Page 188 - ICSE Math 7

Congruence of Two Angles A P

Two angles are said to be congruent if and only if they
have the same measure.
B Q
C R
Symbolically, ∠ABC ≅ ∠PQR if and only if m∠ABC = m∠PQR.

Congruence of Triangles
Maths Info
Two triangles are said to be congruent if they have the same shape
and size. For two triangles to be congruent, each of the six parts (three • Two circles having equal radii
are congruent.
sides and three angles) of one triangle must be congruent (equal) to the • Two squares having equal
corresponding six parts of the other triangle. For two congruent triangles, sides are congruent.
the sides which are equal to each other are known as corresponding • Two rectangles having same
sides and the angles which are equal are known as corresponding angles. length and breadth are
congruent.
A P

B C Q R
To ensure correspondence of equal sides and equal angles, matching of vertices is very important
in symbolic representation of two congruent triangles. For example, in the above triangles since
A ↔ P, B ↔ Q and C ↔ R (i.e., ABC ↔ PQR, which means the two triangles will superimpose only
if A lies on P, B lies on Q and C lies on R) therefore, we write ∆ ABC ≅ ∆ PQR. In fact, the equality of
six parts of one triangle with the corresponding parts of the other triangle can be identified as follows:

∠C = ∠R

∆ ABC ≅ ∆ PQR also ∆ A B C ≅ ∆ P Q R
⇒ AB = PQ, BC = QR and CA = RP ∠A = ∠P
∠B = ∠Q

Any other representation like ∆ ABC ≅ ∆ PRQ to represent the Try This
congruency of the above triangles is incorrect. The notation ∆ ABC and ∆ DEF are congruent
∆ ABC ≅ ∆ PRQ besides other correspondences, also suggests under the correspondence
∠B = ∠R, ∠C = ∠Q, AB = PR, etc. which is incorrect. So, one should ABC ↔ DEF. Write the equivalent
be careful about the order of the letters while representing two congruent parts which correspond to
triangles in symbolic form. (a) EF (b) ∠C
(c) AC (d) ∠E
Example 1: Complete the following statements.
(a) If two plane figures cover each other exactly, they are ______________.
(b) When we write ∠A ≅ ∠B, we actually mean ______________.
(c) If under a given correspondence, two triangles are congruent, then their corresponding
parts that match each other are ______________.
Solution: (a) congruent by superposition (b) m∠A = m∠B (c) equal

174