - x) = 180^\circ$ , ensure signs are distributed properly: $180^\circ - (180^\circ - x) = x$ , not $-x$ .

More questions in IT

Observe the following figures.

Figs. (i), (ii), and (iii) are quadrilaterals, and the others are not. Why?

Are there other ways to define a rectangle?

A Carpenter's Problem

A carpenter needs to put together two thin strips of wood, as shown in Fig. 1, so that when a thread is passed through their endpoints, it forms a rectangle. She already has one 8 cm long strip. What should be the length of the other strip? Where should they both be joined?

Let us first model the structure that the carpenter has to make. The strips can be modelled as line segments. They are the diagonals of the quadrilateral formed by their endpoints. For the quadrilateral to be a rectangle, we need to answer the following questions —

What is the length of the other diagonal?
What is the point of intersection of the two diagonals?
What should the angle be between the diagonals?

Can the following equalities be used to establish that $\Delta AOD \cong \Delta COB$ ?

$AO = CO$ (proved above)
$\angle AOB = \angle COD$ (vertically opposite angles)
$AD = CB$

Context: Let us check what quadrilateral we get if we draw the two diagonals such that their lengths are equal, they bisect each other and have an arbitrary angle, say $60^\circ$ , between them as shown in the figure to the right.

Q. Can you find all the remaining angles?

Context: In $\Delta AOB$ , since $OA = OB$ , the angles opposite them are equal, say $a$ .

Q. Can you find the value of $a$ ?

Can we now identify what type of quadrilateral ABCD is?

Notice that its angles all add up to $90^\circ$ ( $30^\circ + 60^\circ$ ).

What can we say about its sides?

Will $ABCD$ remain a rectangle if the angles between the diagonals are changed? Can we generalise this?

Take one of the angles between the diagonals as $x$ .

Q10

Context: We can compute the four angles between the diagonals to be $x$ , $x$ , $180^\circ - x$ , and $180^\circ - x$ .

Q. Can you find the other angles?

Q11

Context: Since we know that $\triangle AOB$ is isosceles, we can denote the measures of both of its base angles by $a$ .

Q. What is the value of $a$ (in degrees) in terms of $x$ ?

Q12

Context: Thus, all four angles of the quadrilateral $ABCD$ are $90^\circ$ .

Q. What can we say about $AB$ and $CD$ , and $AD$ and $BC$ ?

Q13

In the earlier definition, we stated that a rectangle has (a) opposite sides of equal length, and (b) all angles equal to $90^\circ$ . Would we be wrong if we just define a rectangle as a quadrilateral in which all the angles are $90^\circ$ ?

Q14

If you think that this definition is incomplete, try constructing a quadrilateral in which the angles are all $90^\circ$ but the opposite sides are not equal.

Are you able to construct such a quadrilateral?

Q15

Is it wrong to write $\Delta BAD \cong \Delta CDB$ ? Why?

Q16

Can you similarly show that $\text{AB}$ is parallel to $\text{DC}$ ( $\text{AB} \parallel \text{DC}$ )?

Q17

In the quadrilaterals below, are there any non-rectangles?

Q18

Let us consider the Carpenter's Problem again. If the wooden strips have to be placed such that the thread passing through their endpoints forms a square, what must be done?

Q19

What more needs to be done to get equal sidelengths as well? Can this be achieved by properly choosing the angle between the diagonals? See if you can reason and/or experiment to figure this out!

Q20

Can this be used to find the angles $\angle\text{BOA}$ and $\angle\text{BOC}$ formed by the diagonals?

Q21

Context: The diagonals of a square are of equal lengths and bisect each other at right angles.

Q. Using this fact, construct a square with a diagonal of length $8\text{ cm}$ .

Q22

Context: Since a square is a special type of rectangle, all the properties of a rectangle hold true for a square.

Q. Verify if this is true by going through geometric reasoning in Deduction 1 and Deduction 2, and see if they apply to a square as well.

Q23

Q. Similarly, find $\angle 2$ and $\angle 4$ .

Q24

4.2 Angles in a Quadrilateral

Is it possible to construct a quadrilateral with three angles equal to $90^\circ$ and the fourth angle not equal to $90^\circ$ ?

Q25

But why not?

Q26

Are there quadrilaterals that have parallel opposite sides that are not rectangles?

Q27

Construct such a figure by recalling how parallel lines can be constructed using a ruler and a set-square, or a compass and a ruler.

Q28

Context: Consider a parallelogram $ABCD$ with adjacent sides of lengths $4\text{ cm}$ and $5\text{ cm}$ , and an angle of $30^\circ$ between them.

Q. What are the remaining angles of the parallelogram? What are the lengths of the remaining sides? See if you can reason out and/or experiment to figure these out.

Q29

Deduction 7— What can we say about the sides of a parallelogram?

By looking at a parallelogram, it appears that the opposite sides are equal. Can we again use congruence to show this? Which two triangles can be considered for this?

Q30

Is it wrong to write $\Delta \text{ABD} \cong \Delta \text{CBD}$ ? Why?

Q31

Are the diagonals of a parallelogram always equal? Check with the parallelogram that you have constructed.

Q32

Context: We see that the diagonals of a parallelogram need not be equal.

Q. Do they bisect each other (do they intersect at their midpoints)? Reason and/or experiment to figure this out.

Q33

Is it wrong to write $\Delta \text{AOE} \cong \Delta \text{SOY}$ ? Why?

Q34

Do the diagonals of a parallelogram intersect at a particular angle?

Q35

What are the other angles of the rhombus $ABCD$ that we have constructed? Reason and/or experiment to figure this out.

Q36

It can be seen that $\Delta GAE \cong \Delta MAE$ (How?)

Q37

So a rhombus is a parallelogram, and a rectangle is also a parallelogram. How can this be represented using a Venn diagram?

Q38

Where will the set of squares occur in this diagram?

Q39

Are the diagonals of a rhombus equal?

Q40

Do the diagonals of a rhombus intersect at any particular angle? Reason out and/or experiment to figure this out!

Q41

In the rhombus GAME, we have $\Delta \text{GEO} \cong \Delta \text{MEO}$ (why?).

Q42

In the kite, show that the diagonal $BD$

(i) bisects $\angle ABC$ and $\angle ADC$ ,

(ii) bisects the diagonal $AC$ , that is, $AO = OC$ , and is perpendicular to it.

Hint: Is $\triangle AOB \cong \triangle COB$ ?

Q43

Construct a trapezium. Measure the base angles (marked in the figure).

Q44

Can you find the remaining angles without measuring them?

Q45

How do we construct an isosceles trapezium?

Q46

Construct an isosceles trapezium $UVWX$ , with $UV \parallel XW$ . Measure $\angle U$ .

Q47

Now, it can be shown that $\Delta \text{UXY} \cong \Delta \text{VWZ}$ . (How?)