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10.  Explain, why $\Delta ABC\cong \Delta FED$

Comparing both triangles, we have, So By RHS congruency criterion,  .

8. Draw a rough sketch of two triangles such that they have five pairs of congruent parts but still the triangles are not congruent.

Five pairs of congruent parts can be three pairs of sides and two pairs of angles. In that case, the SAS or ASA criterion would prove them to be congruent. Hence, such a figure is not possible.

7.  In a squared sheet, draw two triangles of equal areas such that
(i) the triangles are congruent.
(ii) the triangles are not congruent.

What can you say about their perimeters?

When two triangles are congruent, the corresponding parts are exactly identical so they have the same area and perimeter. While the triangles are not congruent but have the same area, then the perimeter of both triangles are not equal.

9. If $\Delta ABC$ and $\Delta PQR$ are to be congruent, name one additional pair of corresponding parts. What criterion did you use?

Given  One additional pair which is not given in the figure is  We used the ASA Criterion as the two corresponding angles are given and we figured out the side by congruency.

6. Complete the congruence statement:

Comparing from the figure, we get, So By SSS Congruency Rule,    Also, Comparing from the figure, we get, So By SSS Congruency Rule,   .

5. In the figure, the two triangles are congruent. The corresponding parts are marked. We can write $\Delta RAT\cong$?

Comparing from the figure. By SAS Congruency criterion, we can say that  ]

4. In $\bigtriangleup ABC$,$\angle A= 30^{o}$ , $\angle B= 40^{o}$ and $\angle C= 110^{o}$.

In $\bigtriangleup PQR$, $\angle P=30^{o}$ , $\angle Q=40^{o}$ and $\angle R=110^{o}$. A student says that $\bigtriangleup ABC\cong \bigtriangleup PQR$ by AAA congruence criterion. Is he justified? Why or why not?

No, because it is not necessary that two triangles will be congruent if their all three corresponding angles are equal. in this case, the triangles might be zoomed copy of one another.

2.(c) You want to show that $\Delta ART\cong \Delta PEN,$

(c) If it is given that $AT=PN$ and you are to use ASA criterion, you need to have

$(i)?\; \; \; \; \; \; \; \; \; \; \; \; \; \; \;\; \; \; \; \; (ii)?$

Given,  also,  Now, As we know in the ASA criterion of proving congruency, the one and side two angles are equal to their corresponding parts. So,

2.(b)  You want to show that $\Delta ART\cong \Delta PEN,$

(b) If it is given that $\angle T=\angle N$ and you are to use SAS criterion, you need to have

$(i)RT = \; \; \; \; \; \; \; \; \; \; \; \; \; and \; \; \; \; \; \; \; \; \; \; \; \; \; (ii)PN=$

As we know in SAS  criterion the two sides and one angle are identical to their corresponding parts of another triangle. So to prove congruency we need to prove that,

2.(a)  You want to show that $\Delta ART\cong \Delta PEN,$

(a) If you have to use $SSS$ criterion, then you need to show

$(i)AR=$            $(ii)RT=$                $(iii)AT=$

As we know that in the criterion of proving congruent, all three corresponding sides are equal to another. So to prove the congruency, we kneed to know the following things:

3. You have to show that $\bigtriangleup AMP\cong \bigtriangleup AMQ$. In the following proof, supply the missing reasons.

 Steps Reasons $(i)PM= QM$ $(i).......$ $(ii)\angle PMA = \angle QMA$ $(ii).......$ $(iii)AM= AM$ $(iii).......$ $(iv)\bigtriangleup AMP\cong \bigtriangleup AMQ$ $(iv).......$

Steps     Reasons    Given in the question    Given in the question.    the side which is common in both triangle     By SAS Congruence Rule

1.(c)  Which congruence criterion do you use in the following?

$(c)\; Given:\angle MLN=\angle FGH$

$\angle NML=\angle GFH$

$ML = FG$

$So,\; \Delta LMN\cong \Delta GFH$

Since we are comparing two angles and one side, ASA(Angle, Side, Angle) congruency criterion is used to prove the congruency.

1.(b)   Which congruence criterion do you use in the following?

$(b)\; Given:ZX=RP$

$RQ = ZY$

$\angle PRQ =\angle XZY$

$So, \Delta PQR\cong \Delta XYZ$

Since we are comparing two sides and one angle of the two triangles, the SAS (sie, angle, side) congruent criterion is used to prove them congruent.

1.(a)   Which congruence criterion do you use in the following?

$\\(a)\: Given\; AC=DF$

$AB = DE$

$BC = EF$

$So,\; \; \Delta ABC \cong \Delta DEF$

Since we are comparing all the sides of two triangles, The SSS (side, side, side) Congruent criterion is used.

1.(d) Which congruence criterion do you use in the following?

$(d) Given:\; EB = DB$

$AE = BC$

$\angle A = \angle C = 90^{o}$

$So, \Delta ABE\cong \Delta CDB$

Since we are comparing two sides and one angle of the two triangles, the SSA (Side, Side, Angle) congruent criterion is used to prove the congruency.

4. ABC is an isosceles triangle with $AB= AC$ and $AD$ is one of its altitudes (Fig 7.34).

(i) State the three pairs of equal parts in $\bigtriangleup AD\! B$ and $\bigtriangleup ADC$.
(ii) Is $\bigtriangleup ADB\cong \bigtriangleup ADC$? Why or why not?
(iii) Is $\angle B= \angle C$? Why or why not?
(iv) Is $BD= CD$? Why or why not?

i) Given in and .    ( Common side)    ii) So, by RHS Rule of congruency, we conclude   iii) Since both triangles are congruent all the corresponding parts will be equal. So,   iv) Since both triangles are congruent all the corresponding parts will be equal. So, .

3. In Fig 7.33, $BD$ and $CE$ are altitudes of $\bigtriangleup ABC$ such that $BD= CE$.

(i) State the three pairs of equal parts in $\bigtriangleup CBD$ and $\bigtriangleup BCE$.
(ii) Is $\bigtriangleup CBD\cong \bigtriangleup BCE$ ? Why or why not?
(iii) Is $\angle DCB= \angle EBC$ ? Why or why not?

i) Given,  in and .     ii) So, By RHS Rule of congruency, we conclude:   iii) Since both the triangle are congruent, all parts of one triangle are equal to their corresponding part from another triangle. So. .

2. It is to be established by RHS congruence rule that  $\bigtriangleup ABC\cong \bigtriangleup RPQ$. What additional information is needed, if it is given that

$\angle B= \angle P= 90^{\underline{0}}$  and $AB= RP?$

To prove congruency by RHS (Right angle, Hypotenuse, Side ) rule, we need hypotenuse and side equal to the corresponding hypotenus and side of different angle. So Given   ( Right angle )    ( Side ) So the third information we need is the equality of  Hypotenuse of both triangles. i.e. Hence, if this information is given then we can say, .

1.  In Fig 7.32, measures of some parts of triangles are given.By applying RHS congruence rule, state which pairs of triangles are congruent. In case of congruent triangles, write the result in symbolic form.

i) In  and  Hence they are not congruent. ii)  In  and   ( same side ) So, by RHS congruency rule, iii) In  and   ( same side ) So, by RHS congruency rule, iv)  In  and   ( same side ) So, by RHS congruency rule,

5. In Fig 7.28, ray AZ bisects $\angle DAB$ as well as $\angle DCB$.

(i) State the three pairs of equal parts in triangles $BAC$ and $DAC$.
(ii) Is $\Delta BAC\cong \Delta DAC ?$ Give reasons.
(iii) Is $AB = AD ?$ Justify your answer.
(iv) Is $CD = CB \; ?$ Give reasons.

i) Given in triangles  and   ( common side) ii) So, By ASA congruency criterion,triangles  and  are congruent. iii) Since  , all corresponding parts will be equal. So . iv) Since  , all corresponding parts will be equal. So
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