3.8.1 Congruence
3.8.1 Congruence
3.8.1 Congruence
Congruent is a term used in mathematics to describe two or more shapes or objects that have the same shape and size.
When two shapes or objects are congruent, they have exactly the same dimensions and all corresponding sides and angles are equal.
For example, two triangles are congruent if they have the same three sides and three angles. Similarly, two circles are congruent if they have the same radius.
Congruent figures are often represented using the symbol ≅ (an equal sign with a tilde over it).
Congruent triangles
Congruent triangles are two or more triangles that have exactly the same shape and size. When two triangles are congruent, all their corresponding sides and angles are equal. This means that if you were to superimpose one triangle on top of the other, they would perfectly match up without any gaps or overlap.
There are several ways to show that two triangles are congruent, including:
Side-Side-Side (SSS) Congruence:
If the three sides of one triangle are equal to the three sides of another triangle, then the triangles are congruent.
Side-Angle-Side (SAS) Congruence:
If two sides and the angle between them in one triangle are equal to two sides and the angle between them in another triangle, then the triangles are congruent.
Angle-Side-Angle (ASA) Congruence:
If two angles and the side between them in one triangle are equal to two angles and the side between them in another triangle, then the triangles are congruent.
Angle-Angle-Side (AAS) Congruence:
If two angles and the non-included side of one triangle are equal to two angles and the non-included side of another triangle, then the two triangles are congruent.
Hypotenuse-Leg (HL) Congruence:
If the hypotenuse and one leg of a right triangle are equal to the hypotenuse and one leg of another right triangle, then the triangles are congruent.
Congruent triangles are important in geometry because they have the same properties and measurements, and therefore can be used interchangeably in geometric proofs and calculations.
Worked example:
\( A,B, C \) and \( D \) are points on the circle, center \( O \).
\( M \) is the midpoint of \( AB \) and \( N \) is the midpoint of \( CD \).
\( OM=ON \)
Explain, giving reasons why triangle \( OAB \) is congruent to triangle \( OCD \).
Identify any corresponding sides or corresponding angles that are equal between the triangles \( OAB \) and \( OCD \)
\( OA=OC \)
\( OB=OD \) (As they all are radii)
\( AB \) and \( CD \) are both chords and \( OM \) is perpendicular to \( AB \) and \( ON \) is perpendicular to \( CD \) (The line joining a midpoint of a chord to the center is perpendicular to the chord.)
As \( OM \) and \( ON \) are equal so both chords are the same distance from the center.
\( AB=CD \) (As they are both chords that are equidistant from the center)
\( OAB≅OCD \) by \( SSS \)