Chapter 4 UNIT - 1 AXIOMS, POSTULATES AND THEOREMS I. Choose the correct answers: 1. In the figure a pair of alternate angles are

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1 STD-VIII ST. CLARET SCHOOL Subject : MATHEMATICS Chapter 4 UNIT - 1 AXIOMS, POSTULATES AND THEOREMS I. Choose the correct answers: 1. In the figure a pair of alternate angles are a) and b) and c) and d) and 2. Which one of the following is not an example for reflex angle. a) b) c) d) In the adjoining figure the value of X is a) 30 0 b) 40 0 c) d) AB and CD intersect at O. If = 132 0, then is equal to, a) 48 0 b) 15 0 c) d) One sixth of a right angle is a) 20 b) 15 c) 30 d)25 6. Which one of the following is wrong statement? a) A live can be extended infinitely on either side of it. b) A ray has definite length. c) A line segment has a definite length d) A ray has only one end point. 7. The measure of an angle which is 5 times its supplement is

2 a) 30 0 b) 60 0 c) d) The measure of an angle which is 4 times its compliment is a) 30 0 b) 50 0 c) 60 0 d) 40 0 II Fill up the blanks: 1. The measure of an angle which is equal to its compliment is 2. The measure of an angle which is equal to its supplement is 3. If a transversal cuts two parallel lines of inferior angles on the same side of the transversal are 4. The supplement of 0 0 is 5. The compliment of x 0 is 6. The compliment of 35 0 is 7. The points which lies in the same straight line are called 8. If a transversal cuts two parallel lines, then corresponding angles are III Solve the following(2m) 1. If two straight lines intersect each other, Prove that vertically opposite angles are equal. 2. Find the value of x in the given figure. 3. Find the value of x from the given diagram. 4. Find the value of X from the given figure.

3 5. Find x in the diagram. 6. From the given figure, write a) A pair of alternate angles b) A pair of corresponding angles. 7. From the given figure, write

4 a) A pair of adjacent angles b) A pair of vertically opposite angles. IV Solve: 3 Marks 1. Find all the angles from the given figure. 2. Find the value of x in the diagram given below: 3. In the adjoining figure, find the measure of a) b) c)

5 4. If a transversal cuts two parallel lines, show that angular bisector of a pair of corresponding angles are parallel to each other. 5. Identify and write: a) Vertically opposite angles b) Alternate angles c) Interior angles on the same side and the transversal 6. If a transversal cuts two parallel lines then, prove that interior angles on the same side of the transversal are supplementary. V Solve: (theorem) 4 Marks 1. If two parallel lines are cut by a transversal then prove that each pair of alternate angles are equal. 2. If two parallel lines are cut by a transversal then prove that each pair of corresponding angles are equal. Lesson : Unit - 2 Theorems on triangles I Fill in the blanks 1 Mark(s) 1. In, PQ = QR = RP = 5cm. Then is triangle. 2. In, AB = BC, then it is called triangle 3. If all the sides of a triangle are unequal then it is called triangle. 4. If in a if then 5. Sum of the angles of a triangle is.

6 6. An exterior angle of a triangle is equal to the sum of opposite angles. 7. An exterior angle of a triangle is a ways than either of the interior opposite angles. 8. A triangle cannot have more than right angle. 9. A triangle cannot have more than obtuse angle. II Choose the correct answer 1 Mark(s) 10.In, then the triangle is a) Isosceles triangle b) Right angled triangle c) Equilateral triangle d) Obtuse angled triangle 11.In a triangle, all the angles are acute then it is a a) Equilateral triangle b) Right angled triangle c) Acute angled triangle d) Obtuse angled triangle 12.In an equilateral triangle, each exterior angle is a) 60 b) 90 c) 120 d) Sum of the three exterior angles of a triangle is a) Two right angles b) three right angles c) one right angle d) four right angles. 14.In a triangle ABC,, and AB = AC then B = a) 50 b) 55 c) 75 d) In a triangle ABC, then the triangle is. a) Right angle b) Acute angle c) Obtuse angle d) Equilateral triangle 16.An exterior angle in a triangle is 100 and one of the interior opposite angle is 40. then the remaining angle is a) 40 b) 60 c) 50 d) 80 III Do as directed 2 Mark(s) 17.In the given figure. Find all the angles. 18.In a right angled triangle, if one of the other two angle is 35. Find the remaining angle.

7 19.If the vertex angle of an isosceles triangle is 50. Find the other angles. 20.Compute the value of x. 21.An exterior angle of a triangle is 120 and one of the interior opposite angle is 30. Find the other angles of the triangle. 22.In a triangle ABC, find 23.In a 110 and AB = AC. Find IV Solve the following 3 Mark(s) 24.If three angles of a triangle are in the ratio 2:3:5. Determine three angles. 25.The angles of a triangle are in the ratio 1:2:3. Determine the three angles. 26.In the adjacent triangle ABC, find the value of 'x' and calculate the measure of all the angles of the triangle. 27.In the given fig, sides PQ and RQ of a triangle PQR are produced to the points S and T respectively. If = 135 and find.

8 28.The exterior angles obtained on producing the base of a triangle both angle are 104 and 136. Find the angles of the triangle. 29.The angles of a triangle are arranged in ascending order of magnitude. If the difference between two consecutive angles is 15, Find the three angles. 30.The angles of triangle are arranged in ascending order of their magnitude. If the difference between two consecutive angles is 10. Find the three angles. 31.The sum of two angles of a triangle is equal to its third angle. Determine the measure of third angle. 32.In a triangle ABC, if 2 A = 3 B = 6 C. Determine A, B and C. 34.The sum of two angles of a triangle is 80 and three difference is 20.Find the angles of the triangle. V Simplify 4 Mark(s) 35.If the bisectors of the angle ABC and ACB of a ABC meet at point O, then prove that 36. Sides BC, CA and AB of a triangle ABC are produced in an order forming exterior angles. Show that ACD + BAE + CBF = In the figure QT Find

9 VI Theorems 4 Mark(s) 40.State and prove interior angle theorem. 41.State and prove exterior angle theorem. Lesson : Unit - 3 Congruency of triangles I. Choose the correct Answer: 1 Mark 1. ΔABC ΔRPQ, If RP corresponds to AB, BC corresponds to PQ, then B corresponds to a) R b) P c) Q d) PQR. 2. ABCD is a square. AC is the diagonal, then ABC = a) ACB b) BAC c) ACD d) ADC. 3. Diagonals of a square are equal. This statement can be proved by using a) SAS Postulate b) ASA Postulate c) SSS Postulate d) RHS Theorem. 4. The statement diagonals of a parallelogram bisect each other can be proved by using a) SAS Postulate b) ASA Postulate c) SSS Postulate d) RHS Theorem. 5. To prove that each diagonal divides the parallelogram into two congruent triangles, which one of the following cannot be used a) SAS Postulate b) ASA Postulate c) SSS Postulate d) RHS Theorem 6. AB bisects CD at O. To prove AD = CB. The postulate can be used is a) SAS b) ASA c) SSS d) All of these. 7. PQ and RS intersect at O, such that PR SQ and PR = SQ. To prove PQ and RS bisect each other, the postulate that can be used is a) ASA b) SAS c) SSS d) All of these. 8. In ΔABC, AB = AC, = x, = 2x - 20, the measure of is a) 34 0 b) 44 0 c) 45 0 d) 50 0

10 II Fill in the blanks: 1 Mark 1. In right triangle, the hypotenuse is the side. 2. The sum of three altitudes of a triangle is than its perimeter. 3. The sum of any two sides of a triangle is than the third side. 4. If two angles of a triangle are unequal, then the smaller angle has the side opposite to it. 5. If two sides of a triangle are unequal, then the larger side has the angle opposite to it. 6. Two lines segments are congruent, if they have 7. Two circles are congruent, if they have same 8. Two squares are congruent, if they have of the same length. 9. Two congruent geometrical figures have same III Solve the following: 2 Marks 1. Identify the corresponding sides and corresponding angles in the following: 2. AB and CD bisect at O. Prove that AC = BD 3. In a quadrilateral ABCD, AC=AD, and AB bisect, show that ΔABC is congruent to ΔABD. 4. In ΔABC, AB=AC, and = 50 0, find B and C 5. Find the value of x in the following figure 6. Find X in the following figure.

11 7. Diagonal AC of a quadrilateral ABCD bisects the angles and C. Prove that AB=AD and CB=CD. 8. In the given figure, if AB DC, and P is the midpoint of BD, prove that P is also midpoint of AC 9. In the figure, it is given that AB=CD and AD=BC. Prove that ADC and CBA are congruent In a triangle ABC, AC=AB and the altitude AD bisects BC. Prove that ΔADC ΔADB. 11. Suppose ABC is an isosceles triangle, such that AB=AC and AD is the altitude from A on BC. Prove that AD bisects. 12. Show that in a right angle triangle, hypotenuse is larger than any of the remaining sides. 13. In a triangle ABC, B= 28 0, C = 56 0, find the largest and smallest sides. 14. In a triangle ABC, AB=4cm, BC=5.6cm and CA = 7.6 cm. Write the angles of the triangle in ascending order of measures. 15. In ΔPQR, PQ =QR, L, M and N are the midpoint of the sides PQ, QR and RP respectively. Prove that LN = MN.

12 IV Solve the Problem: 3 Marks. 1. A triangle is equilateral. Prove that it is equiangular. 2. In the figure, CD and BE are altitudes of an isosceles triangle ABC with AC=AB. Prove that AE = AD 3. Suppose the altitude AD, BE and CF of a triangle ABC are equal. Prove that ABC is an equilateral triangle. V. Theorems: 3 Marks 1. Prove that if in a triangle two angles are equal, then the sides opposite to them are equal. 2. Prove that in a triangle the angles opposite to equal sides are equal. V Solve the problem: 4 Marks 1. In a ΔABC, AB=AC and the bisector of angles B and C intersect at O. Prove that BO=CO and AO is the bisector of BAC. Lesson : Unit 4 Construction of triangles Construct the following: 4 Marks 1. Construct Δ ABC in which AB=5cm, BC = 4.6cm and AC= 3.7 cm. 2. Construct an equilateral triangle with side 4.8 cm. 3. Construct Δ PQR in which QR = 4.8cm, 0 and = Construct a right angle triangle ABC in which = 90 0, AB = 5cm and AC = 7cm. 5. Construct an isosceles Δ ABC in which base BC=6.5cm and altitude from A on BC is 4 cm.

13 6. Construct an isosceles Δ XYZ in which base YZ = 5.5cm and altitude from X on YZ is 3.8 cm. 7. Construct an isosceles Δ whose altitude is 4.5 cm and vertex angle is Construct an isosceles Δ whose altitude is 5 cm and vertex angle is Construct a Δ ABC, whose perimeter is 13 cm and whose sides are in the ratio 3:4: Construct a Δ PQR whose perimeter is 14cm and whose sides are in the ratio 2:4: Construct a Δ ABC whose perimeter is 12.5 cm and whose base angles are 60 0 and Construct a Δ PQR whose perimeter is 12 cm and whose base angles are 50 0 and 80 0 Lesson : Unit 5 Quadrilaterals I Multiple Choice: 1. Three angles of a quadrilateral are equal and 4 th angle is Each equal angle measures. a) b) c) 70 0 d) If three angles of quadrilateral measure 110 0, 60 0 and 100 0, the remaining angle is angle. a) Acute b) Obtuse c) Right d) Reflex 3. Angles of a quadrilateral are in the ratio 1: 2: 3 : 4. The greatest angle is a) 172 0, b) c) d) Two angles of a quadrilateral are 89 0 and If other two angles are equal, then each equal angle measure a) b) 80 0 c) 80 0 d) In an isosceles trapezium ABCD, AB CD. If A = 70 0, C = a) 70 0 b) c) d) The perimeter of parallelogram is 26cm. If the base measure 8cm, the other side is cm. a) 5 b) 10 c) 16 d) Which one of the following not true in case of parallelogram. a) Opposite sides are equal and parallel. b) Diagonals are equal and bisect each other. c) Each diagonal divides the parallelogram into 2 triangles. d) Opposite angles are equal. 8. ABCD is a rectangle. Diagonal AC & BD intersect at O. IF AO = 4 cm, then BD = a) 4cm b) 8cm c) 12cm d) 16cm. 9. In a rectangle PQRS, if area of PQR = 60cm 2, then the area of the rectangle is a) 60cm 2 b) 120cm 2 c) 30cm 2 d) 240cm The quadrilateral having unequal perpendicular diagonal is a) Parallelogram b) Square c) rhombus d) Trapezium

14 11. The quadrilateral which is equidiagonal and equilateral is a) square b) rectangle c) parallelogram d) rhombus. II Fill in the blanks: 1. A quadrilateral in which a pair of opposite sides are parallel is. 2. In an isosceles trapezium basic angles are 3. In a rhombus the diagonals bisect each other at angles. 4. In a square all angles are. 5. If one angle of a parallelogram is 90 0, then it is. III Solve the following: 2 Marks 1. Prove that sum of the angles of a quadrilateral is In the figure, P and Q are supplementary angles and R = Find the measure of S. 3. Find all the angles in the given quadrilateral below. 4. The perimeter of square is 60cm.Find its side length. 5. List out the difference between square and rhombus. 6. Three angles of a quadrilateral are in the ratio 2:3:5 and the fourth angle is Find the measures of the other three angles. 7. In a trapezium PQRS, PQ RS and P = 70 0 and Q = Calculate the measures of S and R.

15 8. The adjacent angles of a parallelogram are in the ratio 2:1. Find the measures of all the angles. 9. In a parallelogram KLMN, K = Find the measures of all the angles. 10. The sides of a rectangle are in the ratio 2:1. The perimeter is 30 cm. Calculate the measure of all the sides. 11. In a rectangle RENT, the diagonals meet at O. If OR = 2x+4, and OT = 3x+1, find x. 12. In a rhombus ABCD, C=70 0, find the other angles of the rhombus. 13. In the figure PQRS is a kite. PQ =3cm and QR =6cm. Find the perimeter of PQRS. 14. A field is in the shape of a square with side 20m. A pathway of 2m width is surrounding it. Find the outer perimeter of the pathway.

16 15. In a rhombus PQRS, if PQ = 3x-7 and QR = x+3, find PS. 16. Let ABCD is a rhombus and BAC = Calculate, D and C. Solve the Following: 3 Marks 1. Prove that diagonal divides the parallelogram into two congruent triangles. 2. A square field has side 20m. Find the length of the wire required to fence it four times. 3. Let PQRS be a rhombus with PR=15 cm and QS = 8cm. Find the area of the rhombus. 4. Prove logically that the diagonals of a rectangle are equal. 5. Let ABCD be a quadrilateral in which AB = CD and AD = BC. Prove that ABCD is a parallelogram. 6. A field is in the form parallelogram, whose perimeter is 450 m and one of its sides is larger than the other by 75m. Find the length of all sides. 7. In quadrilateral ABCD, if AC=BD and AD=BC, prove that ABCD is a trapezium. 8. Prove that diagonals of a parallelogram bisect each other. 9. If diagonals of a quadrilateral bisect each other. Prove that it is parallelogram. Solve the following: 4 Marks 1. Rhombus is a parallelogram. Justify. 2. The side of a square ABCD is 5 cm and another square PQRS has perimeter equal to 40cm. Find the ratio of the perimeter of ABCD to perimeter of PQRS. Find the ratio of the area ABCD to the area of PQRS. 3. The sides of a rectangular park are in the ratio 4:3. If the area is 1728 m 2. Find the cost of fencing it at the ratio of Rs. 2.50/m. Lesson : Unit 1 Mensuration Chapter 4 I. Choose the correct answer: 1. A cube has faces. a) 4 b) 2 c) 6 d) 8 2. Faces of cube are a) Rectangle b) Square c) rhombus d) Parallelogram. 3. A cuboid has faces. a) 4 b) 2 c) 6 d) 8 4. The shape of match box is a) Square b) rectangle c) cube d) cuboid. 5. The line along which the surface meet is called a) Vertex b) Edge c) Side d) Face. II Fill in the blanks:

17 1. Finding the area of a planar objects or volume of a three dimensional object is called 2. A cube has pairs of parallel planes. 3. The faces of cuboid are in shape. 4. A solid is dimensional figures. 5. The sum of all the areas of all the six surfaces of a cuboid is called 6. The measure of the space occupied by a solid is called 2 Mark 1. Find the length of each side of a cube having the total surface area 294 cm Find the total surface area of the cuboid with l=4m, b=3m and h=1.5m. 3. Find the area of four walls of a room. Whose length is 3.5m breadth 2.5m and height 3m. 4. A closed box is 40cm long 50cm wide and 60cm deep. Find the area of the foil needed for covering top. 5. Find the lateral surface area of a cuboid of 8m long 5m broad and 3.5m high. 6. Lateral surface area of a cube is 64m 2. Find the length of its side. 7. Find the total surface area of a cuboid whose length; breadth and height are 23cm. 14cm and 15cm respectively. 3 Marks: 1. How many tiles each of 30cm 20cm are required to cover the floor of hall of dimension 15m by 12m? 2. Find the area of a metal sheet required to make a cube of length 2m. Find the cost of metal sheet required at the rate of Rs. 8/m 2 to make the cube. 3. Find the cost of white washing the four walls of cubical room of side 4m at the rate of Rs.20/m A match box measure 4cm, 2.5cm, and volume of the packet containing 12 such boxes. 5. How many 3 meter cubes can be cut from a cuboid measuring 18m 12m 9m. 6. Two cubes, each of volume 512cm 3 are joined end to end. Find the lateral and total surface area of the resulting cuboid. 7. Find the area of four walls of a room having length, breadth and height as 8m, 5m and 3m respectively. Find the cost of white washing the walls at the rate of Rs. 15/m 2. 4 Marks: 1. Find the lateral surface area and total surface area of cuboid which is 8m long. 5m broad and 3.5m high. 2. The total surface area of a cube is 600cm 2. Find the lateral surface area of the cube. 3. The dimensions of a room are l=8m, b=5m, h=4m. find the cost of distempering its four walls at the rate of Rs. 40/m 2 4. A room is 4.8m long, 3.6m broad and 2m high. Find the cost of laying tiles its floor and its four walls at the rate of Rs. 100/m 2

18 5. A cubical box has edge 10cm and another cuboidal box is 12.5cm long, 10cm wide and 8cm high. (i) Which box has smaller total surface area? (ii) If each edge of the cube is doubled, how many times will its T.S.A increase? 6. Three metal cubes whose edges measure 3m, 4cm and 4cm. respectively are melted to form a single cube. Find. (i) Side length (ii) Total surface area of the new cube. What is the difference between the total surface area of the new cube and the sum of total surface areas of the original three cubes? 7. The length, breadth and height of a cuboid are in the ratio 6:5:3. If the total surface area is 504cm 2. Find its dimension. Also find the volume of the cuboid. 8. How many m 3 of soil has to be excavated from a rectangular well 28m deep and whose base dimensions are 10m and 8m. Also find the cost of plastering its vertical walls at the rate of Rs. 15/m A Solid cubical box of fine wood costs Rs.256 at the rate Rs. 500/m 3. Find its volume and length of each side. 10. Find the total surface area and volume of a cube whose length is 12cm. 11. A room is 6m long, 4m broad and 3m high. Find the cost of laying tiles on its floor and four walls the cost of Rs. 80/m Suppose the perimeter of one face of a cube is 24cm. what is its volume.

For all questions, E. NOTA means none of the above answers is correct. Diagrams are NOT drawn to scale.

For all questions, E. NOTA means none of the above answers is correct. Diagrams are NOT drawn to scale. For all questions, means none of the above answers is correct. Diagrams are NOT drawn to scale.. In the diagram, given m = 57, m = (x+ ), m = (4x 5). Find the degree measure of the smallest angle. 5. The