Triangle Properties and Theorems

Reading a triangle before calculating

Begin with the information that is given, not what a sketch appears to show. Matching side marks mean equal lengths. Matching angle arcs mean equal angle measures. A small square marks a right angle. A horizontal-looking base does not establish that another side is perpendicular to it.

Triangles can be classified by sides and by angles at the same time. An isosceles triangle has at least two equal sides; an equilateral triangle has three. Under this inclusive convention, every equilateral triangle is also isosceles. An acute triangle has three acute angles, a right triangle has one right angle, and an obtuse triangle has one obtuse angle. An isosceles right triangle belongs to both a side category and an angle category.

The side opposite $A$ joins the other two vertices, $B$ and $C$. Keeping that correspondence clear prevents errors when comparing side lengths and angles.

Why the angles add to 180 degrees

For every plane triangle,

A useful proof draws a line through $A$ parallel to $BC$. The angle that $AB$ makes with the new line equals the angle at $B$, and the angle that $AC$ makes with it equals the angle at $C$, by alternate interior angles. Those two angles and the original angle at $A$ fill a straight angle. The conclusion follows from parallel-line relationships, not from measuring a drawing. This is the argument formalized in Euclid I.32.

An angle described by algebra

Suppose a triangle has angles $2x$, $3x$ and $4x$. These expressions denote angle measures in degrees.

The angles are $40^\circ$, $60^\circ$ and $80^\circ$. Their sum is $180^\circ$, and all are positive. Both checks matter: a formal algebraic answer that produced a zero or negative interior angle would not describe the intended triangle.

Exterior angles and isosceles triangles

Extend side $BC$ beyond $C$. The exterior angle at $C$ and the interior angle at $C$ form a straight angle. Subtracting the triangle angle-sum equation from that straight-angle relationship gives

The two angles on the right are the remote interior angles, not the angle adjacent to the exterior angle. If they are $46^\circ$ and $71^\circ$, the exterior angle is $117^\circ$ and the remaining interior angle is $63^\circ$.

In an isosceles triangle, equal sides have equal opposite angles. If $AB=AC$, then $\angle B=\angle C$. For example, an apex angle of $44^\circ$ leaves $136^\circ$ for the two equal base angles, so each is $68^\circ$. The converse also holds: equal angles in a triangle have equal opposite sides. Congruence provides a justification for both directions, developed in Congruence in Geometry.

Which side lengths can form a triangle?

Every side is shorter than the sum of the other two. Otherwise the two shorter segments could not reach the endpoints of the longest segment while enclosing an area. Equality produces a straight, degenerate arrangement rather than a triangle.

For fixed positive sides $a$ and $b$, the third side $c$ must satisfy

With sides $5\text{ cm}$ and $8\text{ cm}$, the third side lies strictly between $3\text{ cm}$ and $13\text{ cm}$. If its length is an integer number of centimetres, the possibilities are $4,5,\ldots,12$. A length of $3\text{ cm}$ is excluded, even though it satisfies the non-strict inequality $3\le c$.

Within one triangle, the longest side lies opposite the largest angle. For side lengths $5$, $7$ and $9$, the angle opposite $9$ is largest. This orders the angles without determining their numerical values. Finding the actual measures requires further information or trigonometry.

Medians, altitudes and bisectors are different

A median joins a vertex to the midpoint of the opposite side. An altitude is perpendicular to the line containing the opposite side. An angle bisector splits a vertex angle into two equal angles. A perpendicular bisector passes through a side’s midpoint at right angles to it and need not pass through a vertex.

In a general triangle, these are different lines. In an isosceles triangle, the line from the apex to the base midpoint has several roles at once. That special coincidence is not a rule for every triangle.

The three medians meet at the centroid. It divides each median in the ratio $2:1$, measured from the vertex to the opposite side. A median of length $9\text{ cm}$ is therefore divided into a $6\text{ cm}$ vertex-to-centroid section and a $3\text{ cm}$ centroid-to-midpoint section. The centroid is always inside the triangle.

The perpendicular bisectors meet at the circumcentre, the centre of the circle through the three vertices. Angle bisectors meet at the incentre, which has equal perpendicular distances from the three sides. Altitudes meet at the orthocentre. The circumcentre and orthocentre may lie outside an obtuse triangle; a triangle centre does not have to lie inside its boundary.

A segment joining two midpoints

If $D$ and $E$ are the midpoints of $AB$ and $AC$, then $DE$ is parallel to $BC$ and has half its length. The smaller triangle $ADE$ has the same included angle at $A$ and half the length along both adjacent sides. SAS similarity therefore makes the scale factor $1/2$, giving

If $BC=14\text{ cm}$, then $DE=7\text{ cm}$. This is a relationship between corresponding sides, not a claim that every segment inside a triangle has half the base length. Similarity and Coordinate Proofs give two ways to justify the same result.

Related topics

Use Congruence in Geometry to justify equal lengths and angles, then Similarity in Geometry to compare triangles of different sizes. Lines and Angles provides the parallel-line facts used in the angle-sum proof.