Exercise 7.1
Question 1:
Complete the following statements:
(a) Two line segments are congruent if _______________.
(b) Among two congruent angles, one has a measure of 70^{o}, the measure of other angle is _______________.
(c) When we write âˆ A = âˆ B, we actually mean _______________.
Answer 1:
(a) they have the same length
(b) 70^{o}
(c) mâˆ A = mâˆ B
Question 2:
Give any two real time examples for congruent shapes.
Answer 2:
(i) Two footballs
(ii) Two teacherâ€™s tables
Question 3:
If Î”ABC â‰Œ Î”FED under the correspondence , write all the corresponding congruent parts of the triangles.
Answer 3:
Given: Î”ABC â‰Œ Î”FED.
The corresponding congruent parts of die triangles are:
Question 4:
If Î”DEF â‰Œ Î”BCA, write the part (s) of Î”BCA that correspond to:
Answer 4:
Exercise 7.2
Question 1:
Which congruence criterion do you use in the following?
(a) Given: AC = DF, AB = DE, BC = EF
So Î”ABC â‰Œ Î”DEF
(b) Given: RP = ZX, RQ = ZY, âˆ PRQ = âˆ XZY
So Î”PQR â‰Œ Î”XYZ
(c) Given: âˆ MLN = âˆ FGH, âˆ NML = âˆ HFG, ML = FG
So Î”LMN â‰Œ Î”GFH
(d) Given: EB = BD, AE = CB, âˆ A = âˆ C = 90Â°
So AABE = ACDB
Answer 1:
(a) By SSS congruence criterion, since it is given that AC = DF, AB = DE, BC = EF
The three sides of one triangle are equal to the three corresponding sides of another triangle.
Therefore, Î”ABC â‰Œ Î”DEF
(b) By SAS congruence criterion, since it is given that RP = ZX, RQ = ZY and âˆ PRQ = âˆ XZY
The two sides and one angle in one of the triangle are equal to the corresponding sides and the angle of other triangle.
Therefore, Î”PQR â‰Œ Î”XYZ
(c) By ASA congruence criterion, since it is given that âˆ MLN = âˆ FGH, âˆ NML = âˆ HFG, ML = FG.
The two angles and one side in one of the triangle are equal to the corresponding angles and side of other triangle.
Therefore, Î”LMN â‰Œ Î”GFH
(d) By RHS congruence criterion, since it is given that EB = BD, AE = CB, âˆ A = âˆ C = 90Â°
Hypotenuse and one side of a right angled triangle are respectively equal to the hypotenuse and one side of another right angled triangle.
Therefore, Î”ABE â‰Œ Î”CDB
Question 2:
You want to show that Î”ART â‰Œ Î”PEN:
(a) If you have to use SSS criterion, then you need to show:
(i) AR =
(ii) RT =
(iii) AT =
(b) If it is given that âˆ T = âˆ N and you are to use SAS criterion, you need to have:
(i) RT = and
(ii) PN =
(c) If it is given that AT = PN and you are to use ASA criterion, you need to have:
(i) ?
(ii) ?
Answer 2:
(a) Using SSS criterion, Î”ART â‰Œ Î”PEN
(i) AR = PE
(ii) RT= EN
(iii) AT = PN
(b) Given: âˆ T = âˆ N
Using SAS criterion, Î”ART â‰Œ Î”PEN
(i) RT = EN
(ii) PN = AT
(c) Given: AT = PN
Using ASA criterion, Î”ART â‰Œ Î”PEN
(i) âˆ RAT = âˆ EPN
(ii) âˆ RTA = âˆ ENP
Question 3:
You have to show that Î”AMP = Î”AMQ. In the following proof supply the missing reasons:
Steps | Reasons |
(i) PM = QM | (i) |
(ii) âˆ PMA = âˆ QMA | (ii) |
(iii) AM = AM | (iii) |
(iv) Î”AMP â‰Œ Î”AMQ | (iv) |
Answer 3:
Steps | Reasons |
(i) PM = QM | (i) Given |
(ii) âˆ PMA = âˆ QMA | (ii) Given |
(iii) AM = AM | (iii) Common |
(iv) Î” AMP â‰Œ Î”AMQ | (iv) SAS congruence rule |
Question 4:
In Î”ABC, âˆ A = 30% âˆ B = 40Â° and âˆ C = 110Â°
In Î”PQR, âˆ P = 30% âˆ Q = 40Â° and âˆ R = 110Â°.
A student says that Î”ABC â‰Œ Î”PQR by AAA congruence criterion. Is he justified? Why or why not?
Answer 4:
No, because the two triangles with equal corresponding angles need not be congruent. In such a correspondence, one of them can be an enlarged copy of the other.
Question 5:
In the figure, the two triangles are congruent. The corresponding parts are marked. We can write Î” RAT â‰Œ ?
Answer 5:
In the figure, given two triangles are congruent. So, the corresponding parts are:
We can write, Î”RAT â‰Œ Î”WON [By SAS congruence rule]
Question 6:
Complete the congruence statement:
Answer 6:
In A BAT and ABAC, given triangles are congruent so the corresponding parts are:
Thus, Î”BCA â‰Œ Î”BTA |By SSS congruence rule]
In Î”QRS and Î”TPQ, given triangles are congruent so the corresponding parts are:
Thus, Î”QRS â‰Œ Î”TPQ [By SSS congruence rule]
Question 7:
In a squared sheet, draw two triangles of equal area such that:
(i) the triangles are congruent.
(ii) the triangles are not congruent.
What can you say about their perimeters?
Answer 7:
In a squared sheet, draw Î”ABC and Î”PQR. When two triangles have equal areas and
(i)
In the above figure, Î”ABC and Î”DEF have equal areas.
And also, Î”DEF
So, we can say that perimeters of Î”ABC and Î”DEF are equal.
(ii)
In the above figure, Î”LMN and Î”OPQ
Î”LMN is not congruent to Î”OPQ
So, we can also say that their perimeters are not same.
Question 8:
Draw a rough sketch of two triangles such that they have five pairs of congruent parts but still the triangles are not congruent.
Answer 8:
Let us draw two triangles PQR and ABC.
All angles are equal, two sides are equal except one side. Hence, Î”PQR are not congruent to Î”ABC.
Question 9:
If Î”ABC and Î”PQR are to be congruent, name one additional pair of corresponding parts. What criterion did you use?
Answer 9:
A ABC and A PQR are congruent Then one additional pair is
Given: âˆ B = âˆ Q = 90Â°
Therefore, Î”ABC â‰Œ Î”PQR [By ASA congruence rule]
Question 10:
Explain, why Î”ABC â‰Œ Î”FED.
Answer 10:
Given: âˆ A = âˆ F, BC = ED, âˆ B = âˆ E
In Î”ABC and Î”FED,
âˆ B = âˆ E = 90Â°
âˆ A = âˆ F
BC = ED
Therefore, Î”ABC â‰Œ Î”FED [By RHS congruence rule]