The document discusses the concept of congruence in geometry, stating that two figures are congruent if they are identical in shape and size but may differ in position. It outlines the congruence of line segments, angles, and triangles, introducing various criteria for triangle congruence, including side-side-side (SSS), side-angle-side (SAS), angle-side-angle (ASA), right-angle-hypotenuse-side (RHS), and angle-angle-side (AAS). The document also mentions that corresponding parts of congruent triangles are congruent, alongside providing examples and problems related to congruence.

1. Congruence of Triangles
Congruence of Figures:
Two figures (usually shapes or objects) are considered congruent if they have
the same shape and size. This means that one can be transformed into the
other through translations, rotations, and reflections without changing its size
or shape. It's like saying they are identical in every way, except for their
position in space.
The relation of two objects being congruent is called congruence.
F1 ≅ F2
All geometrical figures are combined to added a method used is called
superposition method.
Congruence of Line Segments:
Line segments are parts of line. If two line segments have the same (ie.,
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 justthat the line segments are equal; and we also write AB = CD
(What we actually meansay overline AB equiv overline CD ).

2. Congruence of Angles:
∠ABC ≅ ∠PQR. ……….(1)
m∠ABC=m∠PQR. (m is measure, in this case m=40°)
Now, ∠ABC ≅ ∠XYZ. …………..(2)
m∠ABC=m∠XYZ
So, we can write from 1 & 2,
∠ABC≅∠PQR≅∠XYZ
Hence, if two angles have the same measure, they are congruent. Also, if two
angles are congruent, their measures are same.
Congruence of Triangles:
△ABC and △PQR have the same size and shape. So, they are congruent.
△ABC≅△PQR
Corresponding vertices: A & P, B & Q, C & R.
Corresponding sides: AB— and PQ—, BC— and QR—, AC— and PR—.
Corresponding angles: ∠A & ∠P, ∠B & ∠Q, ∠C & ∠R.
P falls on A.
Therefore, △PQR ≅△ABC.

3. Criteria for congruent of triangles:
1. Side-Side-Side Criteria (SSS): If under a given correspondence, the
three sides of one triangle are equal to the three corresponding sides of
another triangle, then the triangles are cocongruent.
2. Side-Angle-Side Criteria (SAS): If under a correspondence, two sides
and the angle included between them of a triangle are equal to two
corresponding sides and the angle included between them of another
triangle, then the triangles are congruent.
3. Angle-Side-Angle Criteria (ASA): If under a correspondence, two
angles and the included side of a triangle are equal to two
corresponding angles and the included side of another triangle, then the
triangles are congruent.
4. Right-Angle-Hypotenuse-Side Criteria (RHS): If under a
correspondence, the hypotenuse and one side of a right-angled triangle
are respectively equal to the hypotenuse and one side of another
right-angled triangle, then the triangles are congruent.
5. Angle-Angle-Side Criteria (AAS): If two angles and a side not
included between them in one triangle are congruent to two angles and
the corresponding side in another triangle, then the two triangles are
congruent. AAS is a special case of ASA.

4. ● Corresponding Parts of Congruent Triangle (CPCT): CPCT says if
two or more triangles are congruent to each other, then the
corresponding angles and the corresponding sides of the triangles are
also congruent to each other.
Questions:
1. In the given congruent triangles under ASA, find the value of x and
y, ΔPQR = ΔSTU.
2. In ΔABC, medians BD and CE are equal and intersect each other at
O. Prove that ΔABC is an isosceles triangle.
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