Congruence of Triangles
Congruence of Figures
Two figures are congruent if they have exactly the same shape and size. The symbol for congruence is ≅. Two line segments are congruent if they have the same length, and two angles are congruent if they have the same measure.
Example 1: When are two line segments congruent?
When they have the same length.
Example 2: What does the symbol ≅ mean?
It means 'is congruent to'.
Quick recap
- Congruent = same shape and size.
- The congruence symbol is ≅.
✓ Quick check
The symbol ≅ means ___ ?
≅ means 'is congruent to'.
Two squares are congruent if their ___ are equal.
Equal sides make the squares the same size and shape.
Congruence Criteria (SSS, SAS, ASA, RHS)
Triangles are congruent if: all three sides match (SSS); two sides and the included angle match (SAS); two angles and the included side match (ASA); or, in right triangles, the right angle, hypotenuse and one side match (RHS).
Example 1: Which criterion uses three equal sides?
SSS.
Example 2: Which criterion uses a right angle, the hypotenuse and one side?
RHS.
Quick recap
- SSS, SAS, ASA work for all triangles; RHS for right triangles.
- SSA is NOT a valid criterion.
✓ Quick check
Two sides and the included angle equal — which criterion?
Two sides and the included angle is SAS.
Two angles and the included side equal — which criterion?
Two angles and the included side is ASA.
Correspondence & Congruence Statements
The order of letters in a congruence statement shows which parts match. If △ABC ≅ △DEF, then A matches D, B matches E and C matches F, so side AB corresponds to DE and angle A equals angle D.
Example 1: If △ABC ≅ △DEF, which side corresponds to AB?
DE.
Example 2: If △ABC ≅ △DEF, angle A equals which angle?
Angle D.
Quick recap
- The order of letters gives the correspondence.
- Matching sides and angles are equal.
✓ Quick check
In △ABC ≅ △PQR, side BC corresponds to ___ ?
B matches Q and C matches R, so BC corresponds to QR.
Congruent triangles always have equal ___ ?
Congruent triangles are identical, so their areas are equal.
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