Assertion: triangle | /r/HasProperty | congruent
Facets:
value: in the hl congruence theorem
type: OTHER
value: by sss
type: OTHER
value: mutually
type: MANNER
Source sentences:
text: Two triangles that are congruent have exactly the same size and shape:[note 4] all pairs of corresponding interior angles are equal in measure, and all pairs of corresponding sides have the same length.
text: Side-Side-Angle (or Angle-Side-Side) condition: If two sides and a corresponding non-included angle of a triangle have the same length and measure, respectively, as those in another triangle, then this is not sufficient to prove congruence; but if the angle given is opposite to the longer side of the two sides, then the triangles are congruent.
text: The Side-Side-Angle condition does not by itself guarantee that the triangles are congruent because one triangle could be obtuse-angled and the other acute-angled.
text: Moreover, any two triangles with the given angles are congruent.
text: Two triangles are congruent if and only if they correspond under a finite product of line reflections.
text: Two triangles with corresponding angles equal are congruent (i.e., all similar triangles are congruent).
text: Furthermore, the four triangles that the original triangle is subdivided into by the medial triangle are all mutually congruent by SSS, so their areas are equal and thus the area of each is 1/4 the area of the original triangle.:p.177
text: Two triangles are congruent, if they have accordingly equal: a) two sides and an angle between them; b) two angles and a side, adjacent to them; c) three sides.
text: The AAS congruence theorem states that if two angles and a nonincluded side in one triangle are congruent to two angles and a nonincluded side in another triangle, the two triangles are congruent.
text: In the HL congruence theorem, if the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, the two triangles are congruent.
text: And because the line segment MP is common to both triangles AMP and DMP, the two triangles are congruent by the side-angle-side theorem.
text: Because the two triangles are congruent, the segments AP and DP are also of equal length.