Triangle Theorems

Triangles are the closed figure studied in theoretical geometry. The theorems available for triangle congruence and similarity are used in proving and solving many a geometrical properties and problems. There are quite a number of theorems starting with triangle sum theorem till theorems on the concurrencies. Some of the theorems are given here with brief explanations on them.

Triangle Sum Theorem

The sum of the measures of the angles of a triangle is 180o.

This is a common property of all triangles. This property ensures that any triangle has at least two acute angles.

Exterior Angle Theorem

                                    The measure of an exterior angle of a triangle is equal to the sum of the interior opposite angles.
 The exterior angle of a triangle is the linear pair of an interior angle.
              
 The diagram given here shows the exterior angle measures
 E1, E2 and E3 of the angles A, B and C of the triangle.

 The exterior angle theorem states that

 E1 = m < B + m < C
 E2 =  m < C + m < A
 E3 = m < A + m < B                                       
              Exterior Angle Theorem                    

Side Comparison

What is the relation between the angle measure and side length? How do the sides compare in terms of their lengths?

In a triangle the larger angle has the longer side opposite to it.

Triangle inequality theorem The sum of the lengths of two sides of a triangle is greater than the length of the third side.

Triangle Congruence Theorems

There area number of congruence theorems for triangles based on the criterion considered. All these theorems are commonly used in geometrical problems.

1. SSS Congruence Postulate or SSS Postulate (Side-Side-Side)

    If three sides of a triangle are congruent to the three sides of a triangle then two triangles are congruent.
    In the adjoining diagram, the sides of the triangle ABC are shown to be congruent to the sides of triangle PQR.
    By SSS postulate for  triangle congruency,
    $\bigtriangleup$ ABC $\cong$ $\bigtriangleup$ PQR
SSS Congruence Postulate

The congruence statement should be made maintaining the correct order of the vertices. It is wrong to write $\bigtriangleup$ ABC $\cong$ $\bigtriangleup$ QPR or $\bigtriangleup$ ABC $\cong$ $\bigtriangleup$ RPQ.
 
The position of vertex in the naming of the triangle should match the position of the corresponding vertex in the name of the other triangle.

2. SAS Congruence Postulate or SAS Postulate (Side Angle Side)

    If two sides and the included angle of a triangle are congruent to the two sides and included angle of another triangle, then the two triangles are congruent.

    In the adjoining diagram, the two sides and the angle included by them of triangle DEF are congruent to the two sides and the angle included between them of triangle STU.

    By SAS postulate for triangle congruency, $\bigtriangleup$ DEF $\cong$ $\bigtriangleup$ STU
SAS Congruence Postulate
    The equivalent congruent statements are $\bigtriangleup$ EFD $\cong$ $\bigtriangleup$ TUV and $\bigtriangleup$ FDE $\cong$ $\bigtriangleup$ UST.

In all these statements the order of corresponding vertices are maintained on either side.
3. ASA Congruence Postulate or ASA Postulate ( Angle Side Angle)

    If two angles and the included side of a triangle are congruent to the corresponding two angles and included side of another triangle, then the two triangles are congruent.

     In the adjoining diagram the angles Q and R and the side included QR of triangle PAR are shown to be congruent to the angles V and W and the included side VW of triangle UVW.

     By ASA postulate for triangle congruency

     $\bigtriangleup$ PQR $\cong$ $\bigtriangleup$ UVW
ASA Congruence Postulate
     The congruency also exists even if the side is not included, which is stated by AAS postulate.

4.  AAS Congruence Postulate or AAS Postulate ( Angle Angle Side)

     If two angles and a non included side of one triangle are congruent to two angles and a non included side of another triangle, then the two triangles are congruent.
     In the adjoining diagram angles B and C and the non included side CA of triangle ABC are congruent to the two angles Q and R and the non included side RP of triangle PQR.

     By AAS postulate for triangle congruency

     $\bigtriangleup$ ABC $\cong$ $\bigtriangleup$ PQR
AAS Congruence Postulate
     This theorem is a corollary of ASA theorem. If two angles of a triangle are congruent, by triangle sum property, the third angles are also congruent. That would make angles A and P congruent here.

5.  HL Theorem (Hypotenuse Leg)

     If the hypotenuse and a leg of a right triangle are congruent to the corresponding parts of another right triangle, then the two triangles are congruent.

     In the diagram given Hypotenuse AB and leg CA of right triangle ABC are congruent to hypotenuse LM and leg NL of the second right triangle LMN.

     By HL postulate for triangle congruency

     $\bigtriangleup$ ABC $\cong$ $\bigtriangleup$ LMN
HL Theorem
The fact that the two right angles in the triangles are congruent is implied by the inclusion hypotenuse in the name of the postulate. A general SSA situation does not imply triangle congruence. HL is indeed the version of SSS postulate to be applied to right triangles. If the hypotenuse and a leg are congruent to the corresponding parts of another triangle, by Pythagorean theorem the other legs are also congruent.

Triangle Similarity Theorems

Two triangles are said to be similar if the three angles of one triangle are congruent to three angles of another. The sides in this case will not be congruent but proportional. Triangle congruency is a special case of similarity where this proportion = 1. The theorems stating the types of similarity are given below.

AAA Similarity
Triangle Similarity Theorems
In the diagram it is shown the angles A and B of triangle ABC are congruent to the angles P and Q of triangle PQR. By triangle sum property the third angles C and R are also congruent. Hence by the definition of similar triangles,

$\bigtriangleup$ ABC ||| $\bigtriangleup$ PQR

SSS Similarity
SSS Similarity
The sides of triangle LMN are of half the lengths of the sides of triangle PQR.

$\frac{\overline{PQ}}{\overline{LM}}=\frac{\overline{QR}}{\overline{MN}}=\frac{\overline{RP}}{\overline{NL}}=\frac{2}{1}=2$

The equality of the ratios of the sides determines the similarity.

By SSS similarity

$\bigtriangleup$ PQR ||| $\bigtriangleup$ LMN

SAS similarity

SAS Similarity
In this case the lengths of two sides of the triangle STU are shown proportional to the lengths of two sides of triangle XYZ and the corresponding included angles S and X are congruent.

By SAS similarity

$\bigtriangleup$ STU ||| $\bigtriangleup$ XYZ

Conversely if two triangles are given to be congruent, the lengths of corresponding sides are proportional.

$\Delta PQR |||\Delta LMN\rightarrow \frac{\overline{PQ}}{\overline{LM}}=\frac{\overline{QR}}{\overline{MN}}=\frac{\overline{RP}}{\overline{NL}}$

Mid Point Theorem

This theorem is commonly known as the mid segment theorem. This theorem is proved using similarity of the triangles formed and taking the proportions of the corresponding sides.

The segment joining the midpoints of two sides of a triangle is parallel to the third side, and is half the length of the third side.

In triangle ABC, D and E are the mid points of the sides AC and BC.  The mid segment theorem states that DE is parallel to the third side AB and,

DE = $\frac{1}{2}$ AB

Mid Point Theorem

Concurrency Theorems

The four points of concurrencies related to a triangle are circumcenter, orthocenter, incenter and centroid.

1. Perpendicular Bisector Theorem

Perpendicular Bisector Theorem
The perpendicular bisectors of the three sides of a triangle intersect at a point equidistant from the vertices.
The perpendicular bisectors of the triangle ABC intersect at O.  Point O is equidistant from the three vertices A, B and C. The circle drawn with O as center passing through the vertices is called the circumcircle of the triangle and the point O is called the circumcenter.
The circumcenter is the point of concurrence of the perpendicular bisectors of the triangle.

2. Angle Bisector Theorem

The angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle.
The angle bisectors of triangle ABC intersect at point I. Since this point is equidistant from the sides of the triangle, a circle can be drawn with I as center touching the three sides of the triangle. The circle is called the incircle and I the incenter of the triangle.

Angle Bisector Theorem
Incenter is the point of concurrence of the angle bisectors of a triangle.

3. Altitudes Theorem

The three altitudes of a triangle are concurrent.
In the given diagram for triangle ABC , the altitudes AD, BE and DF of triangle ABC intersect at the point O, which is called the orthocenter of the triangle.
Altitude Theorem
The orthocenter of the triangle is the point of concurrence of the altitudes of a triangle.

4. Centroid Theorem

The Medians of a triangle intersect at a point which is called the centroid or center of gravity of the triangle.
The Medians AD, BE and CF of triangle ABC at G the centroid of the triangle.
The centroid is a point trisection of each median.

$\frac{AG}{GD}=\frac{BG}{GE}=\frac{CG}{GF}=\frac{2}{1}$

Centroid Theorem
The centroid divides each median of the triangle in the ratio 2:1

Topics in Triangle Theorems