Triangles: Similarity and Congruence
Similarity and congruence
Two triangles are said to be similar if every angle of one triangle has the same measure as the corresponding angle in the other triangle. The corresponding sides of similar triangles have lengths that are in the same proportion, and this property is also sufficient to establish similarity.
A few basic theorems about similar triangles: * If two corresponding internal angles of two triangles have the same measure, the triangles are similar. * If two corresponding sides of two triangles are in proportion, and their included angles have the same measure, then the triangles are similar. (The included angle for any two sides of a polygon is the internal angle between those two sides.) * If three corresponding sides of two triangles are in proportion, then the triangles are similar.
Two triangles that are congruent have exactly the same size and shape: all pairs of corresponding interior angles are equal in measure, and all pairs of corresponding sides have the same length. (This is a total of six equalities, but three are often sufficient to prove congruence.)
Some sufficient conditions for a pair of triangles to be congruent are: * SAS Postulate: Two sides in a triangle have the same length as two sides in the other triangle, and the included angles have the same measure. * ASA: Two interior angles and the included side in a triangle have the same measure and length, respectively, as those in the other triangle. (The included side for a pair of angles is the side that is common to them.) * SSS: Each side of a triangle has the same length as a corresponding side of the other triangle. * AAS: Two angles and a corresponding (non-included) side in a triangle have the same measure and length, respectively, as those in the other triangle. (This is sometimes referred to as AAcorrS and then includes ASA above.) * Hypotenuse-Leg (HL) Theorem: The hypotenuse and a leg in a right triangle have the same length as those
Two triangles are said to be similar if every angle of one triangle has the same measure as the corresponding angle in the other triangle. The corresponding sides of similar triangles have lengths that are in the same proportion, and this property is also sufficient to establish similarity.
A few basic theorems about similar triangles: * If two corresponding internal angles of two triangles have the same measure, the triangles are similar. * If two corresponding sides of two triangles are in proportion, and their included angles have the same measure, then the triangles are similar. (The included angle for any two sides of a polygon is the internal angle between those two sides.) * If three corresponding sides of two triangles are in proportion, then the triangles are similar.
Two triangles that are congruent have exactly the same size and shape: all pairs of corresponding interior angles are equal in measure, and all pairs of corresponding sides have the same length. (This is a total of six equalities, but three are often sufficient to prove congruence.)
Some sufficient conditions for a pair of triangles to be congruent are: * SAS Postulate: Two sides in a triangle have the same length as two sides in the other triangle, and the included angles have the same measure. * ASA: Two interior angles and the included side in a triangle have the same measure and length, respectively, as those in the other triangle. (The included side for a pair of angles is the side that is common to them.) * SSS: Each side of a triangle has the same length as a corresponding side of the other triangle. * AAS: Two angles and a corresponding (non-included) side in a triangle have the same measure and length, respectively, as those in the other triangle. (This is sometimes referred to as AAcorrS and then includes ASA above.) * Hypotenuse-Leg (HL) Theorem: The hypotenuse and a leg in a right triangle have the same length as those