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Maths 2 Mark Question

Congruence · Maharashtra Board STD 7 (MSBSHSE - English Medium). 20 questions.

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2 Mark Question

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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 ΔPQR 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, AB = DC and BC = AD.
  1. Is $\triangle\text{ABC}\cong\triangle\text{CBD}?$
  2. What congruence condition have you used?
  3. You have used some fact, not given in the question, what is that?
Answer
We have AB = DC
BC = AD
And AC = AC
  1. Yes by SSS $\triangle\text{ABC}\cong\triangle\text{CBD}$
  2. We have used Side congruence condition with one side common in both the triangles.
  3. Yes, have used the fact that 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 AC = BD and BC = AD as shown in Figure. Which of the following statements is true?
  1. $\triangle\text{ABC}\cong\triangle\text{ABD}$
  2. $\triangle\text{ABC}\cong\triangle\text{ADB}$
  3. $\triangle\text{ABC}\cong\triangle\text{BAD}$
Answer
In $\triangle\text{ABC}$ and $\triangle\text{BAD}$ we have,
AC = BD (given)
BC = AD (given)
and AB = BA (common)
Therefore, by 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 AB = AC, D is the mid-point of base BC.
  1. Is $\triangle\text{ADB}\cong\triangle\text{ADC}?$
  2. State the three pairs of matching parts you use to arrive at your answer.
Answer
  1. We have AB = AC.
Also since D is the midpoint of BC, BD = DC

Also, AD = DA

Therefore by SSS condition,

$\triangle\text{ADB}\cong\triangle\text{ADC}$
  1. We have used AB, AC : BD, DC and AD, DA.
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Question 162 Marks
In Figure, AO = OB and $\angle\text{A}=\angle\text{B}$.
  1. Is $\triangle\text{AOC}\cong\triangle\text{BOD}?$
  2. State the matching pair you have used, which is not given in the question.
  3. Is it true to say that $\angle\text{ACO}=\angle\text{BDO}?$
Answer
  1. Yes, by ASA $\triangle\text{AOC}\cong\triangle\text{BOD}$
  2. $\angle\text{OAC}=\angle\text{OBD}$ and AO = OB
  3. 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 ΔPQR 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 = 4.5cm (side)
BC = EF = 6cm (side)
and AC = DF = 4cm (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?
  1. $\triangle\text{AOC}\cong\triangle\text{DOB}$
  2. $\triangle\text{AOC}\cong\triangle\text{BOD}$
  3. $\triangle\text{AOC}\cong\triangle\text{ODB}$
State the three pairs of matching parts, you have used to arrive at the answer.
Answer
We have,
And, CO = OD
Also, AOC = BOD
Therefore by SAS condition, $\triangle\text{AOC}\cong\triangle\text{BOD}$
Therefore, statement (ii) is true.
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Question 202 Marks
In figure, AD = DC and AB = BC.
  1. Is $\triangle\text{ABD}\cong\triangle\text{CBD}?$
  2. State the three parts of matching pairs you have used to answer (i).
Answer
  1. Yes $\triangle\text{ABD}\cong\triangle\text{CBD}$ by the SSS criterion.
  2. We have used the three conditions in the SSS criterion as follows:
AD = DC

AB = BC

and DB = BD
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