Triangles
Learning objectives
- Classify triangles by their sides and by their angles.
- Use the angle sum and the exterior angle theorem.
- Decide when three lengths can be the sides of a triangle.
- Use the congruence criteria SSS, SAS and ASA.
Triangles and their types
A triangle \(ABC\) is formed by three points that do not lie on one line, joined by three segments called its sides. The three angles at the vertices are its angles.
Drag the vertices (or focus one and use the arrow keys). The triangle snaps to the grid. Watch the three angles and their sum.
By sides: equilateral (three equal sides), isosceles (at least two equal sides), scalene (no equal sides). By angles: acute (all angles acute), right (one right angle), obtuse (one obtuse angle).
The angle sum
The three angles of a triangle add up to \(180^\circ\).
Show proof
Through \(A\) draw the line parallel to \(BC\). The angles it makes with \(AB\) and \(AC\) equal \(\angle B\) and \(\angle C\) (alternate interior angles). Together with \(\angle A\) they fill a straight angle, which is \(180^\circ\). ∎
Two angles of a triangle are \(50^\circ\) and \(60^\circ\). Find the third.
The third angle is \(180^\circ-50^\circ-60^\circ=70^\circ\).
An exterior angle of a triangle equals the sum of the two interior angles that are not adjacent to it.
Show proof
Extend \(BC\) beyond \(C\). The exterior angle at \(C\) and \(\angle C\) form a straight angle, so the exterior angle is \(180^\circ-\angle C\). By the angle sum, \(180^\circ-\angle C=\angle A+\angle B\). ∎
The triangle inequality
Move the vertices and compare the sum of two sides with the third side. Try to put the three points on one line.
In every triangle, each side is shorter than the sum of the other two.
The idea is that a straight segment is the shortest path between two points, so going from \(B\) to \(C\) directly is shorter than going through \(A\).
Can 3, 4 and 8 be the sides of a triangle? What about 5, 7 and 9?
Since \(3+4=7<8\), the lengths 3, 4, 8 cannot. For 5, 7, 9 we have \(5+7>9\), \(5+9>7\) and \(7+9>5\), so they can.
Congruent triangles
Two triangles are congruent if one can be moved onto the other, so that all matching sides and angles are equal.
Two triangles are congruent if:
SSS: the three sides of one equal the three sides of the other;
SAS: two sides and the angle between them are equal;
ASA: two angles and the side between them are equal.
Equal angles alone are not enough: a small and a large equilateral triangle have the same angles but are not congruent. Two sides and a non-included angle (SSA) are not enough either.
In an isosceles triangle, the angles opposite the equal sides are equal.
Show proof
Let \(AB=AC\). Compare triangle \(ABC\) with the same triangle read as \(ACB\). We have \(AB=AC\), \(AC=AB\) and the common angle \(\angle A\), so the two are congruent by SAS. Matching angles give \(\angle B=\angle C\). ∎
Exercises
Two angles of a triangle are \(47^\circ\) and \(68^\circ\). Find the third and classify the triangle by its angles.
Show solution
The third angle is \(180^\circ-47^\circ-68^\circ=65^\circ\). All three angles are acute, so the triangle is acute.
An exterior angle of a triangle is \(125^\circ\) and one of the remote interior angles is \(50^\circ\). Find the other remote interior angle.
Show solution
The exterior angle equals the sum of the remote interior angles, so the other one is \(125^\circ-50^\circ=75^\circ\).
Can 2, 3, 6 be the sides of a triangle? Can 4, 5, 8?
Show solution
No for 2, 3, 6, since \(2+3=5<6\). Yes for 4, 5, 8: \(4+5=9>8\) and the other two sums are larger still.
Prove that every angle of an equilateral triangle is \(60^\circ\).
Show solution
An equilateral triangle is isosceles in every way, so by the base angle theorem all three angles are equal. They add up to \(180^\circ\), so each is \(60^\circ\).
In triangle \(ABC\), \(\angle A=90^\circ\) and \(\angle B=2\angle C\). Find \(\angle B\) and \(\angle C\).
Show solution
We have \(\angle B+\angle C=90^\circ\) and \(\angle B=2\angle C\), so \(3\angle C=90^\circ\). Hence \(\angle C=30^\circ\) and \(\angle B=60^\circ\).
Explain why two triangles with all three angles equal need not be congruent.
Show solution
Take an equilateral triangle with side 1 and another with side 2. All their angles are \(60^\circ\), but their sides differ, so they are not congruent. Equal angles fix the shape but not the size.
Summary
- Triangles are classified by sides (equilateral, isosceles, scalene) and by angles (acute, right, obtuse).
- The angles of a triangle add up to \(180^\circ\), and an exterior angle equals the sum of the two remote interior angles.
- Each side is shorter than the sum of the other two.
- SSS, SAS and ASA prove triangles congruent; equal angles alone do not.