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Quadrilaterals

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Page 82: Observe the following figures. Figs. (i), (ii), and (iii) are quadrilaterals, and the others are not. Why?
Page 83: We know what rectangles are. Let us define them. Are there other ways to define a rectangle? Let us consider the…
Page 84: Deduction 1 — What is the length of the other diagonal? Since ABCD is a rectangle, we have So, ΔADC ≅ ΔDAB by the…
Page 84: Deduction 2 — What is the point of intersection of the two diagonals?
Page 85: Can the following equalities be used to establish that ΔAOD ≅ ΔCOB?
Page 85: Deduction 3 — What are the angles between the diagonals? Can you find all the remaining angles? In ΔAOB, since OA =…
Page 86: Can we now identify what type of quadrilateral ABCD is? Notice that its angles all add up to 90° (30° + 60°). What…
Page 89: In the earlier definition, we stated that a rectangle has (a) opposite sides of equal length, and (b) all angles…
Page 89: Deduction 4 — What is the shape of a quadrilateral with all the angles equal to 90°? Consider a quadrilateral ABCD…
Page 90: Are the opposite sides of a rectangle parallel? Can you similarly show that AB is parallel to DC (AB ∥ DC)?
Page 90: In the quadrilaterals below, are there any non-rectangles?
Page 92: Let us consider the Carpenter’s Problem again. If the wooden strips have to be placed such that the thread passing…
Page 93: Using this fact, construct a square with a diagonal of length 8 cm.
Page 93: Verify if this is true by going through the geometric reasoning in Deduction 1 and Deduction 2, and see if they…
Page 93: There is one more special property of a square. What are the measures of ∠1, ∠2, ∠3, and ∠4? See if you can reason…
Page 94: Find all the other angles inside the following rectangles.
Page 94: Draw a quadrilateral whose diagonals have equal lengths of 8 cm that bisect each other, and intersect at an angle of
Page 94: Consider a circle with centre O. Line segments PL and AM are two perpendicular diameters of the circle. What is the…
Page 94: We have seen how to get 90° using paper folding. Now, suppose we do not have any paper but two sticks of equal…
Page 94: We saw that one of the properties of a rectangle is that its opposite sides are parallel. Can this be chosen as a…
Page 94: Is it possible to construct a quadrilateral with three angles equal to 90° and the fourth angle not equal to 90°?…
Page 95: Are there quadrilaterals that have parallel opposite sides that are not rectangles? Let us try constructing one.…
Page 95: Is a rectangle a parallelogram?
Page 96: To understand the relations between the sides and the angles of a parallelogram, let us construct the following…
Page 97: Deduction 6 — What can we say about the angles of a parallelogram? In the parallelogram ABCD, AB ∥ CD, and AD is a…
Page 98: Deduction 7 — What can we say about the sides of a parallelogram? By looking at a parallelogram, it appears that the…
Page 98: Are the diagonals of a parallelogram always equal? Check with the parallelogram that you constructed. Do they bisect…
Page 99: Are squares the only quadrilaterals that have equal sidelengths? Let us explore this question through construction.…
Page 100: What are the other angles of the rhombus ABCD that we have constructed? Reason and/or experiment to figure this out.…
Page 101: So a rhombus is a parallelogram, and a rectangle is also a parallelogram. How can this be represented using a Venn…
Page 102: Are the diagonals of a rhombus equal? Are they always equal in every rhombus you can draw? (Compare with a square,…
Page 102: Find the remaining angles in the following quadrilaterals.
Page 102: Using the diagonal properties, construct a parallelogram whose diagonals are of lengths 7 cm and 5 cm, and intersect…
Page 102: Using the diagonal properties, construct a rhombus whose diagonals are of lengths 4 cm and 5 cm.
Page 103: Place two rubber bands perpendicular to each other, forming diagonals of equal length. Join the ends. What is the…
Page 103: Extend one of the diagonals on both sides by 2 cm. What quadrilateral will you get now? Justify your answer.
Page 104: Take two cardboard cutouts of an equilateral triangle of sidelength 8 cm. What type of a quadrilateral is this?…
Page 104: Take two cardboard cutouts of an isosceles triangle with sidelengths 8 cm, 8 cm, and 6 cm. What are the different…
Page 105: Take two cardboard cutouts of a scalene triangle with sides 6 cm, 9 cm, and 12 cm. What are the different ways they…
Page 105: Property 1: In the kite, show that the diagonal BD
Page 106: Construct a trapezium. Measure the base angles (marked in the figure). Can you find the remaining angles without…
Page 106: How do we construct an isosceles trapezium? Construct an isosceles trapezium UVWX, with UV ∥ XW. Measure ∠U. Can you…
Page 107: Find all the sides and the angles of the quadrilateral obtained by joining two equilateral triangles with sides 4 cm.
Page 107: Construct a kite whose diagonals are of lengths 6 cm and 8 cm.
Page 107: Find the remaining angles in the following trapeziums —
Page 107: Draw a Venn diagram showing the set of parallelograms, kites, rhombuses, rectangles, and squares. Then, answer the…
Page 107: If PAIR and RODS are two rectangles, find ∠IOD.
Page 108: Construct a square with a diagonal 6 cm, without using a protractor.
Page 108: CASE is a square. The points U, V, W and X are the midpoints of the sides of the square. What type of quadrilateral…
Page 108: If a quadrilateral has four equal sides and one angle of 90°, will it be a square? Find the answer using geometric…
Page 108: What type of a quadrilateral is one in which the opposite sides are equal? Justify your answer.
Page 108: Will the sum of the angles in a quadrilateral such as the following one also be 360°? Find the answer using…
Page 108: State whether the following statements are true or false. Justify your answers.
Page 111: 1. Fold a sheet into half. 2. Now, fold it once more into a quarter. 3. Make a triangular crease at the corner that…
Page 111: 5. How would you fold the quarter paper to get the kinds of creases shown in the following image (several nested…
Page 111: 6. How would you fold the quarter paper such that a square is formed?

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