Chapter-7 Congruence of Triangles

1. Complete the following statements:

 Two-line segments are congruent if

Explanation:


Two line segments are said to be congruent if their lengths are equal. In other words, if you can place one line segment directly on top of the other without any gaps or overlaps, then the two-line segments are congruent. 


2.Among two congruent angles, one has a measure of 70"; the measure of the other angle is

Explanation:


If two angles are congruent, it means that they have the same measure. So if one of the congruent angles has a measure of 70 degrees, then the other angle must also have a measure of 70 degrees. This is because congruent angles have the same measure, and if two angles are congruent, then their measures are equal. Therefore, if one angle has a measure of 70 degrees, the other angle must 


3.When we write <A= <B, we actually mean also have a measure of 70 degrees to be congruent.

Explanation:


When we write AB, we actually mean that the two angles are equal in measure. However, since angles are measured in degrees, we can also write this statement as m Am B. where "m" stands for the measure of the angle in degrees. Therefore, if we say that A=B, we are also saying that the measures of the two angles are equal. 


4. Give any two real-life examples of congruent shapes.

Explanation:


Two real life examples of congruent triangles are:


1. Identical rectangular tiles: Tiles that are used to cover the floor or walls of a room are often cut into identical rectangular shapes, which are congruent to each other.


2. Identical doors: Many homes or offices have doors that are identical in shape and size, which means they are congruent to each other.

3. Identical windowpanes: Windowpanes that are used in a building can be congruent to each other if they are identical in size and shape


5. If AABC  ≅AFED under the correspondence ABC FED, write all the corresponding congruent parts of the triangles.

Explanation:


If AABC =AFED under the correspondence ABC-FED, it means that the two triangles are congruent. This implies that their corresponding parts, including sides and angles, are equal in measure or length. Therefore, the corresponding congruent parts of the triangles are:

ZA ZF

<C D

ABFE

BC - ED

AC - FD


In other words, the corresponding vertices of the two triangles


(AF, BE, CD) define pairs of congruent angles.

while the corresponding sides (AB FE, BC ED, AC FD) define pairs of segments.

congruent line



6. If ADEF≅ABCA, write the part(s) of ABCA that correspond to (i) 48 (ii) EF (iii) <F (iv) DF

Explanation:


From the above figure, we can say that,


If ADEF = ABCA, it means that the two triangles are congruent to each other. Therefore, the corresponding parts of ABCA are as follows:


(i) E corresponds to B in ABCA. This is because corresponding angles of congruent triangles are equal.


(ii) The line segment bar over (EF) corresponds to the line segment bar over (BC) in ABCA. This


EF → CA

angles of congruent triangles are equal.

This is because corresponding sides of congruent triangles are equal. 


(iii) <F corresponds to C in ABCA. This is because corresponding

LE LA


(iv) The line segment har over (DF) corresponds to the line segment bar over (CA) in ABCA. This is because corresponding sides of congruent triangles are equal.

DF → BA


7. Which congruence criterion do you use in the following?  Given: AC = DF


AB = DE


BC= EF


So, AABC≅ADEF

Explanation:


The congruence criterion used in this case is the Side-Side-Side (SSS) criterion,


Two triangles are congruent if and only if their corresponding sides are congruent. If all three sides of one triangle are congruent to the corresponding sides of another triangle, then the triangles are congruent by the SSS criterion


In this case, we are given that AB-DE, BC-EF, and AC-DF. According to the SSS criterion, if the three sides of triangle ABC are equal in length to the three corresponding sides of triangle DEF, then the two triangles are congruent. Since this condition is met, we can conclude that the

triangles are congruent



8.Given: ZX = RP

Explanation:


RQ=ZY


<PRQ= <XZY


So, APQR≅AXYZ



The congruence criterion used in this case is the Side-Angle-Side (SAS) criterion.


The SAS criterion for congruence states that if two sides and the included angle of one triangle are


congruent to two sides and the included angle of another triangle, then the two triangles are congruent.


In this case, we are given that


ZX= RP. RQZY, and PRQ=XZY. Given that the two sides and the included angle of triangle PQR are congruent to the corresponding parts of triangle XYZ, we can apply the SAS criterion to conclude that the two triangles are congruent.


Hence, we can write APQR AXYZ.


9.<MLN = <FGH


<NML = <GFH


<ML = <FG


Explanation:


The congruence criterion used in this case is the Angle-Side-Angle (ASA) criterion


According to the ASA criterion, if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.


In this case, we are given that


<MLN = <FGH, <NML = <GFH. and ML = /FG.


The two triangles LMN and GFH are congruent because they share two corresponding angles and an included side that have the same measure, as required by the ASA congruence criterion


Hence, we can write ALMN ≅ AGFH. 


10. EB = DB

AEBC

<A=4C = 90 

So, AABE≅AACD

Explanation: The congruence criterion used in this case is the Hypotenuse-Leg (HL) enterin


If the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, then the two triangles are congruent.


In this case, we are given that


EB = DB, AE = BC, and both A and C are right angles.


Therefore, we can conclude that AABE and AACD are right triangles with hypotenuses AB and


AC, respectively, and legs AE and BC congruent. Thus, hypotemuse AB of AABE is congruent to hypotenuse AC of AACD, and leg AE of AABE is congruent to leg BC of AACD.


Therefore, the triangles AABE and AACD satisfy the conditions of the HL criterion, and we can conclude that they are congruent. Hence, we can write AABE ≅ AACEX


11. You want to show that AART MAPEN  If you have to use the SSS criterion, then you need to show (i) AR(ii) RT=(iii) AT=


Explanation:


To show that AART≅APEN using the SSS criterion, we need to show that the three pairs of


corresponding sides of the two triangles are congruent. Therefore, we need to show:


ARPE (ii) RT-EN


(AT=PN


To prove (1), we can use the given information that segment AR is congruent to segment PE


To prove (ii), we can use the given information that segment RT is congruent to segment EN.


To prove (i), we can use the fact that AT AR+RT and PN=PE+EN. Since we know that AR = PE and RT = EN from (1) and (ii), we can substitute these values to get


AT=AR+RT=PE+EN=PN


Therefore, we have shown that all three pairs of corresponding sides of the two triangles are congruent, and hence, we can conclude that AART≅APEN by the SSS criterion


12. It is given that <T=<N and you are to use the SAS criterion, you need to have (i) RT = and (ii) PN=


Explanation:


To show that AART RAPEN using the SSS criterion, we just need to show that the three pairs of corresponding sides of the two triangles are congruent, which are


(i) AR-PE


(i) RT-EN


(iii) AT=PN


To show that AART≅APEN using the SAS criterion and given that TN, we jug need to show that


(i) RT=PN


(ii) <RAT - NEP


13.If it is given that AT=PN and you are to use the ASA criterion, you need to have

(i)=? (ii)=?

Explanation:


To use the ASA (Angle-Side-Angle) criterion to prove that AART =APEN given that AT- PN, we need to show that


(1)ART≅PEN, which can be proved using the fact that AT PN and AR PE

Therefore, the requirements for the ASA criterion are just these two pairs of congruent angles.


ATR  ≅PNE which can be proved using the fact that AT = PN and ARTPEN, and the fact that the sum of angles in a triangle is 180


14. You have to show that AAMP AAMQ.

In the following proof, supply the missing reasons.


 Explanation:


15. In AABC, <A 30°, <B = 40° and <C = 110° In APQR, P= 30", <Q= 40° and R = 110° A student says that AABC≅APQR by AAA congruence criterion. Is he justified? Why or Why not?

Explanation:


No, the student is not justified in saying that AABC ≅  APQR by AAA congruence criterion because AAA (Angle-Angle-Angle) is not a valid congruence criterion.

The AAA congruence criterion states that if two triangles have their corresponding angles equal. then they are congruent. However, this is not always true because two triangles can have the same angles but different side lengths, which means they are not congruent.


16. In the figure, the two triangles are congruent. The corresponding parts are marked. Can we write triangle RAT ≅ ?


Explanation:


In the given figure, we can write ARAT APCQ.


This is because the two triangles have corresponding congruent parts:


RA = PC (corresponding sides)


AT=CQ (corresponding sides)


<RAT = <PCQ = 90° (both right angles)


Using the Side-Angle-Side (SAS) congruence criterion, we can conclude that the two triangles are congruent. Therefore, we can write ARAT  ≅ APCQ.


17. Complete the congruence statement:

ABAC≅AQRS≅

Explanation:


The given figure shows two triangles, ABCA and ABTA, such that BT-BC and BA is their


common side. Therefore, we can write:


ABCA≅ ABTA (by SAS congruence criterion)


Similarly, the figure also shows two triangles, AQRS and ATPQ, such that PT - QR, TQ = QS. and PQ = RS.


Hence, we can write:


AQRS ≅ATPQ (by SSS congruence criterion)


In a squared sheet, draw two triangles of equal areas such that

18. The triangles are congruent 

Explanation:

Two triangles APQR and AMNO are equal in area, then:


Area(APQR) = Area(AMNO)


Also,


PQ = MN


QR = NO


Height of APQR-Height of AMNO


ΔΡΟΗ≅ ΕΔΜΝΟ


Therefore. By the Side-Angle-Side (SAS) congruence criterion


Perimeter of APQR=PQ+QR + PR 

Perimeter of AMNO=MN+ON+OM


Because each side are equal, therefore:


Perimeter of APQR = Perimeter of AMNO


 19. The triangles are not congruent

What can you say about their perimeters?

Explanation:

Area(APQR) ≠  Area (AMNO)

Also,

PQ  ≠ MN

QR ≠  NO

Height of APQR Height of AMNO

Therefore, APQR is not congruent to AMNO

 Perimeter of APQR = PQ+QR + PR

Perimeter of AMNO = MN+ON+OM

Because each side are not equal, therefore: Perimeter of APQR ≠  Perimeter of AMNO


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

Explanation:


Let's draw AABC and APQR.



In the triangle drawn above AABC and APQR.


AC PR


AB = QR


BC ≠  PQ


Also,


ABC = QRP = 90 degree


<BAC = 2 PQR


< BCA = < QPR


But side BC and PQ are not equal therefore AABC and APQR. Are not congruent.


21. If AABC and APQR are to be congruent, name one additional pair of corresponding parts. What criterion did you use?


Explanation:


In this case, we have:


BC= QR (given)


ZABC = PQR (given)


< BCA = PRQ (given)


We have three pairs of congruent parts, and we can use the Side-Angle-Side (SAS) criterion to conclude that the triangles are congruent.


The SAS criterion states that if two pairs of sides and the included angle between them are


congruent in two triangles, then the triangles are congruent.


: AABC = APQR


22. Explain, why AABC AFED


Explanation:


From the question it is given:


ZABC = DEF = 90%


<BAC = <DFE


And side, BC = DE


we have two pairs of congruent angles (angle BAC is congruent to angle DFE, and angle ABC is congruent to angle DEF because they are both right angles), and one pair of congruent sides (BC is congruent to DE).


Using the Angle-Side-Angle (ASA) congruence criterion, we can conclude that triangles ABC and DEF are congruent.



AABC≅ AFED