Congruence of Plane Figures

Congruence of plane figures

 

You can use the method of superposition. Take a trace-copy of one of them and place it over the other. If the figures cover each other completely, they are congruent. Alternatively, you may cut out one of them and place it over the other. Beware! You are not allowed to bend, twist or stretch the figure that is cut out (or traced out). In Fig 7.3, if figure F1 is congruent to figure F2 , we write F1 ≅ F2.

 

Congruence among line segments

When are two line segments congruent? Observe the two pairs of line segments given here (Fig 7.4).

Use the ‘trace-copy’ superposition method for the pair of line segments in [Fig 7.4(i)].
    Copy «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mover»«mrow»«mi»C«/mi»«mi»D«/mi»«/mrow»«mo»§#175;«/mo»«/mover»«/math»and place it on «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mover»«mrow»«mi»A«/mi»«mi»B«/mi»«/mrow»«mo»§#175;«/mo»«/mover»«/math» You find that «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mover»«mrow»«mi»C«/mi»«mi»D«/mi»«/mrow»«mo»§#175;«/mo»«/mover»«/math» covers «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mover»«mrow»«mi»A«/mi»«mi»B«/mi»«/mrow»«mo»§#175;«/mo»«/mover»«/math», with C on A and D on B.
Hence, the line segments are congruent. We write«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mover»«mrow»«mi»A«/mi»«mi»B«/mi»«/mrow»«mo»§#175;«/mo»«/mover»«/math»«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mover»«mrow»«mi»C«/mi»«mi»D«/mi»«/mrow»«mo»§#175;«/mo»«/mover»«/math»

Repeat this activity for the pair of line segments in [Fig 7.4(ii)]. What do you find? They are not congruent. How do you know it? It is because the line segments do not coincide when placed one over other.

You should have by now noticed that the pair of line segments in [Fig 7.4(i)] matched with each other because they had same length; and this was not the case in [Fig 7.4(ii)].

If two line segments have the same (i.e., equal) length, they are congruent. Also, if two line segments are congruent, they have the same length.

In view of the above fact, when two line segments are congruent, we sometimes just say that the line segments are equal; and we also write AB = CD. (What we actually mean is «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mover»«mrow»«mi»A«/mi»«mi»B«/mi»«/mrow»«mo»§#175;«/mo»«/mover»«/math»«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mover»«mrow»«mi»C«/mi»«mi»D«/mi»«/mrow»«mo»§#175;«/mo»«/mover»«/math»).

 

Congruence of angles

 

 

Make a trace-copy of ∠PQR. Try to superpose it on ∠ABC. For this, first place Q
on B and«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mover accent=¨true¨»«mrow»«mi»Q«/mi»«mi»P«/mi»«/mrow»«mo»§#8594;«/mo»«/mover»«/math» along «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mover accent=¨true¨»«mrow»«mi»A«/mi»«mi»B«/mi»«/mrow»«mo»§#8594;«/mo»«/mover»«/math» Where does fall? It falls on

Thus, ∠PQR matches exactly with ∠ABC.
That is, ∠ABC and ∠PQR are congruent.
(Note that the measurement of these two congruent angles are same).
We write                     ∠ABC ≅ ∠PQR                                                          (i)
or                              m∠ABC = m ∠PQR(In this case, measure is 40°).
Now, you take a trace-copy of ∠LMN. Try to superpose it on ∠ABC. Place M on B and ML along BA Does MN fall on ? No, in this case it does not happen. You find
that ∠ABC and ∠LMN do not cover each other exactly. So, they are not congruent.
(Note that, in this case, the measures of ∠ABC and ∠LMN are not equal).
What about angles ∠XYZ and ∠ABC? The rays YX and YZ respectively appear
[in Fig 7.5 (iv)] to be longer than
and
. You may, hence, think that ∠ABC is
‘smaller’ than ∠XYZ. But remember that the rays in the figure only indicate the direction
and not any length. On superposition, you will find that these two angles are also congruent.
We write                            ∠ABC ≅ ∠XYZ (ii)
or                                     m∠ABC = m∠XYZ
In view of (i) and (ii), we may even write
                                           ∠ABC ≅ ∠PQR ≅∠XYZ
If two angles have the same measure, they are congruent. Also, if two angles are congruent, their measures are same.

As in the case of line segments, congruency of angles entirely depends on the equality of their measures. So, to say that two angles are congruent, we sometimes just say that the angles are equal; and we write
                                           ∠ABC = ∠PQR (to mean ∠ABC ≅ ∠PQR).

 

Congruence of Triangles

We saw that two line segments are congruent where one of them, is just a copy of the other. Similarly, two angles are congruent if one of them is a copy of the other. We extend this idea to triangles.

Two triangles are congruent if they are copies of each other and when superposed, they cover each other exactly

 

 

ΔABC and ΔPQR have the same size and shape. They are congruent. So, we would express this as

                                                                                        ΔABC ≅ ΔPQR

This means that, when you place ΔPQR on ΔABC, P falls on A, Q falls on B and R falls on C, also falls along «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mover»«mrow»«mi»A«/mi»«mi»B«/mi»«/mrow»«mo»§#175;«/mo»«/mover»«/math» , «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mover»«mrow»«mi»Q«/mi»«mi»R«/mi»«/mrow»«mo»§#175;«/mo»«/mover»«/math» falls along«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mover»«mrow»«mi»B«/mi»«mi»C«/mi»«/mrow»«mo»§#175;«/mo»«/mover»«/math»and «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mover»«mrow»«mi»P«/mi»«mi»R«/mi»«/mrow»«mo»§#175;«/mo»«/mover»«/math» falls along AC. If, under a given correspondence, two triangles are congruent, then their corresponding parts
(i.e., angles and sides) that match one another are equal. Thus, in these two congruent triangles, we have:
                                       Corresponding vertices : A and P, B and Q, C and R.
                                       Corresponding sides : «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mover»«mrow»«mi»A«/mi»«mi»B«/mi»«/mrow»«mo»§#175;«/mo»«/mover»«/math»and «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mover»«mrow»«mi»P«/mi»«mi»Q«/mi»«/mrow»«mo»§#175;«/mo»«/mover»«/math»,«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mover»«mrow»«mi»B«/mi»«mi»C«/mi»«/mrow»«mo»§#175;«/mo»«/mover»«/math» and «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mover»«mrow»«mi»Q«/mi»«mi»R«/mi»«/mrow»«mo»§#175;«/mo»«/mover»«/math» , AC and «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mover»«mrow»«mi»P«/mi»«mi»R«/mi»«/mrow»«mo»§#175;«/mo»«/mover»«/math» .
                                       Corresponding angles : ∠A and ∠P, ∠B and ∠Q, ∠C and ∠R.
If you place ΔPQR on ΔABC such that P falls on B, then, should the other vertices also correspond suitably? It need not happen! Take trace, copies of the triangles and try to find out. This shows that while talking about congruence of triangles, not only the measures of angles and lengths of sides matter, but also the matching of vertices. In the above case, the correspondence is
                                                                     A ↔P, B ↔ Q, C ↔ R
We may write this as                                     ABC ↔PQR

 

EXAMPLE 

ΔABC and ΔPQR are congruent under the correspondence:
                   ABC ↔RQP
Write the parts of ΔABC that correspond to
(i)              ∠P (ii) ∠Q (iii) RP
SOLUTION For better understanding of the correspondence, let us use a diagram (Fig 7.7).

The correspondence is ABC ↔ RQP. This means
                                                                  A ↔ R ; B ↔ Q; and C ↔ P.
So,             (i) «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mover»«mrow»«mi»P«/mi»«mi»Q«/mi»«/mrow»«mo»§#175;«/mo»«/mover»«/math»«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mover»«mrow»«mi»C«/mi»«mi»B«/mi»«/mrow»«mo»§#175;«/mo»«/mover»«/math»             (ii) ∠Q ↔ ∠B and              (iii) «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mover»«mrow»«mi»R«/mi»«mi»P«/mi»«/mrow»«mo»§#175;«/mo»«/mover»«/math»«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mover»«mrow»«mi»A«/mi»«mi»C«/mi»«/mrow»«mo»§#175;«/mo»«/mover»«/math»

 

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