Triangle A closed figure consisting of three line segments linked end-to-end. A 3-sided polygon.



Vertex The vertex (plural: vertices) is a corner of the triangle. Every triangle has three vertices.


Base The base of a triangle can be any one of the three sides, usually the one drawn at the bottom.

• You can pick any side you like to be the base.
• Commonly used as a reference side for calculating the area of the triangle.
• In an isosceles triangle, the base is usually taken to be the unequal side.


Altitude The altitude of a triangle is the perpendicular from the base to the opposite vertex. (The base may need to be extended).


• Since there are three possible bases, there are also three possible altitudes.
• The three altitudes intersect at a single point, called the orthocenter of the triangle.


Median The median of a triangle is a line from a vertex to the midpoint of the opposite side.


• The three medians intersect at a single point, called the centroid of the triangle.
• Each median divides the triangle into two smaller triangles which have the same area.
• Because there are three vertices, there are of course three possible medians.
• No matter what shape the triangle, all three always intersect at a single point. This point is called the centroid of the triangle.
• The three medians divide the triangle into six smaller triangles of equal area.
• The centroid (point where they meet) is the center of gravity of the triangle
• Two-thirds of the length of each median is between the vertex and the centroid, while one-third is between the centroid and the midpoint of the opposite side.
• \(m=\sqrt{\frac{2b^2+2c^2-a^2}{4}}\), where \(a\), \(b\) and \(c\) are the sides of the triangle and \(a\) is the side of the triangle whose midpoint is the extreme point of median \(m\).


Area The number of square units it takes to exactly fill the interior of a triangle.

Usually called "half of base times height", the area of a triangle is given by the formula below.
• \(A=\frac{hb}{2}\)

Other formula:
• \(A=\frac{P*r}{2}\)

• \(A=\frac{abc}{4R}\)

Where \(b\) is the length of the base, \(a\) and \(c\) the other sides; \(h\) is the length of the corresponding altitude; \(R\) is the Radius of circumscribed circle; \(r\) is the radius of inscribed circle; P is the perimeter

• Heron's or Hero's Formula for calculating the area \(A = \sqrt{s(s-a)(s-b)(s-c)}\) where \(a,b,c\) are the three sides of the triangle and \(s = \frac{a+b+c}{2}\) which is the semi perimeter of the triangle.


Perimeter The distance around the triangle. The sum of its sides.

• For a given perimeter equilateral triangle has the largest area.
• For a given area equilateral triangle has the smallest perimeter.


Relationship of the Sides of a Triangle

• The length of any side of a triangle must be larger than the positive difference of the other two sides, but smaller than the sum of the other two sides.


Interior angles The three angles on the inside of the triangle at each vertex.

• The interior angles of a triangle always add up to 180°
• Because the interior angles always add to 180°, every angle must be less than 180°
• The bisectors of the three interior angles meet at a point, called the incenter, which is the center of the incircle of the triangle.


Exterior angles The angle between a side of a triangle and the extension of an adjacent side.


• An exterior angle of a triangle is equal to the sum of the opposite interior angles.
• If the equivalent angle is taken at each vertex, the exterior angles always add to 360° In fact, this is true for any convex polygon, not just triangles.


Midsegment of a Triangle A line segment joining the midpoints of two sides of a triangle


• A triangle has 3 possible midsegments.
• The midsegment is always parallel to the third side of the triangle.
• The midsegment is always half the length of the third side.
• A triangle has three possible midsegments, depending on which pair of sides is initially joined.


Relationship of sides to interior angles in a triangle

• The shortest side is always opposite the smallest interior angle
• The longest side is always opposite the largest interior angle


Angle bisector An angle bisector divides the angle into two angles with equal measures.


• An angle only has one bisector.
• Each point of an angle bisector is equidistant from the sides of the angle.
• The angle bisector theorem states that the ratio of the length of the line segment BD to the length of segment DC is equal to the ratio of the length of side AB to the length of side AC: \(\frac{BD}{DC}=\frac{AB}{AC}\)
• The incenter is the point where the angle bisectors intersect. The incenter is also the center of the triangle's incircle - the largest circle that will fit inside the triangle.


Similar Triangles Triangles in which the three angles are identical.

• It is only necessary to determine that two sets of angles are identical in order to conclude that two triangles are similar; the third set will be identical because all of the angles of a triangle always sum to 180º.
• In similar triangles, the sides of the triangles are in some proportion to one another. For example, a triangle with lengths 3, 4, and 5 has the same angle measures as a triangle with lengths 6, 8, and 10. The two triangles are similar, and all of the sides of the larger triangle are twice the size of the corresponding legs on the smaller triangle.
• If two similar triangles have sides in the ratio \(\frac{x}{y}\), then their areas are in the ratio \(\frac{x^2}{y^2}\)


Congruence of triangles Two triangles are congruent if their corresponding sides are equal in length and their corresponding angles are equal in size.

1. SAS (Side-Angle-Side): If two pairs of sides of two triangles are equal in length, and the included angles are equal in measurement, then the triangles are congruent.

2. SSS (Side-Side-Side): If three pairs of sides of two triangles are equal in length, then the triangles are congruent.

3. ASA (Angle-Side-Angle): If two pairs of angles of two triangles are equal in measurement, and the included sides are equal in length, then the triangles are congruent.

So, knowing SAS or ASA is sufficient to determine unknown angles or sides.

NOTE IMPORTANT EXCEPTION:
The SSA condition (Side-Side-Angle) which specifies two sides and a non-included angle (also known as ASS, or Angle-Side-Side) does not always prove congruence, even when the equal angles are opposite equal sides.

Specifically, SSA does not prove congruence when the angle is acute and the opposite side is shorter than the known adjacent side but longer than the sine of the angle times the adjacent side. This is the ambiguous case. In all other cases with corresponding equalities, SSA proves congruence.

The SSA condition proves congruence if the angle is obtuse or right. In the case of the right angle (also known as the HL (Hypotenuse-Leg) condition or the RHS (Right-angle-Hypotenuse-Side) condition), we can calculate the third side and fall back on SSS.

To establish congruence, it is also necessary to check that the equal angles are opposite equal sides.

So, knowing two sides and non-included angle is NOT sufficient to calculate unknown side and angles.

Angle-Angle-Angle
AAA (Angle-Angle-Angle) says nothing about the size of the two triangles and hence proves only similarity and not congruence.

So, knowing three angles is NOT sufficient to determine lengths of the sides.