Three noncolinear points determine a triangle. Draw the three segments connecting the three pairs of points and find three sides, and 3 interior angles, hence the name triangle.
Two triangles are congruent if one can be moved on top of the other, so that edges and vertices coincide. The corresponding sides have the same lengths, and corresponding angles are congruent.
Assume the edges of one triangle are the same lengths as the corresponding edges of another triangle. Move the first triangle onto the second so that the bases coincide. Do the apexes also coincide? Both apexes are x units away from the left end of the base, and y units from the right end of the base. Since the triangles are oriented the same way, both apexes lie above the base. Draw a circle of radius x centered at the left end of the base, and a circle of radius y centered at the right end of the base. These circles intersect in precisely two points, one above the base and one below. Thus there is only one possible location for the apex. Both apexes coincide, and the first triangle lies directly on top of the second. Corresponding angles coincide, and are congruent. This method of proving triangle congruence is called SSS, for side side side.
Next assume the angle of a triangle, and the adjacent sides that form that angle, are congruent to the angle and adjacent sides of a second triangle. Move the first triangle onto the second so that side angle side lines up with side angle side. If the left sides line up, and the right sides don't, one angle is inside the other, hence smaller than the other. Since the angles are congruent, both pairs of sides line up. Does the vertex at the end of the left side of the first triangle coincide with the vertex at the end of the left side of the second triangle? Both are a fixed distance from the common apex, along a common line, hence they are the same point. By the same reasoning, the third vertex of the first triangle coincides with the third vertex of the second triangle. The sides and angles all coincide and the triangles are congruent. This method of proving congruence is called SAS, for side angle side.
Assume the base and base angles of two triangles are congruent. Place one on top of the other so that the bases coincide. The apex is now the intersection of two lines, at specific angles to the base. These lines are the same in both triangles, and their intersection is the same point, hence both triangles have the same apex, and coincide. This method of proving congruence is called ASA, for angle side angle.
Assume the base, a base angle, and the apex angle of one triangle are congruent to their counterparts in another. Later on we will prove that the angles of a triangle add up to 180°. This means the remaining base angle in the first triangle has the same measure as the remaining base angle in the second. The triangles are congruent by ASA. This method of proving congruence is called SAA, for side angle angle.
| In general, the method of SSA does not guarantee congruence. Place a 5 12 13 right triangle in the corner of a 9 12 15 right triangle, where 12 is the common side. Let the acute base angle be θ. Subtract the first region from the second and find a triangle with sides 13 4 and 15, and angle θ. Now reflect the smaller right triangle through the common side of length 12. This gives a large triangle with sides 13 15 and 14, and angle θ. These are different triangles, yet they have a side side angle congruence. The problem is, we can flip the segment of length 13 back and forth, creating two different triangles. In general, we have congruence when the second side of SSA (not incident to the angle) is at least as long as the first. Then the aforementioned flip is not possible. |
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A right triangle contains a right angle,
whence the other two angles are acute and complementary.
The side opposite the right angle is the hypotenuse.
The sides adjacent to the right angle are called the legs.
We usually orient the triangle so that one of the legs is horizontal,
the base, and the other is vertical, the altitude.
Assume the leg and hypotenuse of one triangle are equal to the leg and hypotenuse of another. Like SAA above, this relies on a theorem we haven't proved yet, but I wanted to keep all the triangle correspondence results together. The Pythagorean theorem tells us the remaining legs in the two triangles are equal. Hence they are congruent by SSS. This method of proving congruence is called LH, for leg hypotenuse. |
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Two triangles are similar if corresponding angles are equal. The sides of the first triangle are multiplied by a ratio to get the sides of the second. In other words, the second is a scale model of the first. It may be larger, or smaller, or the same size (whence the two triangles are congruent). The fact that equal angles, and proportional lengths, are really saying the same thing, is true, but the proof involves linear transformations, and a more rigorous definition of angle; so I'm not going to go into it here. (It's not true in spherical geometry.) As an exercise, draw a right triangle with lengths 3 4 and 5, then another right triangle with lengths 6 8 and 10. Notice that corresponding angles are the same. One looks like the other, only bigger. These are similar triangles.