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Page No 123:
Question 1:
Mark the correct alternative in each of the following:
If ABC ΔLKM, then side of ΔLKM equal to side AC of ΔABC is
(a) LK
(b) KM
(c) LM
(d) None of these
Answer:
It is given that
As triangles are congruent, same sides will be equal.
So
Hence (c).
Page No 123:
Question 2:
If ΔABC ΔABC is isosceles with
(a) AB = AC
(b) AB = BC
(c) AC = BC
(d) None of these
Answer:
It is given that and is isosceles
Since triangles are congruent so as same side are equal
Hence (a).
Page No 123:
Question 3:
If ΔABC ΔPQR and ΔABC is not congruent to ΔRPQ, then which of the following is not true:
(a) BC = PQ
(b) AC = PR
(c) AB = PQ
(d) QR = BC
Answer:
If and is not congruent to
Since and compare corresponding sides you will see
(As )
Hence (a) , is not true.
Page No 123:
Question 4:
In triangles ABC and PQR three equality relations between some parts are as follows:
AB = QP, ∠B = ∠P and BC = PR
State which of the congruence conditions applies:
(a) SAS
(b) ASA
(c) SSS
(d) RHS
Answer:
In and
It is given that
Since two sides and an angle are equal so it obeys
Hence (a).
Page No 123:
Question 5:
In triangles ABC and PQR, if ∠A = ∠R, ∠B = ∠P and AB = RP, then which one of the following congruence conditions applies:
(a) SAS
(b) ASA
(c) SSS
(d) RHS
Answer:
In and
It is given that
Since given two sides and an angle are equal so it obeys
Hence (b).
Page No 124:
Question 6:
In ΔPQR ΔEFD then ED =
(a) PQ
(b) QR
(c) PR
(d) None of these
Answer:
If
We have to find
Since, as in congruent triangles equal sides are decided on the basis of “how they are named”.
Hence (c).
Page No 124:
Question 7:
If ΔPQR ΔEFD, then ∠E =
(a) ∠P
(b) ∠Q
(c) ∠R
(d) None of these
Answer:
If
Then we have to find
From the given congruence, as equal angles or equal sides are decided by the location of the letters in naming the triangles.
Hence (a)
Page No 124:
Question 8:
In a ΔABC, if AB = AC and BC is produced to D such that ∠ACD = 100°, then ∠A =
(a) 20°
(b) 40°
(c) 60°
(d) 80°
Answer:
In the triangle ABC it is given that
We have to find
Now (linear pair)
Since
So, (by isosceles triangle)
This implies that
Now,
(Property of triangle)
Hence (a).
Page No 124:
Question 9:
In an isosceles triangle, if the vertex angle is twice the sum of the base angles, then the measure of vertex angle of the triangle is
(a) 100°
(b) 120°
(c) 110°
(d) 130°
Answer:
Let be isosceles triangle
Then
Now it is given that vertex angle is 2 times the sum of base angles
(As)
Now
(Property of triangle)
(Since, and )
Hence (b).
Page No 124:
Question 10:
Which of the following is not a criterion for congruence of triangles?
(a) SAS
(b) SSA
(c) ASA
(d) SSS
Answer:
(b) ,as it does not follow the congruence criteria.
Page No 124:
Question 11:
In the given figure, the measure of ∠B'A'C' is
(a) 50°
(b) 60°
(c) 70°
(d) 80°
Answer:
We have to find
Since triangles are congruent
So
Now in
(By property of triangle)
Hence (b) .
Page No 124:
Question 12:
If ABC and DEF are two triangles such that ΔABC ΔFDE and AB = 5cm, ∠B = 40°
(a) DF = 5cm, ∠F = 60°
(b) DE = 5cm, ∠E = 60°
(c) DF = 5cm, ∠E = 60°
(d) DE = 5cm, ∠D = 40°
Answer:
It is given thatand
So and
Now, in triangle ABC,
Therefore,
Hence the correct option is (c).
Page No 124:
Question 13:
In the given figure, AB ⊥ BE and FE ⊥ BE. If BC = DE and AB = EF, then ΔABD is congruent to
(a) ΔEFC
(b) ΔECF
(c) ΔCEF
(d) ΔFEC
Answer:
It is given that
And
(Given)
(Given)
So (from above)
Hence
From (d).
Page No 124:
Question 14:
In the given figure, if AE || DC and AB = AC, the value of ∠ABD is
(a) 70°
(b) 110°
(c) 120°
(d) 130°
Answer:
We have to find the value of in the following figure.
It is given that
(Vertically apposite angle)
Now (linear pair) …… (1)
Similarly (linear pair) …… (2)
From equation (1) we have
Now (same exterior angle)
(Interior angle)
Now
So
Since
Hence (b).
Page No 124:
Question 15:
In the given figure, ABC is an isosceles triangle whose side AC is produced to E. Through C, CD is drawn parallel to BA. The value of x is
(a) 52°
(b) 76°
(c) 156°
(d) 104°
Answer:
We are given that;
, is isosceles
And
We are asked to find angle x
From the figure we have
Therefore,
Since, so
Now
Hence (d) .
Page No 124:
Question 16:
In the given figure, if AC is bisector of ∠BAD such that AB = 3 cm and AC = 5 cm, then CD =
(a) 2 cm
(b) 3 cm
(c) 4 cm
(d) 5 cm
Answer:
It is given that
, is bisector of
We are to find the side CD
Analyze the figure and conclude that
(As in the two triangles are congruent)
In
So
Hence (c) .
Page No 125:
Question 17:
D, E, F are the mid-point of the sides BC, CA and AB respectively of ΔABC. Then ΔDEF is congruent to triangle
(a) ABC
(b) AEF
(c) BFD, CDE
(d) AFE, BFD, CDE
Answer:
It is given that, and are the mid points of the sides, andrespectively of
(By mid point theorem)
(As it is mid point)
Now in and
(Common)
(Mid point)
(Mid point)
Hence (d)
Page No 125:
Question 18:
ABC is an isosceles triangle such that AB = AC and AD is the median to base BC. Then, ∠BAD =
(a) 55°
(b) 70°
(c) 35°
(d) 110°
Answer:
It is given that, AB=AC and Ad is the median of BC
We know that in isosceles triangle the median from he vertex to the unequal side divides it into two equal part at right angle.
Therefore,
(Property of triangle)
Hence (a) .
Page No 125:
Question 19:
In the given figure, X is a point in the interior of square ABCD. AXYZ is also a square. If DY = 3 cm and AZ = 2 cm, then BY =
(a) 5 cm
(b) 6 cm
(c) 7 cm
(d) 8 cm
Answer:
In the following figure we are given
Where ABCD is a square and AXYZ is also a square
We are asked to find BY
From the above figure we have XY=YZ=AZ=AX
Now in the given figure
So,
Now inn triangle ΔAXB
So
Hence (c) .
Page No 125:
Question 20:
In the given figure, ABC is a triangle in which ∠B = 2∠C. D is a point on side BC such that AD bisects ∠BAC and AB = CD. BE is the bisector of ∠B. The measure of ∠BAC is
(a) 72°
(b) 73°
(c) 74°
(d) 95°
Answer:
It is given that
,
AB = CD
We have to find
Now AB = CD
AB = BD
Now the triangle is isosceles
Let
So
Now
Since
Hence (a) .
Page No 125:
Question 21:
Each of the following questions contains STATEMENT-1 (Assertion) and STATEMENT-2 (Reason) and has following four choices (a), (b), (c) and (d), only one of which is the correct answer. Mark the correct choice.
(a) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
(b) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
(c) Statement-1 is true, Statement-2 is false.
(d) âStatement-1 is false, Statement-2 is true.
Statement-1 (Assertion): If AD is median of âABC , then perimeter of âABC is greater than 2 AD.
Statement-2 (Reason): The sum of any two sides of a triangle is greater than the third side.
Answer:
Statement-1
Given: AD is median of
In ,
.....(1) [sum of any two sides of a triangle is greater than the third side]
Similarly, in
AC + CD > AD .....(2)
Now, on adding (1) and (2), we get
AB + BD + AC + CD > 2AD
AB + BC + CA > 2AC ()
Thus, it is true.
Statement-2: True, the sum of any two sides of a triangle is greater than the third side.
So, Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
Hence, the correct answer is option (a).
Page No 125:
Question 22:
Each of the following questions contains STATEMENT-1 (Assertion) and STATEMENT-2 (Reason) and has following four choices (a), (b), (c) and (d), only one of which is the correct answer. Mark the correct choice.
(a) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
(b) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
(c) Statement-1 is true, Statement-2 is false.
(d) âStatement-1 is false, Statement-2 is true.
Statement-1 (Assertion): In an equilateral triangle ABC, if AD is the median, then AB + AC > 2 AD.
Statement-2 (Reason): In a right triangle, hypotenuse is the longest side.
Answer:
Statement-1
Given that AD is the median of
In ,
.....(1) [sum of any two sides of a triangle is greater than the third side]
Similarly, in
AC + CD > AD .....(2)
Now, on adding (1) and (2), we get
AB + BD + AC + CD > 2AD
AB + BC + CA > 2AC ()
Thus, it is true.
Statement-2:
Let us consider a right-angled triangle ABC, right-angled at B.
In âABC,
∠A + ∠B + ∠C = 180° (Angle sum property)
∠A + 90o + ∠C = 180°
∠A + ∠C = 90°
Hence, the other two angles have to be acute (i.e., less than 90o ).
Thus, ∠B is the largest angle in âABC.
So, ∠B > ∠A and ∠B > ∠C
Therefore, AC > BC and AC > AB [Using theorem 7.7 of triangles, in any triangle, the side opposite to the larger (greater) angle is longer.]
Therefore, AC is the largest side in âABC.
However, AC is the hypotenuse of âABC.
Therefore, the hypotenuse is the longest side in a right-angled triangle.
Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
Page No 125:
Question 23:
Each of the following questions contains STATEMENT-1 (Assertion) and STATEMENT-2 (Reason) and has following four choices (a), (b), (c) and (d), only one of which is the correct answer. Mark the correct choice.
(a) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
(b) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
(c) Statement-1 is true, Statement-2 is false.
(d) âStatement-1 is false, Statement-2 is true.
Statement-1 (Assertion): The sum of any two sides of a triangle is greater than the third side.
Statement-2 (Reason): It is possible to construct a triangle with lengths of its sides as 4 cm, 3 cm, and 7 cm.
Answer:
Statement-1
True, the sum of any two sides of a triangle is greater than the third side.
Statement-2
Let AB = 4cm, BC = 3 cm and CA = 7 cm
Now, AB + BC = 4 + 3 = 7 cm = CA
Thus, it is not possible to construct a triangle with sides of 4 cm, 3 cm, and 7 cm.
Hence, the correct answer is option (a).
Page No 125:
Question 24:
Each of the following questions contains STATEMENT-1 (Assertion) and STATEMENT-2 (Reason) and has following four choices (a), (b), (c) and (d), only one of which is the correct answer. Mark the correct choice.
(a) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
(b) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
(c) Statement-1 is true, Statement-2 is false.
(d) âStatement-1 is false, Statement-2 is true.
Statement-1 (Assertion): If âABC ≅ âRPQ, then BC = QR
Statement-2 (Reason): Corresponding parts of two congruent triangle are equal.
Answer:
Statement-1
If âABC ≅ âRPQ.
We know that the corresponding parts of two congruent triangles are equal.(Statement-2)
∴ AB = RP,
BC = PQ
and CA = QR
Thus, Statement-1 is false.
So, âStatement-1 is false, Statement-2 is true.
Hence, the correct answer is option (d).
Page No 125:
Question 25:
Each of the following questions contains STATEMENT-1 (Assertion) and STATEMENT-2 (Reason) and has the following four choices (a), (b), (c) and (d), only one of which is the correct answer. Mark the right choice.
(a) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
(b) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
(c) Statement-1 is true, Statement-2 is false.
(d) âStatement-1 is false, Statement-2 is true.
Statement-1 (Assertion): If any two angles and a non-included side of one triangle are equal to the corresponding angles and side of another triangle, then the two triangles are concurrent.
Statement-2 (Reason): If two angles and a side of one triangle are equal to two angles and a side of another triangle, then the two triangles must be congruent.
Answer:
Statement-1:
Yes, the statement is true because the two triangles can be congruent by either AAS or ASA congruence criteria.
If any two angles and a non-included side of one triangle are equal to the corresponding angles and side of the other triangle, then the two triangles are concurrent by AAS criteria.
Statement-2:
True, If two angles and a side of one triangle are equal to two angles and a side of another triangle, then the two triangles must be congruent by AAS or ASA criteria.
Hence, the correct answer is an option (a).
Page No 126:
Question 26:
Each of the following questions contains STATEMENT-1 (Assertion) and STATEMENT-2 (Reason) and has following four choices (a), (b), (c), and (d), only one of which is the correct answer. Mark the correct choice.
(a) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
(b) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
(c) Statement-1 is true, Statement-2 is false.
(d) âStatement-1 is false, Statement-2 is true.
Statement-1 (Assertion): If two sides and the included angle of one triangle are equal to the corresponding sides and the included angle of the other triangle, then the two triangles are concurrent
Statement-2 (Reason): If two sides and an angle of one triangle are equal to two sides and an angle of another triangle, then the two triangles are congruent.
Answer:
Statement-1:
If two sides and the included angle of one triangle are equal to the corresponding sides and the included angle of the other triangle, then the two triangles are concurrent by SAS criteria.
Thus, it is true
Statement-2:
The two sides and the included angle of one triangle are equal to the two sides and the included angle of the other triangle, i.e., the SAS rule.
According to the SAS rule: If any two sides and the angle included between the sides of one triangle are equivalent to the corresponding two sides and the angle between the sides of the second triangle, then the two triangles are said to be congruent by the SAS rule.
Therefore, the angles should be included between the angles of their two given sides.
Thus, the given Statement is false.
Hence, the correct answer is option (c).
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