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Maths 2 Marks Questions

Congruence · Jharkhand Board (JAC) STD 7 (Jharkhand - English Medium). 20 questions.

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Question 12 Marks
In each of the following pairs of right triangles, the measures of some parts are indicated along side. State by the application of $RHS$ congruence condition which are congruent. State each result in symbolic form.
Answer
In figure, $BC = DC$ hypoteuse $AC$ = hypoteuse $CA$ $\angle\text{ABC}=\angle\text{ADC}=90^\circ$
Therefore, by $RHS$, $\triangle\text{ABC}\cong\triangle\text{ADC}$.
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Question 22 Marks
In the following pair of triangle (Fig.), the lengths of the sides are indicated along sides. By applying $SSS$ condition, determine which are congruent. State the result in symbolic form.
Answer


In $\triangle\text{ACB}$ and $\triangle\text{ADB},$
$AC = AD$ (side)
$BC = BD$ (side)
and $AB = AB$ (side)
Therefore, by $SSS$ criterion of congruence, $\triangle\text{ACB}\cong\triangle\text{ADB}$
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Question 32 Marks
Draw any triangle $ABC$. Use $ASA$ condition to construct other triangle congruent to it.
Answer


We have drawn, $\triangle\text{ABC}$ with $\angle\text{ABC}=65^\circ$ and $\angle\text{ACB}=70^\circ$
We now construct $\triangle\text{PQR}\cong\triangle\text{ABC}$ has $\angle\text{PQR}=65^\circ$ and $\angle\text{PRQ}=70^\circ$
Also we construct $\triangle\text{PQR}$ such that $BC = QR$
Therefore by $ASA$ the two triangles are congruent.
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Question 42 Marks
In each of the following pairs of right triangles, the measures of some parts are indicated along side. State by the application of $RHS$ congruence condition which are congruent. State each result in symbolic form.
Answer
In figure, $\angle\text{ADC}=\angle\text{BCA}=90^\circ$ $AD = BC$ and
hypoteuse $AB$ = hypoteuse $AB$
Therefore, by $RHS$ $\triangle\text{ADB}\cong\triangle\text{ACB}$.
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Question 52 Marks
In Figure, $AX$ bisects $\angle\text{BAC}$ as well as $∠BDC$. State the three facts needed to ensure that $\triangle\text{ACD}\cong\triangle\text{ABD}$.
Answer
As per the given conditions,
$\angle\text{CAD}=\angle\text{BAD}$ and
$\angle\text{CDA}=\angle\text{BDA}$ (because $AX$ bisects $\angle\text{BAC}$)
$AD = DA$ (common)
Therefore, by $ASA$, $\triangle\text{ACD}\cong\triangle\text{ABD}$.
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Question 62 Marks
In $\triangle\text{PQR}\cong\triangle\text{CBD},$ Which side of $\triangle PQR$ equals $ED$?
Answer


In $\triangle\text{PQR}\cong\triangle\text{CBD},$
Therefore $PR = ED$ since the corresponding sides of congruent triangles are equal.
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Question 72 Marks
In the following pair of triangle (Fig.), the lengths of the sides are indicated along sides. By applying $SSS$ condition, determine which are congruent. State the result in symbolic form.
Answer
In $\triangle\text{ABO}$ and $\triangle\text{DOC},$ $AB = DC$ (side) $AO = OC$ (side) $BO = OD$ (side)
Therefore, by $SSS$ criterion of congruence, $\triangle\text{ABO}\cong\triangle\text{ODC}$
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Question 82 Marks
Line-segments $AB$ and $CD$ bisect each other at $O$.
$AC$ and $BD$ are joined forming triangles $AOC$ and $BOD$. State the three equality relations between the parts of the two triangles that are given or otherwise known. Are the two triangles congruent? State in symbolic form, which congruence condition do you use?
Answer
We have $AO = OB$ and $CO = OD$ since $AB$ and $CD$ bisect each other at $0$.
Also $\angle\text{AOC}=\angle\text{BOD}$ since they are opposite angles on the same vertex.
Therefore by $SAS$ congruence condition, $\triangle\text{AOC}\cong\triangle\text{BOD}$.
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Question 92 Marks
In Figure, $\ce{AB = DC}$ and $\ce{BC = AD}$.
$i.$ Is $\triangle\text{ABC}\cong\triangle\text{CBD}?$
$ii.$ What congruence condition have you used?
$iii.$ You have used some fact, not given in the question, what is that?
Answer
We have $\ce{AB = DC}$
$\ce{BC = AD}$
And $\ce{AC = AC}$
$i.$ Yes by $\ce{SSS}$ $\triangle\text{ABC}\cong\triangle\text{CBD}$
$ii.$ We have used Side congruence condition with one side common in both the triangles.
$iii.$ Yes, have used the fact that $\ce{AC = CA}.$
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Question 102 Marks
Which of the following pairs of triangle are congruent by $ASA$ condition?
Answer


We have,
Since $\angle\text{ABO}=\angle\text{CDO}=45^\circ$ and both are alternate angles.
$AB\ ||\ DC$, $\angle\text{BAO}=\angle\text{DCO}$ (alternate angle, $AB\ ||\ CD$ and $AC$ is a transversal line)
$\angle\text{ABO}=\angle\text{CDO}=45^\circ$ (given in the figure)
Also, $AB = DC$ (Given in the figure)
Therefore, by $ASA$ $\triangle\text{AOB}\cong\triangle\text{DOC}$.
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Question 112 Marks
In the following pair of triangle (Fig.), the lengths of the sides are indicated along sides. By applying $SSS$ condition, determine which are congruent. State the result in symbolic form.
Answer
In $\triangle\text{ABD}$ and $\triangle\text{FEC},$ $AB = FE$ (side) $AD = FC$ (side) and $BD = CE$ (side)
Therefore, by $SSS$ criterion of congruence, $\triangle\text{ABD}\cong\triangle\text{FEC}$
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Question 122 Marks
Triangles $ABC$ and $DBC$ have side $BC$ common, $AB = BD$ and $AC = CD$. Are the two triangles congruent? State in symbolic form, which congruence do you use? Does $\angle\text{ABD}$ equal $\angle\text{ACD}?$ Why or why not?
Answer
Yes, Given, $\triangle\text{ABC}$ and $\triangle\text{DBC}$ have side $BC$ common,
$AB = BD$ and $AC = CD$ By
$SSS$ criterion of congruency,
$\triangle\text{ABC}\cong\triangle\text{DBC}$ No,
$\angle\text{ABD}$ and $\angle\text{ACD}$ are not equal because $\text{AB}\ne\text{AC}$
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Question 132 Marks
In each of the following pairs of right triangles, the measures of some parts are indicated along side. State by the application of RHS congruence condition which are congruent. State each result in symbolic form.
Answer
In figure, $BO = CO$ hypoteuse $AO = $hypoteuse $DO$ $\angle\text{B}=\angle\text{C}=90^\circ$
Therefore, by $RHS$, $\triangle\text{AOB}\cong\triangle\text{DOC}$.
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Question 142 Marks
$\triangle\text{ABC}$ and $\triangle\text{ABD}$ are on a common base $AB$, and $\ce{AC = BD}$ and $\ce{BC = AD}$ as shown in Figure. Which of the following statements is true?
$i. \triangle\text{ABC}\cong\triangle\text{ABD}$
$ii. \triangle\text{ABC}\cong\triangle\text{ADB}$
$iii. \triangle\text{ABC}\cong\triangle\text{BAD}$
Answer
In $\triangle\text{ABC}$ and $\triangle\text{BAD}$
we have, $\ce{AC = BD} ($given$) \ce{BC = AD} ($given$)$ and $\ce{AB = BA}($common$)$
Therefore, by $\ce{SSS}$ criterion of congruency,
$\triangle\text{ABC}\cong\triangle\text{BAD}$ There option $(iii)$ is true.
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Question 152 Marks
In Figure, $\triangle\text{ABC}$ is isosceles with $\ce{AB = AC, D}$ is the mid$-$point of base $BC$.
$i.$ Is $\triangle\text{ADB}\cong\triangle\text{ADC}?$
$ii.$ State the three pairs of matching parts you use to arrive at your answer.
Answer
$i.$ We have $\ce{AB = AC}$.
Also since $D$ is the midpoint of $\ce{BC, BD = DC}$
Also, $\ce{AD = DA}$
Therefore by $\ce{SSS}$ condition,
$\triangle\text{ADB}\cong\triangle\text{ADC}$
$ii.$ We have used $\ce{AB, AC : BD, DC}$ and $\ce{AD, DA}$.
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Question 162 Marks
In Figure, $\ce{AO = OB}$ and $\angle\text{A}=\angle\text{B}$.
$i.$ Is $\triangle\text{AOC}\cong\triangle\text{BOD}?$
$ii.$ State the matching pair you have used, which is not given in the question.
$iii.$ Is it true to say that $\angle\text{ACO}=\angle\text{BDO}?$

Answer
$i.$ Yes, by $\ce{ASA}, \triangle\text{AOC}\cong\triangle\text{BOD}$
$ii. \angle\text{OAC}=\angle\text{OBD}$ and $\ce{AO = OB}$
$iii.$ Yes, $\angle\text{AOC}=\angle\text{BOD} ($Opposite angles on same vertex$)$
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Question 172 Marks
In $\triangle\text{PQR}\cong\triangle\text{CBD},$ Which angle of $\triangle PQR$ equals angle $E$?
Answer


In $\triangle\text{PQR}\cong\triangle\text{CBD},$
$\angle\text{QPR}=\angle\text{FED}$ since the corresponding angles of congruent triangles are equal.
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Question 182 Marks
In the following pair of triangle (Fig.), the lengths of the sides are indicated along sides. By applying $SSS$ condition, determine which are congruent. State the result in symbolic form.
Answer


In $\triangle\text{ABC}$ and $\triangle\text{DEF},$
$AB = DE = 4.5\ cm$ (side)
$BC = EF = 6\ cm$ (side)
and $AC = DF = 4\ cm$ (side)
Therefore, by $SSS$ criterion of congruence, $\triangle\text{ABC}\cong\triangle\text{DEF}$
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Question 192 Marks
In figure, line segments $AB$ and $CD$ bisect each other at $O$. Which of the following statements is true?
$i. \triangle\text{AOC}\cong\triangle\text{DOB}$
$ii. \triangle\text{AOC}\cong\triangle\text{BOD}$
$iii. \triangle\text{AOC}\cong\triangle\text{ODB}$
State the three pairs of matching parts, you have used to arrive at the answer.
Answer
We have, And, $\ce{CO = OD}$ Also, $\ce{AOC = BOD}$
 Therefore by $\ce{SAS}$ condition,
$\triangle\text{AOC}\cong\triangle\text{BOD}$
Therefore, statement $(ii)$ is true.
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Question 202 Marks
In figure, $\ce{AD = DC}$ and $\ce{AB = BC}$.
$i.$ Is $\triangle\text{ABD}\cong\triangle\text{CBD}?$
$ii.$ State the three parts of matching pairs you have used to answer $(i)$.

Answer
$i.$ Yes $\triangle\text{ABD}\cong\triangle\text{CBD}$ by the $\ce{SSS}$ criterion.
$ii.$ We have used the three conditions in the $\ce{SSS}$ criterion as follows:
$\ce{AD = DC}$
$\ce{AB = BC}$
and $\ce{DB = BD}$
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