AP 8th Maths Textbook Solutions – 4. Quadrilaterals (2026-27)

4 QUADRILATERALS

4.0 Introduction

• 1. Observe the following figures. Figs. (i), (ii), and (iii) are quadrilaterals, and the others are not. Why?
  Quadrilaterals are closed figures bounded by exactly four straight line segments. Figures (i), (ii), and (iii) have four straight sides and are closed. Figure (iv) has a curved boundary, and figure (v) has five sides along with a curved boundary.

4.1 Rectangles and Squares

• 1. Are there other ways to define a rectangle?
  Yes, a rectangle can also be defined as a parallelogram in which one angle is a right angle, or a quadrilateral whose diagonals are equal and bisect each other.
• 2. A Carpenter's Problem: A carpenter needs to put together two thin strips of wood so that when a thread is passed through their endpoints, it forms a rectangle. He already has one 8 cm long strip. What should be the length of the other strip? Where should they both be joined?
  The other strip should also be exactly 8 cm long. They should be joined exactly at their midpoints (4 cm from each end).
• 3. What is the length of the other diagonal?
  The length of the other diagonal is 8 cm.
• 4. What is the point of intersection of the two diagonals?
  The point of intersection is the exact midpoint of both the diagonals.
• 5. What should the angle be between the diagonals?
  The angle between the diagonals depends on the ratio of the sides of the rectangle; they do not necessarily intersect at right angles (unless it is a square).

• 1. In order to show congruence, consider ∠1 and ∠2. Are they equal?
  Yes, ∠1 and ∠2 are equal because they are alternate interior angles formed when the transversal intersects the parallel opposite sides of the rectangle.

• 1. Can the following equalities be used to establish that ΔAOD ≅ ΔCOB? AO = CO, ∠AOB = ∠COD, AD = CB
  No, these equalities represent the SSA (Side-Side-Angle) condition, which is not a valid criterion for congruence.

• 1. Can you find all the remaining angles?
  The remaining angles between the diagonals are 120°, 60°, and 120°.
• 2. Can you find the value of a?
  Yes. In ΔAOB, the sum of interior angles is 180°. So, a + a + 60° = 180° ⇒ 2a = 120° ⇒ a = 60°.

• 1. Can we now identify what type of quadrilateral ABCD is?
  Since all its interior angles are exactly 90° (30° + 60°) and its opposite sides are equal, quadrilateral ABCD is a rectangle.
• 2. What can we say about its sides?
  Its opposite sides are equal in length (AB = CD and AD = CB).
• 3. 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.
  Yes, ABCD will remain a rectangle regardless of the angle between the diagonals, as long as the diagonals are of equal length and bisect each other.
• 4. Can you find the other angles?
  The other angles between the diagonals are x, 180° - x, and 180° - x.

• 1. What is the value of a (in degrees) in terms of x?
  a = (180 - x) / 2 = 90 - (x/2).
• 2. What can we say about AB and CD, and AD and BC?
  AB = CD and AD = BC because they are corresponding parts of congruent triangles (ΔAOB ≅ ΔCOD and ΔAOD ≅ ΔCOB).

• 1. Are you able to construct such a quadrilateral? (a quadrilateral in which the angles are all 90° but the opposite sides are not equal)
  No, it is impossible. If all four angles of a quadrilateral are 90°, the opposite sides must be parallel and equal.
• 2. Two equalities can be directly seen in the triangles. What can we say about ∠1 and ∠2?
  ∠1 and ∠2 are alternate interior angles and are equal to each other because they are formed between parallel lines AB and DC with BD as the transversal.

• 1. Is it wrong to write ΔBAD ≅ ΔCDB? Why?
  Yes, it is wrong because the order of vertices matters in congruence. The correct correspondence is ΔBAD ≅ ΔDCB. Writing ΔCDB would incorrectly pair vertex A with D, which are not corresponding equal parts.
• 2. Are the opposite sides of a rectangle parallel?
  Yes, the opposite sides of a rectangle are always parallel.
• 3. Can you similarly show that AB is parallel to DC (AB || DC)?
  Yes, AD is a transversal to lines AB and DC. Since ∠A + ∠D = 90° + 90° = 180°, the interior angles on the same side of the transversal are supplementary. Therefore, line AB is parallel to line DC.

• 1. In the quadrilaterals below, are there any non-rectangles?
  No, all the given quadrilaterals (i, ii, iii, iv) are rectangles because all their interior angles are 90°. Figure (iv) is a special rectangle called a square.

• 1. 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?
  The wooden strips (diagonals) must be of equal length, they must bisect each other exactly at their midpoints, and they must intersect at right angles (90°).

• 1. What more needs to be done to get equal sidelengths as well? Can this be achieved by properly choosing the angle between the diagonals?
  Yes, equal side lengths can be achieved by choosing the angle between the diagonals to be exactly 90° (right angles).
• 2. To find the angle formed by the diagonals, what are the two triangles we should consider for congruence?
  We should consider adjacent triangles sharing a side, such as ΔBOA and ΔBOC.
• 3. Can this be used to find the angles ∠BOA and ∠BOC formed by the diagonals?
  Yes. Since ΔBOA ≅ ΔBOC by SSS congruence, ∠BOA = ∠BOC. Since they form a linear pair on the straight line AC, ∠BOA + ∠BOC = 180°. Therefore, 2∠BOA = 180°, which means ∠BOA = 90° and ∠BOC = 90°.

• 1. 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.
  Yes, since a square is a special type of rectangle, all properties deduced for a rectangle in Deduction 1 and 2 (diagonals are equal and bisect each other) apply perfectly to a square as well.
• 2. What are the measures of ∠1, ∠2, ∠3, and ∠4?
  In a square, the diagonal bisects the 90° interior angles. Therefore, ∠1, ∠2, ∠3, and ∠4 are all exactly 45°.
• 3. Similarly, find ∠2 and ∠4.
  In ΔABC, AB = BC, so the angles opposite them are equal (∠2 = ∠4). Since ∠B = 90°, the sum of the remaining angles is ∠2 + ∠4 = 90°. Thus, each is 45°.

Figure it Out - 4.1

• 1. Find all the other angles inside the following rectangles.
  (i) The angle vertically opposite to 110° is 110°. The other two angles at the intersection are 180° - 110° = 70°. The base angles of the isosceles triangles formed by the diagonals are (180° - 110°)/2 = 35° and (180° - 70°)/2 = 55°. Thus, the 90° corner angles are split into 35° and 55°.
  
  (ii) The angle given at the corner is 30°. The other angle at that vertex is 90° - 30° = 60°. Because the diagonals bisect each other and are equal, isosceles triangles are formed. The angles at the central intersection are 180° - (30°+30°) = 120° and 180° - (60°+60°) = 60°.

• 2. Draw a quadrilateral whose diagonals have equal lengths of 8 cm that bisect each other, and intersect at an angle of (i) 30° (ii) 40° (iii) 90° (iv) 140°
  (i), (ii), and (iv) will result in rectangles of varying widths. (iii) will result in a perfect square.
• 3. Consider a circle with centre O. Line segments PL and AM are two perpendicular diameters of the circle. What is the figure APML? Reason and/or experiment to figure this out.
  The figure APML is a square. The diagonals (which are the diameters) are equal in length, bisect each other perfectly at the centre of the circle, and are perpendicular to each other. These properties define a square.
• 4. We have seen how to get 90° using paper folding. Now, suppose we do not have any paper but two sticks of equal length, and a thread. How do we make an exact 90° using these?
  Cross the two sticks exactly at their midpoints. Tie the thread around their four endpoints to form a quadrilateral. Adjust the angle between the sticks until all four sides formed by the thread are perfectly equal in length (making it a square). The angle between the sticks will then be exactly 90°.
• 5. We saw that one of the properties of a rectangle is that its opposite sides are parallel. Can this be chosen as a definition of a rectangle? In other words, is every quadrilateral that has opposite sides parallel and equal, a rectangle?
  No, because a standard parallelogram also has opposite sides that are parallel and equal, but its interior angles are not necessarily 90°. A rectangle must be defined with the condition that all its angles are equal to 90°.
• 6. Is it possible to construct a quadrilateral with three angles equal to 90° and the fourth angle not equal to 90°?
  No, it is not possible.
• 7. But why not?
  The sum of the interior angles of any quadrilateral is always 360°. If three angles are exactly 90°, their sum is 270°. The fourth angle must be 360° - 270° = 90°.
• 8. What do we get when we add all six angles?
  We get ∠1 + ∠2 + ∠3 + ∠4 + ∠5 + ∠6 = 180° + 180° = 360°.

• 1. Are there quadrilaterals that have parallel opposite sides that are not rectangles?
  Yes, parallelograms and rhombuses have parallel opposite sides but do not have to be rectangles (as their interior angles don't have to be 90°).
• 3. Is a rectangle a parallelogram?
  Yes, a rectangle is a special kind of parallelogram because its opposite sides are parallel to each other.

• 1. What are the remaining angles of the parallelogram? What are the lengths of the remaining sides?
  The remaining angles are 150°, 30°, and 150°. The remaining sides are 4 cm and 5 cm respectively.

• 1. What about the opposite angles? Will they be equal in all parallelograms? If yes, how can we be sure?
  Yes, opposite angles are always equal. If one angle is x, the adjacent angle is supplementary (180° - x) due to parallel lines. The angle opposite to the adjacent one is 180° - (180° - x) = x.
• 2. Let us take one of the angles to be x. What are the other angles?
  The other angles will be 180° - x, x, and 180° - x.
• 3. What can we say about the sides of a parallelogram?
  The opposite sides of a parallelogram are always equal in length.
• 4. Which two triangles can be considered for this?
  We can consider the two triangles formed by drawing a diagonal, such as ΔABD and ΔCDB.
• 5. Is it wrong to write ΔABD ≅ ΔCBD? Why?
  Yes, it is wrong because when naming congruent triangles, the corresponding vertices must match exactly. Vertex A corresponds to C, B corresponds to D, and D corresponds to B. Therefore, it must be written as ΔABD ≅ ΔCDB.

• 1. Are the diagonals of a parallelogram always equal? Check with the parallelogram that you have constructed.
  No, the diagonals of a standard parallelogram are generally not equal (unless the parallelogram is a rectangle).
• 2. Do they bisect each other (do they intersect at their midpoints)? Reason and/or experiment to figure this out.
  Yes, the diagonals of a parallelogram always bisect each other, intersecting exactly at their midpoints.

• 1. Is it wrong to write ΔAOE ≅ ΔSOY? Why?
  Yes, it is wrong because corresponding vertices must be correctly matched. Vertex A matches with Y, O matches with O, and E matches with S. It must be written as ΔAOE ≅ ΔYOS.
• 2. Do the diagonals of a parallelogram intersect at a particular angle?
  No, they do not intersect at a specific fixed angle unless the parallelogram is a rhombus or a square (where they intersect at exactly 90°).
• 3. Are squares the only quadrilaterals that have equal sidelengths?
  No, a rhombus also has four equal side lengths but does not necessarily have 90° interior angles like a square.
• 4. Can we complete this quadrilateral so that all its sides are of the same length?
  Yes, by using a compass set to the length of AB, and drawing intersecting arcs from points B and D to find vertex C.
• 5. What are the other angles of the rhombus ABCD that we have constructed?
  Since one angle is 50°, the opposite angle is 50°. The adjacent angles are 180° - 50° = 130°. So the remaining angles are 130°, 50°, and 130°.

• 1. So a rhombus is a parallelogram, and a rectangle is also a parallelogram. How can this be represented using a Venn diagram?
  Both the Rhombus and Rectangle circles will be placed completely inside a larger Parallelogram circle.
• 2. Where will the set of squares occur in this diagram?
  The set of squares will be the intersection (the overlapping region) of the Rectangle set and the Rhombus set.
• 3. Are the diagonals of a rhombus equal?
  No, they are not necessarily equal (unless the rhombus also happens to be a square).
• 4. Do the diagonals of a rhombus intersect at any particular angle? Reason out and/or experiment to figure this out!
  Yes, they always intersect exactly at right angles (90°).
• 5. What can we say about the angles formed by the diagonals of a rhombus at their point of intersection?
  The angles formed at the intersection are all exactly 90° (they bisect each other perpendicularly).

Figure it Out - 4.2

• 1. Find the remaining angles in the following quadrilaterals.
  (i) (Rhombus) The opposite angle is 40°. The adjacent angles are 180° - 40° = 140°. The remaining angles are 140°, 40°, and 140°.
  
  (ii) (Parallelogram) The adjacent angle is 180° - 110° = 70°. The remaining angles are 70°, 110°, and 70°.
  
  (iii) (Rectangle) All four corner angles are 90°. In the triangles formed by the diagonals, the base angles are 30° and 60° (since 90-30=60). The angles at the central intersection are 120° and 60°.
  
  (iv) (Rhombus) The diagonals bisect the angles at exactly 90°. In the right-angled triangles formed, the remaining angles are 90° - 20° = 70°. Therefore, the full corner angles are 40° and 140°.

• 1. What is the quadrilateral that you get? Justify your answer. (Geoboard Activity)
  You get a square because the diagonals (rubber bands) are of equal length and bisect each other exactly at right angles.
• 2. Extend one of the diagonals on both sides by 2 cm. What quadrilateral will you get now? Justify your answer.
  You will get a rhombus. The diagonals still bisect each other at right angles, but they are no longer equal in length.
• 3. Can you join them to get a quadrilateral? (2 equilateral triangles of 8cm)
  Yes, by joining them along one of the 8 cm sides, you get a rhombus with all sides equal to 8 cm and interior angles of 60° and 120°.

• 1. What are the different ways they can be joined to get a quadrilateral? (Isosceles triangles 8, 8, 6)
  They can be joined along the 6 cm side or along one of the 8 cm sides.
• 2. What quadrilaterals are these? Justify your answers.
  Joining along the 6 cm side creates a Rhombus (all four sides are 8 cm). Joining along the 8 cm side creates a Kite (two pairs of equal adjacent sides: 6 cm and 8 cm).
• 3. What are the different ways they can be joined to get a quadrilateral? (Scalene triangles 6, 9, 12)
  They can be joined along the 6 cm side, the 9 cm side, or the 12 cm side.
• 4. Are you able to identify the different quadrilaterals that are obtained by joining the triangles? Justify your answer whenever you identify a quadrilateral.
  Joining along the 12 cm side gives a kite (sides 6 cm and 9 cm). Joining along the 9 cm side gives a kite (sides 6 cm and 12 cm). Joining along the 6 cm side gives a kite (sides 9 cm and 12 cm). In all cases, two pairs of adjacent sides are equal.

• 1. Property 1: In the kite, show that the diagonal BD (i) bisects ∠ABC and ∠ADC, (ii) bisects the diagonal AC, that is, AO = OC, and is perpendicular to it.
  (i) Yes, ΔABD ≅ ΔCBD by SSS congruence (AB=CB, AD=CD, BD is common). Therefore, ∠ABD = ∠CBD and ∠ADB = ∠CDB.
  (ii) In ΔABO and ΔCBO, AB=CB, ∠ABO = ∠CBO, and BO is common. So ΔABO ≅ ΔCBO by SAS. Thus, AO=OC and ∠AOB = ∠COB = 90°.
• 2. Construct a trapezium. Measure the base angles. Can you find the remaining angles without measuring them?
  Yes, the interior angles on the same side of the transversal (between the parallel lines) sum to 180°. So, ∠P = 180° - ∠S and ∠Q = 180° - ∠R.

• 1. How do we construct an isosceles trapezium?
  Draw two parallel lines. Mark a segment UV on one, and a shorter segment XW on the other such that they are perfectly centered relative to each other, resulting in equal non-parallel sides UX and VW.
• 2. Can you find the remaining angles without measuring them? Does it appear that the angles opposite to the equal sides - ∠U and ∠V - are also equal?
  Yes, in an isosceles trapezium, the base angles opposite the equal sides are equal to each other (∠U = ∠V). The upper angles are 180° - ∠U.
• 3. What type of quadrilateral is XWZY?
  XWZY is a rectangle because XW || YZ, and XY ⊥ UV, WZ ⊥ UV, making all interior angles 90°.
• 4. Now, it can be shown that ΔUXY ≅ ΔVWZ. (How?)
  Since XWZY is a rectangle, XY = WZ. We are given UX = VW (hypotenuse). ∠XYU = ∠WZV = 90°. So, ΔUXY ≅ ΔVWZ by the RHS congruence condition.

Figure it Out - 4.3

• 1. Find all the sides and the angles of the quadrilateral obtained by joining two equilateral triangles with sides 4 cm.
  All four sides are 4 cm. The interior angles are 60°, 120° (60°+60°), 60°, and 120°. The quadrilateral is a rhombus.
• 2. Construct a kite whose diagonals are of lengths 6 cm and 8 cm.
  Draw a line segment of 8 cm. Draw its perpendicular bisector. On the bisector, mark points at 2 cm and 4 cm (or any lengths summing to 6cm) from the intersection to form the vertices of a kite.
• 3. Find the remaining angles in the following trapeziums.
  For the trapezium with upper angles 135° and 100°, the adjacent bottom angles are 180° - 135° = 45° and 180° - 100° = 80°.
• 4. Draw a Venn diagram showing the set of parallelograms, kites, rhombuses, rectangles, and squares. Then, answer the following questions: (i) What is the quadrilateral that is both a kite and a parallelogram? (ii) Can there be a quadrilateral that is both a kite and a rectangle? (iii) Is every kite a rhombus? If not, what is the correct relationship between these two types of quadrilaterals?
  (i) A quadrilateral that is both a kite and a parallelogram is a rhombus.
  (ii) Yes, a square is both a kite and a rectangle.
  (iii) No, not every kite is a rhombus. A rhombus is a special type of kite where all four sides are equal, whereas a standard kite only needs two pairs of adjacent sides to be equal.
• 5. If PAIR and RODS are two rectangles, find ∠IOD.
  Since diagonals of a rectangle bisect each other, ΔOIR is isosceles. ∠IRO = 30°, so ∠IOR = 180° - (30°+30°) = 120°. ∠IOD and ∠IOR are vertically opposite angles. Therefore, ∠IOD = 120°.
• 7. CASE is a square. The points U, V, W and X are the midpoints of the sides of the square. What type of quadrilateral is UVWX? Find this by using geometric reasoning, as well as by construction and measurement.
  UVWX is a square. (The four triangles formed at the corners are congruent right isosceles triangles, making the inner sides equal and interior angles exactly 90°).
• 8. If a quadrilateral has four equal sides and one angle of 90°, will it be a square? Find the answer using geometric reasoning as well as by construction and measurement.
  Yes. Four equal sides make it a rhombus. A rhombus with one 90° angle must have all 90° angles (since opposite angles are equal and adjacent angles sum to 180°). Thus, it is a square.
• 9. What type of a quadrilateral is one in which the opposite sides are equal? Justify your answer.
  It is a parallelogram. If opposite sides are equal (AB=CD, AD=BC), drawing a diagonal AC creates two triangles that are congruent by SSS. This means alternate interior angles are equal, proving the opposite sides are parallel.
• 10. Will the sum of the angles in a quadrilateral such as the following one also be 360°?
  Yes. Even for a concave quadrilateral, drawing a diagonal connecting the interior reflex angle splits the figure into two triangles. The sum of the angles is 180° + 180° = 360°.
• 11. State whether the following statements are true or false. Justify your answers.
  (i) False. It is a rectangle, but not necessarily a square unless the diagonals are also perpendicular.
  (ii) True. The sum is 360°, so the fourth angle must be 360° - 270° = 90°, making it a rectangle.
  (iii) True. If diagonals bisect each other, the quadrilateral is always a parallelogram.
  (iv) False. A kite's diagonals are perpendicular, but it is not a rhombus unless they also bisect each other.
  (v) True. If opposite angles are equal, adjacent angles sum to 180°, meaning opposite sides are parallel.
  (vi) True. If all four angles are equal, they must be 360°/4 = 90°, which defines a rectangle.
  (vii) False. An isosceles trapezium has one pair of parallel sides and one pair of non-parallel sides, so it cannot be a parallelogram.

Puzzle Time

• 1. Open the sheet. What is the shape formed by the creases?
  It forms a rhombus (or a square, depending on the dimensions of the initial paper sheet).
• 2. How would you fold the quarter paper to get the kinds of creases shown in the following image.
  By folding the quarter piece of paper diagonally from corner to corner.
• 3. How would you fold the quarter paper such that a square is formed?
  By ensuring the folded diagonal makes exactly a 45° angle with the edges, which creates equal length sides from the center when unfolded.

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