Mathematics: Geometry - Triangle Classification and Properties

Classifications of Triangles
A scalene triangle is a triangle with no congruent sides. A scalene triangle will also have three angles of different measures.
The angle with the largest measure is opposite the longest side, and the angle with the smallest measure is opposite the shortest side. An acute triangle is a triangle whose three angles are all less than 90°. If two of the angles are equal, the acute triangle is also an isosceles triangle. An isosceles triangle will also have two congruent angles opposite the two congruent sides. If the three angles are all equal, the acute triangle is also an equilateral triangle. An equilateral triangle will also have three congruent angles, each 60°. All equilateral triangles are also acute triangles. An obtuse triangle is a triangle with exactly one angle greater than 90°. The other two angles may or may not be equal. If the two remaining angles are equal, the obtuse triangle is also an isosceles triangle. A right triangle is a triangle with exactly one angle equal to 90°. All right triangles follow the Pythagorean theorem. A right triangle can never be acute or obtuse.



Similarity and Congruence Rules
Similar triangles are triangles whose corresponding angles are equal and whose corresponding sides are proportional. Represented by AAA. Similar triangles whose corresponding sides are congruent are also congruent triangles.

Triangles can be shown to be congruent in 5 ways:
SSS: Three sides of one triangle are congruent to the three corresponding sides of the second triangle.
SAS: Two sides and the included angle (the angle formed by those two sides) of one triangle are congruent to the corresponding two sides and included angle of the second triangle.
ASA: Two angles and the included side (the side that joins the two angles) of one triangle are congruent to the corresponding two angles and included side of the second triangle.
AAS: Two angles and a non-included side of one triangle are congruent to the corresponding two angles and non-included side of the second triangle.
HL: The hypotenuse and leg of one right triangle are congruent to the corresponding hypotenuse and leg of the second right triangle.

General Rules for Triangles
The triangle inequality theorem states that the sum of the measures of any two sides of a triangle is always greater than the measure of the third side. If the sum of the measures of two sides were equal to the third side, a triangle would be impossible because the two sides would lie flat across the third side and there would be no vertex. If the sum of the measures of two of the sides was less than the third side, a closed figure would be impossible because the two shortest sides would never meet. In other words, for a triangle with sides lengths A, B, and C:
,
, and

The sum of the measures of the interior angles of a triangle is always 180°. Therefore, a triangle can never have more than one angle greater than or equal to 90°.
In any triangle, the angles opposite congruent sides are congruent, and the sides opposite congruent angles are congruent. The largest angle is always opposite the longest side, and the smallest angle is always opposite the shortest side.
The line segment that joins the midpoints of any two sides of a triangle is always parallel to the third side and exactly half the length of the third side.

P1. Given the following pairs of triangles, determine whether they are similar, congruent, or neither (note that the figures are not drawn to scale):
(a).

(b).

(c).

P1. (a). Neither: We are given that two sides lengths and an angle are equal, however, the angle given is not between the given side lengths. That means there are two possible triangles that could satisfy the given measurements. Thus, we cannot be certain of congruence:

(b) Similar: Since we are given a side-angle-side of each triangle and the side lengths given are scaled evenly

and the angles are equal. Thus, . If the side lengths were equal, then they would be congruent.

(c) Congruent: Even though we aren't given a measurement for the shared side of the figure, since it is shared it is equal. So, this is a case of SAS. Thus,