1 Marks Question

Question 11 Mark

By applying $SAS$ congruence condition, state which of the following pairs of triangle are congruent. State the result in symbolic form.

Answer

We have $OA = OC, OB = OD$ and
$\angle\text{AOB}=\angle\text{COD}$ which are vertically opposite angles.
Therefore by $SAS$ condition,$\triangle\text{AOC}\cong\triangle\text{BOD}$

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Question 21 Mark

Fill in the blanks:
Two rectangles are congruent if .

Answer

Two rectangles are congruent if their lengths are equal and their breadths are also equal. The opposite sides of a rectangle are equal. So if two rectangles have lengths of the same size and breadths of the same size, then they are congruent to each other.

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Question 31 Mark

In $\triangle\text{ABC},$ it is known that $\angle\text{B}=\text{C}$. Imagine you have another copy of $\triangle\text{ABC}$. Is $\triangle\text{ACB}\cong\triangle\text{ACB}?$

Answer

Yes $\triangle\text{ABC}\cong\triangle\text{ACB}$.

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Question 41 Mark

State the condition by which the following pairs of triangles are congruent.

Answer

$AD = BC$ and $\angle\text{DAC}=\angle\text{BCA}$
Therefore, by $SAS$ condition $\triangle\text{ABC}\cong\triangle\text{ADC}$.

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Question 51 Mark

Following statement are true or false;
If two triangles are equal in area, they are congruent.

Answer

False.
Area of a triangle $= 12\ \times $ base $\times $ height
Two triangles can have the same area but the lengths of their sides can vary and hence they cannot be congruent.

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Question 61 Mark

$\triangle\text{ABC}$ is isosceles with $AB = AC. AD$ is the altitude from $A$ on $BC$.
Is $\triangle\text{ABD}\cong\triangle\text{ACD}?$

Answer

Yes, $\triangle\text{ABD}\cong\triangle\text{CBD}$ by $RHS$ congruence condition.

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Question 71 Mark

In $\triangle\text{ABC},$ it is known that $\angle\text{B}=\text{C}$. Imagine you have another copy of $\triangle\text{ABC}$. Is it true to say that $AB = AC$?

Answer

Yes it is true to say that $AB = AC$ since $\angle\text{ABC}=\angle\text{ACB}$.

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Question 81 Mark

$\triangle\text{ABC}$ is isosceles with $AB = AC$. $AD$ is the altitude from $A$ on $BC$. Is it true to say that $BD = DC$?

Answer

Yes, it is true to say that $BD = DC$ $(C.P.C.T.)$ since we have already proved that the two triangles are congruent.

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Question 91 Mark

$\triangle\text{ABC}$ is isosceles with $AB = AC$. Line segment $AD$ bisects $\angle\text{A}$ and meets the base $BC$ in $D$.
Is it true to say that $BD = DC$?

Answer

Now, $\triangle\text{ABD}\cong\triangle\text{ACD}$ therefore by $c.p.c.t.$, $BD = DC$.

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Question 101 Mark

By applying $SAS$ congruence condition, state which of the following pairs of triangle are congruent. State the result in symbolic form.

Answer

We have $BD = DC$ and
$\angle\text{ADB}=\angle\text{ADC}=90^\circ$
Therefore, by SAS condition $\triangle\text{ADB}\cong\triangle\text{ADC}$.

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Question 111 Mark

By applying $SAS$ congruence condition, state which of the following pairs of triangle are congruent. State the result in symbolic form.

Answer

We have $AB = DC$ and
$\angle\text{ABD}=\angle\text{CDB}$
Therefore, by $SAS$ condition $\triangle\text{ABD}\cong\triangle\text{CBD}$.

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Question 121 Mark

State the condition by which the following pairs of triangles are congruent.

Answer

$AC = BD, AD =BC$ and $AB = BA$
Therefore, by SAS condition $\triangle\text{ABD}\cong\triangle\text{BAC}$.

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Question 131 Mark

In $\triangle\text{ABC},$ it is known that $\angle\text{B}=\text{C}$. Imagine you have another copy of $\triangle\text{ABC}$. State the three pairs of matching parts you have used to answer $(i)$.

Answer

We have used $\angle\text{ABC}=\angle\text{ACB}$ and $\angle\text{ACB}=\angle\text{ABC}$ again.
Also $BC = CB$

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Question 141 Mark

$\triangle\text{ABC}$ is isosceles with $AB = AC$. Line segment $AD$ bisects $\angle\text{A}$ and meets the base $BC$ in $D$. Is $\triangle\text{ADB}\cong\triangle\text{ADC}?$

Answer

We have $AB = AC$ (Given) $\angle\text{BAD}=\angle\text{CAD}$ ($AD$ bisects $\angle\text{BAC}$ ) Therefore by $SAS$ condition of congruence, $\triangle\text{ABD}\cong\triangle\text{ACD}$

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Question 151 Mark

Following statement are true or false; If two squares have equal areas, they are congruent.

Answer

True.
Area of a square = side $\times $ side
Therefore, two squares that have the same area will have sides of the same lengths. Hence they will be congruent.

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Question 161 Mark

Fill in the blanks: Two angles are congruent if .

Answer

Two angles are congruent if their measures are the same. On superposition, we can see that the angles are equal.

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Question 171 Mark

Fill in the blanks:Two squres are congruent if .

Answer

Two squres are congruent if their sides are equal. All the sides of a square are equal and if two squares have equal sides, then all their sides are of the same length. Also angles of a square are $90^{\circ}$ which is also the same for both the squares.

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Question 181 Mark

Fill in the blanks: Two line segments are congruent if .

Answer

Two line segments are congruent if they have the same length, since they can superpose on each other.

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Question 191 Mark

Which of the following pairs of triangle are congruent by $ASA$ condition?

Answer

We have only $BC = QR$ but none of the angles of $\triangle\text{ABC} $ and $\triangle\text{PQR}$ are equal.
Therefore, $\triangle\text{ABC}\ncong\triangle\text{PQR}$.

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Question 201 Mark

By applying $SAS$ congruence condition, state which of the following pairs of triangle are congruent. State the result in symbolic form.

Answer

We have $BC = QR$,
$\angle\text{ABC}=\angle\text{PQR}=90^\circ$ and $AB = PQ$
Therefore, by $SAS$ condition $\triangle\text{ABC}\cong\triangle\text{PQR}$.

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Question 211 Mark

State the condition by which the following pairs of triangles are congruent.

Answer

$AB = AD$ and $\angle\text{BAC}=\angle\text{DAC}$
Therefore, by $SAS$ condition $\triangle\text{BAC}\cong\triangle\text{DAC}$.

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Question 221 Mark

State the condition by which the following pairs of triangles are congruent.

Answer

$AB = AD, BC =CD$ and $AC = CA$
Therefore, by $SAS$ condition $\triangle\text{ABC}\cong\triangle\text{ADC}$.

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Question 231 Mark

$\triangle\text{ABC}$ is isosceles with $AB = AC$. $AD$ is the altitude from $A$ on $BC$. State the pairs of matching parts you have used to answer $(i)$.

Answer

We have used, hypotenuse $AB =$ hypotenuse$ AC$ $AD = DA$ and $\angle\text{ADB}=\angle\text{ADC}=90^\circ$ ($\text{AD}\perp\text{BC}$ at point $D$)

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Question 241 Mark

Following statement are true or false; All squares are congruent.

Answer

False.
All the sides of a square are of equal length.
However, different squares can have sides of different lengths. Hence all squares are not congruent.

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Question 251 Mark

Fill in the blanks: Two circles are congruent if .

Answer

Two circles are congruent if their radii are of the same length. Then the circles will have the same diameter and thus will be congruent to each other.

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Question 261 Mark

Following statement are true or false; If two rectangles have equal area, they are congruent.

Answer

False.
Area of a rectangle = length $\times$ breadth
Two rectangles can have the same area.
However, the lengths of their sides can vary and hence they are not congruent.
Example: Suppose rectangle $1$ has sides $8m$ and $8m$ and area $64$ meter square.
Rectangle $2$ has sides $16m$ and $4m$ and area $64$ meter square.
Then rectangle $1$ and $2$ are not congruent.

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Question 271 Mark

In figure, $\triangle\text{ABC}$ is isosceles with $AB = AC$. State if $\triangle\text{ABC}\cong\triangle\text{ACB}.$ If yes, state three relations that you use to arrive at your answer.

Answer

Yes, $\triangle\text{ABC}\cong\triangle\text{ACB}$ by $SSS$ condition.
Since, $ABC$ is an isosceles triangle, $AB = BC, BC = CB$ and $AC = AB$.

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Question 281 Mark

$\triangle\text{ABC}$ is isosceles with $AB = AC$. Line segment $AD$ bisects $\angle\text{A}$ and meets the base $BC$ in $D$. State the three pairs of matching parts used to answer $(i)$.

Answer

We have used $AB, AC$; $\angle\text{BAD}=\angle\text{CAD}$; $AD, DA.$

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