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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Question 511 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

$∆\text{ABD} ≅ ∆\text{CDB}$

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Question 521 Mark

Two angles are said to be, ________ if they have equal measures.

Answer

Two angles are said to be, congruent if they have equal measures.Solution:
Two angles are said to be congruent, if they have equal measures.

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Question 531 Mark

Measures of each of the angles of an equilateral triangle is ________.

Answer

Measures of each of the angles of an equilateral triangle is $60^\circ$ as all the angles in an equilateral triangle are equal. Let $x$ be the angle of equilateral. According to the angle sum property of a triangle.
$x + x + x = 180^\circ $
$\Rightarrow \ 3\text{x}={180^{\circ}}$
$\Rightarrow \ 3\text{x}=\frac{180^{\circ}}{3}$
$\Rightarrow \ \text{x}=60^{\circ}$

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Question 541 Mark

Answer

$\triangle\text{STU} \cong \triangle\text{PQR}$

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Question 551 Mark

Two acute angles are congruent.

Answer

False. Solution: Because the measure of two acute angles may be different.

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Question 561 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

Not possible, because the side is not included between two angles.

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Question 571 Mark

A one rupee coin is congruent to a five rupee coin.

Answer

False. Solution: Because they don’t have same shape and same size.

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Question 581 Mark

Answer

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

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Question 591 Mark

If the areas of two circles are the same, they are congruent.

Answer

True.
Solution:
Because areas of two circles will be equal only if their radii are equal and circle with same radii will superimpose to each other.

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Question 601 Mark

Every triangle has at least _________ acute angle (s).

Answer

Every triangle has atleast two acute angle (s). Solution: Every triangle has atleast two acute angles.

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Question 611 Mark

Answer

$\triangle\text{STU} \cong \triangle\text{SVU}$

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Question 621 Mark

If the areas of two rectangles are same, they are congruent.

Answer

False. Solution: Because rectangles with the different length and breadth may have equal areas. But, they will not superimpose to each other.

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Question 631 Mark

The ________ triangle always has altitude outside itself.

Answer

The obtuse triangle always has altitude outside itself. Solution: The obtuse angled triangle always has altitude outside itself.

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Question 641 Mark

$ABC$ is an isosceles triangle with $AB = AC$ and $D$ is the mid-point of base $BC$ Figure. Is $Δ\text{ABD} ≅ Δ\text{ACD}?$ If so why?

Answer

Given, $AB = AC$ and $BD = CD$ Yes, by SSS congruence criterion, $Δ\text{ABD} ≅ Δ\text{ACD}$

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Question 651 Mark

In Figure. $AB = AD$ and $\angle\text{BAC}=\angle\text{DAC}.$ then
$i. \triangle.........\cong\triangle\text{ABC}.$
$ii. \text{BC} =.................$
$iii. \angle\text{BCA}=..........$
$iv.$ Line segment $AC$ bisects $..........$ and $..........$

Answer

In $\triangle\text{ABC}$ and $\triangle\text{ADC},$
$\text{AB} = \text{AD} [$given$]$
$AC = AC [$common side$]$
$\angle\text{BAC}=\angle\text{DAC} [$given$]$
By $\ce{SAS}$ congruence criterion,
$\triangle\text{ADC}\cong\triangle\text{ABC}$
Now,$\text{ BC} =\text{ DC} [$by $\ce{CPCT}]$
Also, $\angle\text{BCA}=\angle\text{DCA} [$by $\ce{CPCT}]$
Line segment $AC$ bisects $\angle\text{BAD}$ and $\angle\text{BCD}.$
Since, $\angle\text{BAC}=\angle\text{DAC}$
and $\angle\text{BCA}=\angle\text{DCA}$

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Question 661 Mark

Sum of any two angles of a triangle is always greater than the third angle.

Answer

It is not necessary that sum of any two angles of a triangle is always greater than the third angle,
e.g. Let the angles of a triangle be $20^\circ , 50^\circ$ and $110^\circ ,$ respectively.
Hence, $20^\circ + 50^\circ = 70^\circ ,$ which is less than $110^\circ .$

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Question 671 Mark

If two legs of a right triangle are equal to two legs of another right triangle, then the right triangles are congruent.

Answer

True. Solution: If two legs of a right angled triangle are equal to two legs of another right angled triangle, then their third leg will also be equal. Hence, they will have same shape and same size.

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Question 681 Mark

If the areas of two squares is same, they are congruent.

Answer

True.
Solution:
Because two squares will have same areas only if their sides are equal and squares with same sides will superimpose to each other.

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Question 691 Mark

In given pairs of triangles of Figure. using only $RHS$ congruence criterion, determine which pairs of triangles are congruent. In case of congruence, write the result in symbolic form:

Answer

Not possible, because there is not any right angle.

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Question 701 Mark

In Figure.$AD \bot BC$ and $AD$ is the bisector of angle $BAC.$ Then, $\triangle\text{ABD} \cong\triangle\text{ACD}$ by $RHS.$

In $\triangle\text{ABD}$ and $\triangle\text{ACD},$
$\text{AD} = \text{AD}$ [common side] $\angle\text{BAD}=\angle\text{CAD}$
$[\because AD$ is the bisector of $\angle\text{BAC}]$ By $ASA$ congruence criterion, $\triangle\text{ABD}\cong\triangle\text{ACD}$

Answer

In $\triangle\text{ABD}$ and $\triangle\text{ACD},$
$\text{AD} = \text{AD}$ [common side]
$\angle\text{BAD}=\angle\text{CAD}$
$[\because AD$ is the bisector of $\angle\text{BAC}]$
By $ASA$ congruence criterion, $\triangle\text{ABD}\cong\triangle\text{ACD}$

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Question 711 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 721 Mark

Two rectangles are congruent, if they have same ________ and _________.

Answer

Two rectangles are congruent, if they have same length and breadth. Solution: Two rectangles are congruent, if they have same length and breadth.

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Question 731 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

$∆\text{MNO} ≅ ∆\text{PON}$

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Question 741 Mark

In $\triangle\text{ABC}, AB = 3.5\ cm, AC = 5\ cm, BC = 6\ cm$ and in $\triangle\text{PQR},$ $PR= 3.5\ cm, PQ = 5\ cm, RQ = 6\ cm.$ Then $\triangle\text{ABC} \cong \text{PQR}.$

Answer

In $\triangle\text{ABC}$ and $\triangle\text{PRQ},$
$AB = PR = 3.5\ cm, BC = RQ = 6\ cm$ and $AC = PQ = 5\ cm$
By $SSS$ congruence criterion, $\triangle\text{ABC}\cong\triangle\text{PRQ}$

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Question 751 Mark

Two right angles are congruent.

Answer

True. Solution: Since, the measure of right angles is always same.

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Question 761 Mark

The difference between the lengths of any two sides of a triangle is smaller than the length of third side.

Answer

The difference between the lengths of any two sides of a triangle is smaller than the length of third side. $AB - BC < AC$

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Question 771 Mark

If hypotenuse and an acute angle of one right triangle are equal to the hypotenuse and an acute angle of another right triangle, then the triangles are congruent.

Answer

In $\triangle\text{ABC}$ and $\triangle\text{PQR},$
$\angle\text{B}=\angle\text{Q}=90^{\circ}$
$\angle\text{C}=\angle\text{R}$ [given] $\angle\text{A}=\angle\text{P}$
Now, In $\triangle\text{ABC}$ and $\triangle\text{PQR},$
$\angle\text{A}=\angle\text{P}$
$\text{AC} =\text{ PR}$
$\angle\text{C}=\angle\text{R}$
By $ASA$ congruene criterian, $\triangle\text{ABC}\cong\triangle\text{PQR}$

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Question 781 Mark

It is possible to have a right-angled equilateral triangle.

Answer

In a right angled triangle, one angle is equal to $90^\circ$ and in equilateral triangle, all angles are equal to $60^\circ .$

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Question 791 Mark

Two circles having same circumference are congruent.

Answer

True.
Solution:
If two circles have same circumference, then their radii will be equal. Hence, the circles will superimpose to each other.

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Question 801 Mark

In a triangle, sum of squares of two sides is equal to the square of the third side.

Answer

False. Solution: Only in a right angled triangle, the sum of two shorter sides is equal to the square of the third side.

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Question 811 Mark

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

Answer

We know that, corresponding parts of congruent triangles are equal. $\triangle\text{STU} \cong \triangle\text{DEF}$ $\angle\text{S}=\angle\text{D},\angle\text{T}=\angle\text{E}$ and $\angle\text{U}=\angle\text{F},\text{ST}=\text{DE},\text{TU}=\text{EF}$ and $\text{SU}=\text{DF}$

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