1 Marks Question

Question 11 Mark

In figure. $\triangle\text{PQR}\cong\triangle .........$

$i.$

$ii.$

Answer

In $\triangle\text{PQR}$ and $\triangle\text{XYZ},$
$PQ = XY = 3.5\ cm $
$QR = ZY = 5\ cm$
$\angle\text{PQR}=\angle\text{XYZ}=45^{\circ}$
By $\ce{SAS}$ congruence criterion, $\triangle\text{PQR}\cong\triangle\text{XZY}.$

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Question 21 Mark

In Figure. which pairs of triangles are congruent by $SAS$ congruence criterion (condition)? If congruent, write the congruence of the two triangles in symbolic form.

Answer

Not congruent, because angle is not included between two sides.

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Question 31 Mark

In an isosceles triangle, two angles are always ________.

Answer

In an isosceles triangle, two angles are always equal. Solution: In an isosceles triangle, two angles are always equal. Since, if two sides are equal, then the angles opposite them are equal.

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Question 41 Mark

It is possible to have a triangle in which each angle is equal to $60^\circ .$

Answer

The triangle in which each angle is equal to $60^\circ $ is called an equilateral triangle.

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Question 51 Mark

It is possible to have a triangle in which two angles are acute.

Answer

True.Solution:
In a triangle, atleast two angles must be acute angle.

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Question 61 Mark

If two sides and one angle of a triangle are equal to the two sides and angle of another triangle, then the two triangles are congruent.

Answer

False. Solution: Because if two sides and the angle included between them of the other triangle, then the two triangles will be congruent.

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Question 71 Mark

It is possible to have a triangle in which two of the angles are right angles.

Answer

If in a triangle two angles are right angles, then third angle $= 180^\circ - (90^\circ + 90^\circ ) = 0^\circ ,$ which is not possible.

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Question 81 Mark

Two line segments are congruent, if they are of _________ lengths.

Answer

Two line segments are congruent, if they are of equal lengths.
Solution:
Two line segments are congruent, if they are of equal lengths.

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Question 91 Mark

In an isosceles triangle, angles opposite to equal sides are __________.

Answer

In an isosceles triangle, angles opposite to equal sides are equal.Solution:
In an isosceles triangle, angles opposite to equal sides are equal. Since, if two angles are equal then the sides opposite to them are also equal.

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Question 101 Mark

The sum of an exterior angle of a triangle and its adjacent angle is always ________.

Answer

The sum of an exterior angle of a triangle and its adjacent angle is always, $180^\circ$ because they form a linear pair.

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Question 111 Mark

If one angle of a triangle is equal to the sum of other two, then the measure of that angle is ________.

Answer

Let the angles of a triangle be $a, b$ and $c.$
It is given that, $a = b + c$
we also know that, $a + b + c = 180^\circ [$angle sum property of atriangle$]$
$\Rightarrow a + a = 180^\circ $
$\Rightarrow 2a = 180^\circ $
$\Rightarrow \ \text{a}=\frac{180^{\circ}}{2}$
$\Rightarrow \text{a}=90^{\circ}$
Hence, the measure of that angle is $90^\circ .$

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Question 121 Mark

It is possible to have a triangle in which two of the angles are obtuse.

Answer

Obtuse angles are those angles which are greater than $90^\circ .$
So, sum of two obtuse angles will be greater than $180^\circ ,$
which is not possible as the sum of all the angles of a triangle is $180^\circ .$

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Question 131 Mark

In Figure. wo triangles are congruent by $\ce{RHS}.$
$i.$

$ii.$

Answer

True. Solution:
$i.$

$ii.$

In $\triangle\text{ABC}, \text{AC}=\sqrt{\text{AB}^{2}+\text{BC}^{2}} =\sqrt{\text{4}^{2}+\text{5}^{2}}=\sqrt{41\ \text{cm}} [$by Pythagoras theoram$]$
In $\triangle\text{PQR}, \text{PR}=\sqrt{\text{PQ}^{2}+\text{QR}^{2}} =\sqrt{\text{4}^{2}+\text{5}^{2}}=\sqrt{41\ \text{cm}} [$by Pythagoras theoram$]$
Now, in $\triangle\text{ABC}$ and $\triangle\text{PQR},$
$\text{AB}=\text{PQ}=4\ \text{cm}$
$\text{AC}=\text{PR}=\sqrt{41\ \text{cm}}$
$\angle\text{ABC}=\angle\text{PQR}=90^{\circ}$
By $\ce{RHS}$ congruence criterian, $\triangle\text{ABC}\cong\triangle\text{PQR}$

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Question 141 Mark

If two angles of a triangle are equal, the third angle is also equal to each of the other two angles.

Answer

False. Solution: In an isosceles triangle, always two angles are equal and not the third one.

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Question 151 Mark

Two figures are congruent, if they have the same shape.

Answer

False. Solution: Two figures are congruent, if they have the same shape and same size.

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Question 161 Mark

$AAS$ congruence criterion is same as ASA congruence criterion.

Answer

In $ASA$ congruence criterion, the side $‘S’$ included between the two angles of the triangle. In $AAS$ congruence criterion, side ‘S’ is not included between two angles.

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Question 171 Mark

In the following pair of triangles of Figure. the lengths of the sides are indicated along the sides. By applying $SSS$ congruence criterion, determine which triangles are congruent. If congruent, write the results in symbolic form.

Answer

$\triangle\text{OAB} \cong \triangle\text{DOE}$

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Question 181 Mark

It is possible to have a triangle in which each angle is less than $60^\circ .$

Answer

The sum of all angles in a triangle is equal to $180^\circ .$ So, all three angles can never be less than $60^\circ .$

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Question 191 Mark

Answer

$\triangle\text{ABC} \cong \triangle\text{NLM}$

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Question 201 Mark

Two squares are congruent, if they have same __________.

Answer

Two squares are congruent, if they have same side.
Solution:
Two squares are congruent, if they have same side.

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Question 211 Mark

Find the measure of $\angle\text{A}$ In Figure.

Answer

As we know, the measure of exterior angle is equal to the sum of opposite interior angles. $\therefore \ 115^{\circ}=65^{\circ}+\angle\text{A}$ $\Rightarrow \ \angle\text{A}=115^{\circ}-65^{\circ}=50^{\circ}$

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Question 221 Mark

In Figure.
$i. \angle\text{TPQ}=\angle.........+ \angle.........$
$ii. \angle\text{UQR}=\angle.........+ \angle.........$
$iii. \angle\text{PRS}=\angle.........+ \angle.........$

Answer

Exterior angle property,
The measure of an exterior angle is equal to the sum of the two opposite interior angles.
$\angle\text{TPQ}=\angle\text{PQR}+\angle\text{PRQ}$
$\angle\text{UQR}=\angle\text{QRP}+\angle\text{QPR}$
$\angle\text{PRS}=\angle\text{RPQ}+\angle\text{RQP}$

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Question 231 Mark

Median is also called ________ in an equilateral triangle.

Answer

Median is also called an altitude in an equilateral triangle. Solution: Median is also called an altitude in an equilateral triangle.

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Question 241 Mark

A right-angled triangle may have all sides equal.

Answer

False.
Solution:
Hypotenuse is always the greater than the other two sides of the right angled triangle.

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Question 251 Mark

Answer

$\triangle\text{PSR} \cong \triangle\text{RQP}$

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Question 261 Mark

The longest side of a right angled triangle is called its _______.

Answer

The longest side of a right angled triangle is called its Hypontenuse. Solution: Hypotenuse is the longest side of a right angled triangle.

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Question 271 Mark

Answer

$\triangle\text{LMN} \cong \triangle\text{LON}$

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Question 281 Mark

If $\triangle\text{PQR}$ and $ \triangle\text{XYZ}$ are congruent under the correspondence $QPR \leftrightarrow XYZ,$ then:
$i. \angle\text{R} = ..........$
$ii. \text{QR}=..........$
$iii. \angle\text{P} = ..........$
$iv. \text{QP}=..........$
$v. \angle\text{Q}=..........$
$vi. \text{RP}=..........$

Answer

$i. \angle\text{R} = \angle\text{Z}$
$ii. \text{QR}=\text{XZ}$
$iii. \angle\text{P} = \angle\text{Y}$
$iv. \text{QP}=\text{XY}$
$v. \angle\text{Q}=\angle\text{X}$
$vi. \text{RP}=\text{ZY}$

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Question 291 Mark

In Figure. $\triangle\text{ARO}\cong\triangle$ _________.

Answer

In Figure. $\triangle\text{ARO}\cong\triangle\text{PQO}$ Solution: In $\triangle\text{ARO}$ and $\triangle\text{PQO},$ $\angle\text{AOR}=\angle\text{POQ}$ [vertically opposite angles] $\angle\text{ARO}=\angle\text{PQO}=55^{\circ}$ [given] $\Rightarrow \ \angle\text{RAO}=\angle\text{QPO}$ Now, in $\triangle\text{ARO}$ and $\triangle\text{PQO},$ $ \ \angle\text{AOR}=\angle\text{POQ}$ [vertically opposite angles] $\text{AO} = \text{PO} = 2.5\text{cm}$ $ \ \angle\text{RAO}=\angle\text{QPO}$ By Ass congruence criterion, $\triangle\text{ARO}\cong\triangle\text{PQO}$ [proved above]

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Question 301 Mark

Two squares having same perimeter are congruent.

Answer

True. Solution: If two squares have same perimeter, then their sides will be equal. Hence, the squares will superimpose to each other.

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Question 311 Mark

If two angles and a side of a triangle are equal to two angles and a side of another triangle, then the triangles are congruent.

Answer

False.Solution:
if two angles and the side included between them of a triangle are equal to two angles and included a side between them of the other triangle, then triangles are congruent.

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Question 321 Mark

Without drawing the triangles write all six pairs of equal measures in following pairs of congruent triangles.
$\triangle\text{ABC} \cong \triangle\text{LMN}$

Answer

We know that, corresponding parts of congruent triangles are equal.
$\triangle\text{ABC} \cong \triangle\text{LMN}$
$\angle\text{A}=\angle\text{L},\angle\text{B}=\angle\text{M}$ and $\angle\text{C}=\angle\text{N},\text{AB}=\text{LM},\text{BC}=\text{MN}$ and $\text{AC}=\text{LN}$

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Question 331 Mark

In given pairs of triangles of Figure. applying only $\ce{ASA}$ congruence criterion, determine which triangles are congruent. Also, write the congruent triangles in symbolic form.
$i.$

$ii.$

Answer

$\triangle \text{XYZ} ≅ \triangle \text{LMN}$

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Question 341 Mark

In given pairs of triangles of Figure. applying only $\ce{ASA}$ congruence criterion, determine which triangles are congruent. Also, write the congruent triangles in symbolic form.
$i.$

$ii.$

Answer

Not possible, because there is not any included side equal.

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Question 351 Mark

In Figure. $\triangle$ _______ $\cong\triangle\text{PQR}.$

Answer

From the given figure, In $\triangle\text{DRQ}$ and $\triangle\text{PQR}$
$QR = QR [$common side$]$
$\angle\text{DRQ}=\angle\text{PQR}=70^{\circ}$
$\angle\text{DQR}=\angle\text{PRQ}=40^{\circ}$
By $ASA$ congruence criterion, $\triangle\text{DRQ}=\triangle\text{PQR}$

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Question 361 Mark

If three angles of two triangles are equal, triangles are congruent.

Answer

False. Solution: Consider two equilateral triangles with different sides.

Both $Δ\text{ABC}$ and $Δ\text{DEF}$ have same angles but their size is different. So, they are not congruent

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Question 371 Mark

Answer

$\triangle\text{ZYX} \cong \triangle\text{WXY}$

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Question 381 Mark

In given pairs of triangles of Figure. applying only $ASA$ congruence criterion, determine which triangles are congruent. Also, write the congruent triangles in symbolic form.

Answer

$∆\text{AOD} ≅ ∆\text{BOC}$

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Question 391 Mark

Sum of two sides of a triangle is greater than or equal to the third side.

Answer

False. Solution: Sum of two sides of a triangle is greater than the third side.

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Question 401 Mark

The congruent figures super impose each other completely.

Answer

True. Solution: Because congruent figures have same shape and same size.

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Question 411 Mark

If two triangles are congruent, then the corresponding angles are equal.

Answer

True. Solution: Because if two triangles are congruent, then their sides and angles are equal.

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Question 421 Mark

Without drawing the triangles write all six pairs of equal measures in following pairs of congruent triangles. $\triangle\text{XYZ} \cong \triangle\text{MLN}$

Answer

We know that, corresponding parts of congruent triangles are equal. $\triangle\text{XYZ} \cong \triangle\text{MLN}$ $\angle\text{X}=\angle\text{M},\angle\text{Y}=\angle\text{L}$ and $\angle\text{Z}=\angle\text{N},\text{XY}=\text{ML},\text{YZ}=\text{LN}$ and $\text{XZ}=\text{MN}$

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Question 431 Mark

$ABC$ is an isosceles triangle with $AB = AC$ and $D$ is the mid-point of base $BC$ Figure.
State three pairs of equal parts in the triangles $ABD$ and $ACD.$

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Question 441 Mark

The sum of the measures of three angles of a triangle is greater than $180^\circ .$

Answer

The sum of the measures of three angles of a triangle is always equal to $180^\circ .$

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Question 451 Mark

The top and bottom faces of a kaleidoscope are congruent.

Answer

True. Solution: Because they superimpose to each other.

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Question 461 Mark

In figure. $\triangle\text{PQR}\cong\triangle$ _______.

Answer

In $\triangle\text{PQR}$ and $\triangle\text{RSP},$
$QR = SP = 4.1\ cm$
$PR = PR [$common side$]$
$\angle\text{SPR}=\angle\text{QRP}=45^{\circ}$
By SAS congruence criterion, $\triangle\text{PQR}\cong\triangle\text{RSP}.$

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Question 471 Mark

If $M$ is the mid-point of a line segment $AB,$ then we can say that $AM$ and $MB$ are congruent.

Answer

Given that, $m$ is mid-point of line segment $AB,$
i.e. $AM = MB$ We know that,
two line segments are congruent that's why they are of same legnths.

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Question 481 Mark

Without drawing the triangles write all six pairs of equal measures in following pairs of congruent triangles. $\triangle\text{YZX} \cong \triangle\text{PQR}$

Answer

We know that, corresponding parts of congruent triangles are equal. $\triangle\text{YZX} \cong \triangle\text{PQR}$ $\angle\text{T}=\angle\text{P},\angle\text{Z}=\angle\text{Q}$ and $\angle\text{X}=\angle\text{R},\text{YZ}=\text{PQ},\text{ZX}=\text{QR}$ and $\text{YX}=\text{PR}$

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Question 491 Mark

If the hypotenuse of one right triangle is equal to the hypotenuse of another right triangle, then the triangles are congruent.

Answer

False. Solution: Two right angled triangles are congruent, if the hypotenuse and a side of one of the triangle are equal to the hypotenuse and one of the side of the other triangle.

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Question 501 Mark

It is possible to have a triangle in which each angle is greater than $60^\circ .$

Answer

If all the angles are greater than $60^\circ$ in a triangle, then the sum of all the three angles with exceed $180^\circ ,$ which cannot be possible in case of triangle

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