Parts of a triangle
A triangle has three corners, three sides and three angles. How do you point at exactly the part you mean, however the triangle is turned on the page?
Where you are 18 in Triangles
Why this exists
Turn a triangle over and “the top corner” becomes the bottom one. Stretch it and “the long side” becomes the short one. Any description that leans on how a triangle happens to be drawn breaks the moment the triangle is drawn again.
Geometry’s fix is to put a letter on each corner and to describe every other part using those letters. A part of the triangle then stops being “this one, here” and becomes a fact about letters: the side across from A, or the angle between \overline{AB} and \overline{AC}. Those descriptions survive turning, stretching and being copied into a proof.
Much of what comes later is written in exactly this language. The rules that say when two triangles must match are sentences such as “two sides and the angle between them” and “two angles and the side between them” (see the congruence criteria), and a sentence like that is only exact if you know which side or angle “between” points to. This lesson makes the vocabulary exact.
Definition
Take three points that do not lie on one line. Join each pair of them with a segment. The figure you have drawn is a triangle.
- The three points are its vertices (one of them is a vertex, singular).
- The three segments are its sides.
- At each vertex, the two sides that meet there make an angle that opens toward the inside of the triangle. These are the interior angles, and “the angles of a triangle” always means these three.
Naming. The triangle with vertices A, B and C is written \triangle ABC and read “triangle A B C”. The segment from A to B is \overline{AB}, and AB with no bar is its length, which is a number. A segment has no direction, so \overline{AB} and \overline{BA} are the same side.
It is common to write the length of a side with the lowercase letter of the vertex it is across from. In Figure 1 that gives
so the side called a sits across from the vertex A and does not touch it. This is the convention the OpenStax Prealgebra text uses for triangles, and it is the one here.
An angle of the triangle can be named by its vertex, \angle A, or by three letters with the vertex in the middle, \angle BAC (see naming angles). The one-letter name is safe only when the figure holds a single angle at that vertex. Its measure in degrees is m\angle A.
Opposite. Each side has two endpoints, so exactly one vertex is not on it. That side and that vertex are opposite each other, and so are the side and the angle at that vertex. In Figure 1, \overline{BC} is opposite A and opposite \angle A, which is why its length is called a.
Included. Any two angles of a triangle sit at two vertices, and exactly one side joins those two vertices. That side is included between the two angles. In Figure 2, in the upper copy, \overline{BC} is the included side of \angle B and \angle C. In the lower copy, \overline{CA} is not included between those two angles, because its far end is at A.
The same word works the other way round. Any two sides meet at exactly one vertex, and the angle at that vertex is included between the two sides. In Figure 3, \angle A is the included angle of \overline{AB} and \overline{AC}.
“Included” always describes a part for a chosen pair. The side \overline{BC} is included between \angle B and \angle C, and it is not included between \angle A and \angle B.
Two kinds of triangle name their sides. A triangle with one 90^\circ angle is a right triangle. The two sides that make the right angle are its legs, and the side opposite the right angle is its hypotenuse. A triangle with at least two equal sides is isosceles. Two equal sides are its legs, the third side is its base, the angle between the legs is the vertex angle, and the two angles at the ends of the base are the base angles. If all three sides are equal, any two of them may be taken as the legs.
The word “leg” is used in both, and it means a different side in each: in a right triangle it is a side of the right angle, and in an isosceles triangle it is one of the equal sides.
Intuition
Everything in a triangle is made from three letters. A vertex is one letter. A side is a pair of letters, two vertices joined. An angle belongs to one letter, the vertex it sits at. So every question about parts turns into a question about which letters are in and which are out.
Stand at vertex A. Two sides start from where you are standing, \overline{AB} and \overline{AC}. The third side, \overline{BC}, is across the triangle from you, and it is the only side you do not touch. That is what “opposite” means.
Now walk from A to B along the one side that joins them. That is the side included between the angles at A and at B, and its name is exactly the two letters you used. Two sides always meet at one corner, and the angle there is the one included between them. So a side is across from one corner and between the other two, all at once.
Results
Fact 1 (counting). A triangle has exactly three vertices, three sides and three angles.
Fact 2 (opposite). Every side has exactly one opposite angle, and every angle has exactly one opposite side. To find the opposite part, leave out the letter or letters you are given.
| Angle | Opposite side | Its lowercase name |
|---|---|---|
| \angle A | \overline{BC} | a |
| \angle B | \overline{CA} | b |
| \angle C | \overline{AB} | c |
Fact 3 (included). Any two angles have exactly one included side, named by the letters of their two vertices. Any two sides have exactly one included angle, at the vertex they share.
| Two angles | Included side |
|---|---|
| \angle A and \angle B | \overline{AB} |
| \angle B and \angle C | \overline{BC} |
| \angle C and \angle A | \overline{CA} |
| Two sides | Included angle |
|---|---|
| \overline{AB} and \overline{AC} | \angle A |
| \overline{BA} and \overline{BC} | \angle B |
| \overline{CA} and \overline{CB} | \angle C |
The two facts meet in one more statement: the side opposite \angle A is the side included between \angle B and \angle C. Both descriptions pick out \overline{BC}.
Fact 4 (names). The triangle with vertices A, B, C has six names, one for each order of the three letters:
Why it is true
Each fact comes from counting letters.
Counting. A triangle is three vertices and the three segments that join them in pairs. That is three vertices, three sides, and one angle at each vertex.
Opposite. A side is fixed by its two endpoints, which are two of the three vertices. Choosing a side is the same as choosing the one vertex to leave out. Leave out A and the side is \overline{BC}. Leave out B and it is \overline{CA}. Leave out C and it is \overline{AB}. Three vertices to leave out, three different sides, and each side comes from exactly one choice. So each vertex pairs with one side, and each side with one vertex, and the same holds for the angles at those vertices.
Included side. Take the angles at two vertices, say A and B. The segment from A to B is a side of the triangle, so there is an included side. There is only one, because two points are the ends of only one segment, and the other two sides each end at C, so neither has both A and B as ends.
Included angle. Two different sides of the triangle are two different pairs of vertices drawn from three letters. Two different pairs cannot share both letters, or they would be the same pair. They must share at least one, because between them they fill four places with only three letters, so some letter is used twice. They therefore share exactly one letter. That letter is the vertex where the two sides meet, and the angle there is the included angle.
Names. The first letter of a name can be chosen in three ways, the second in two, and the third is what is left: 3 \cdot 2 \cdot 1 = 6 orders. All six describe the same three points joined the same way, because a triangle is three vertices and the sides between them, with no order built in.
The remark after Fact 3 follows at once. Leaving out A leaves the pair B, C, and a pair of vertices names both a side and the two angles that side is between.
Worked examples
Reading every part from a figure
Figure 5 shows a triangle.
(a) Name its vertices, its sides and its angles. (b) Name the side opposite \angle L and the angle opposite \overline{KM}. (c) Name the side included between \angle K and \angle M, and the angle included between \overline{LK} and \overline{LM}.
Solution(a) The vertices are K, L and M. Each pair of vertices is joined by a side, so the sides are \overline{KL}, \overline{LM} and \overline{MK}. There is one angle at each vertex: \angle K, \angle L and \angle M. The triangle is \triangle KLM, or any of its other five names.
(b) The side opposite \angle L is the one not touching L, which is \overline{KM}. The angle opposite \overline{KM} is at the vertex left out when K and M are taken, which is L, so it is \angle L.
(c) The included side of \angle K and \angle M is the segment between their vertices, \overline{KM}. The two sides \overline{LK} and \overline{LM} share the letter L, so their included angle is \angle L.
Parts (b) and (c) both ended at \overline{KM} and \angle L, as the remark after Fact 3 says they must.
Using the lowercase letters
In \triangle ABC the side lengths are a = 12, b = 9 and c = 10, as in Figure 1. (a) Name, by its endpoints, the side of length 10. (b) Name the angle opposite the side of length 9. (c) Name the side included between \angle A and \angle C, and give its length. (d) Name the angle included between the sides of length 12 and 10.
Solution(a) The lowercase c is the side across from C, which is the side without C: \overline{AB}. So \overline{AB} has length 10.
(b) The side of length 9 is b, and b is across from B. The angle opposite it is \angle B.
(c) The included side of \angle A and \angle C joins A to C, so it is \overline{CA}. That is the side called b, so its length is 9.
(d) The side of length 12 is a = BC and the side of length 10 is c = AB. They share the letter B, so the included angle is \angle B.
A side that is not between the given angles
In \triangle ABC, m\angle A = 50^\circ, m\angle B = 60^\circ and BC = 8. (a) Is \overline{BC} the side included between the two given angles? (b) Find m\angle C. (c) Name two angles that have \overline{BC} included between them.
Solution(a) The given angles are at A and B, so their included side joins A to B: it is \overline{AB}. The side \overline{BC} has one end at B but its other end is at C, where no angle was given, so it is not included between them. It is the side opposite \angle A.
(b) The angles of a triangle add to 180^\circ (see the triangle angle sum):
(c) \overline{BC} joins B to C, so it is included between \angle B and \angle C, which measure 60^\circ and 70^\circ.
So two angles and a side that is not between them become, once the third angle is found, two angles and the side that is between them.
Parts of the small triangles in a figure
In Figure 6, the point D lies on \overline{BC} and the segment \overline{AD} is drawn.
(a) List every triangle in the figure. (b) Name the three angles of \triangle ABD and of \triangle ADC. (c) Which side do \triangle ABD and \triangle ADC share? (d) Why is the name \angle A no longer safe? (e) Which angle of \triangle ABD and which angle of \triangle ADC lie at D, and what do they make together?
Solution(a) A triangle needs three vertices that are not on one line. Of the four points A, B, C, D, the points B, D, C are on one line, so they make nothing. The other three choices work: \triangle ABC, \triangle ABD and \triangle ADC.
(b) Name each angle with three letters, vertex in the middle. For \triangle ABD they are \angle BAD, \angle ABD and \angle ADB. For \triangle ADC they are \angle DAC, \angle ACD and \angle ADC. Because D is on \overline{BC}, the ray from B through D is the ray from B through C, so \angle ABD and \angle ABC are the same angle. In the same way \angle ACD is \angle ACB.
(c) Both triangles have A and D as vertices, so they share \overline{AD}.
(d) The figure now has three angles at A: \angle BAD, \angle DAC and \angle BAC, which is the two together. The letter A alone could mean any of them, so here every angle at A needs three letters.
(e) In \triangle ABD the angle at D is \angle ADB, and in \triangle ADC it is \angle ADC. They sit side by side along the straight line BC, so they make a linear pair and add to 180^\circ.
A part belongs to a triangle, not to the whole figure. The same segment can be a side of one triangle and a cut through another.
An isosceles triangle with expressions
In isosceles \triangle PQR the equal sides are \overline{PQ} and \overline{PR}. The lengths are
Find x and the three side lengths, and name the legs, the base, the vertex angle and the base angles.
SolutionThe legs are the equal sides, so set their lengths equal:
Then PQ = 4(6) - 3 = 21, PR = 2(6) + 9 = 21 and QR = 6 + 4 = 10.
The legs are \overline{PQ} and \overline{PR}. The base is the third side, \overline{QR}. The vertex angle is the angle between the legs, at the vertex the legs share: \angle P. The base angles are at the ends of the base: \angle Q and \angle R. Each base angle is opposite a leg, since \angle Q is across from \overline{PR} and \angle R is across from \overline{PQ}.
Check. The lengths 21, 21, 10 really do make a triangle: any two of them add to more than the third, since 21 + 10 > 21 and 21 + 21 > 10.
A right triangle
In \triangle XYZ the angle at Y is a right angle and m\angle Z = 28^\circ. Name the legs and the hypotenuse, and find m\angle X.
SolutionThe right angle is at Y, so its two sides meet there: the legs are \overline{XY} and \overline{YZ}. The hypotenuse is the side opposite the right angle, the one without Y: \overline{XZ}.
The three angles add to 180^\circ, and one of them is 90^\circ:
The leg \overline{YZ} is opposite \angle X, and the leg \overline{XY} is opposite \angle Z. The hypotenuse is opposite the right angle. A right triangle has only one right angle, because another would leave nothing for the third angle, so “the hypotenuse” always names a single side.
Counting triangles in a fan
In Figure 7 the points B, D, E and C lie on one line, in that order, and A is joined to each of them.
(a) How many triangles does the figure contain? (b) If there were n points on the line, each joined to A, how many triangles would there be?
Solution(a) A triangle is three vertices not on one line, with all three sides drawn. The five points are A, B, D, E, C. Three points taken from the four on the line never make a triangle, since they are on one line. So every triangle uses A and two of the four points on the line. The sides are all there: the segment from A to each point, and the pieces of the line between two points.
The six choices of two points on the line give six triangles:
(b) The count is the number of ways to choose two of n points. There are n choices for the first and n - 1 for the second, which counts each pair twice, once for each order, so the number of triangles is
For n = 4 this gives \tfrac{4 \cdot 3}{2} = 6, the same as the list.
Boundaries
The three points must not lie on one line. Three points on a line, joined in pairs, give one long segment with a point on it. There is no inside and there are no three angles, so none of the words in this lesson has anything to point at.
Not every triple of lengths is a triangle. Three sides of lengths 3, 4 and 8 would need 3 + 4 to reach across 8, and it does not. The sides of a real triangle have to satisfy the triangle inequality.
“The angles of a triangle” are the interior angles. Extending one side past a vertex makes an angle outside the triangle, an exterior angle. It is not one of the three angles that belong to the triangle, and it gets its own lesson, the exterior angle theorem.
Special names need the special shape. Only a triangle with a right angle has a hypotenuse, and only one with at least two equal sides has legs and a base in the sense used here. A triangle with no right angle and no equal sides has none of these names. (In area, any side may be called the base, with the height measured to it; that is a different use of the word.) Do not read either shape off a drawing: a right angle or a pair of equal sides counts only if it is marked or stated (what a diagram may assume).
Where this goes next
The three angles at the three vertices add up to 180^\circ, and that one fact does a great deal of work: the triangle angle sum. The same fact gives a quick rule for the exterior angles that appear when a side is extended.
“Opposite” is what ties sides to angles. The longest side sits across from the largest angle (sides and angles in order). It also names the special segments of a triangle. A median runs from a vertex to the midpoint of the opposite side. An altitude runs from a vertex perpendicular to the line containing the opposite side, and in an obtuse triangle the altitudes from the two acute angles meet that line outside the triangle, past the end of the side.
“Included” is the heart of the congruence criteria. SAS uses two sides and the angle included between them, ASA uses two angles and the side included between them, and AAS uses two angles and a side that is not between them. Example 3 shows why that last case is still enough once the third angle is known.
Writing a congruence statement brings back the letters. \triangle ABC has six names, but in \triangle ABC \cong \triangle DEF the order pairs up the parts, so there it matters (congruence statements and correspondence). Legs and hypotenuse return in the Pythagorean theorem, and legs and base in isosceles and equilateral triangles.
Exercises
Reading the names
Exercises 1 to 5 are about \triangle UVW. No figure is needed: the letters carry everything.
- Name the three vertices, the three sides and the three angles of the triangle.
- Which side is opposite \angle V?
- Which angle is opposite \overline{UV}?
- Which side is included between \angle V and \angle W? Which angle is included between \overline{UV} and \overline{UW}?
- Which of these are names for the triangle: \triangle WUV, \triangle UVW, \triangle VUU, \triangle WVU?
Across and between
- In \triangle DEF, name the side opposite \angle E and the angle opposite \overline{DE}.
- In \triangle GHJ, name the side included between \angle G and \angle J, and the angle included between \overline{HG} and \overline{HJ}.
- In \triangle ABC the lowercase letters a, b, c are the lengths of the sides opposite A, B, C, and a = 8, b = 6, c = 7. (a) Name the side of length 8 by its endpoints. (b) Name the angle opposite the side of length 6. (c) Name the angle included between the sides of length 6 and 7.
Measures and expressions
- In \triangle ABC, m\angle A = 35^\circ and m\angle B = 80^\circ. Find m\angle C and name the side opposite \angle C.
- In \triangle LMN, m\angle L = 38^\circ and m\angle N = 94^\circ. Find m\angle M and name the side included between \angle L and \angle M.
- In \triangle DEF, m\angle D = 48^\circ, m\angle E = 67^\circ and EF = 10. Is \overline{EF} included between the two given angles? Find m\angle F, and name two angles that have \overline{EF} between them.
- The angles of a triangle measure (2x)^\circ, (3x + 10)^\circ and (5x - 30)^\circ. Find x and the three angles.
- In isosceles \triangle JKL the equal sides are \overline{JK} and \overline{JL}, with JK = 3x + 2, JL = 5x - 8 and KL = 2x + 1. Find x and the three lengths, and name the base, the vertex angle and the base angles.
- In \triangle STU the angle at S is a right angle and m\angle T = 41^\circ. Name the hypotenuse and the two legs, and find m\angle U.
Parts that belong to a triangle
Use Figure 8 for Exercises 15 to 18.
- Name the two smaller triangles into which \overline{BD} cuts \triangle ABC.
- Which side do the two smaller triangles share?
- Name every angle with its vertex at B in the figure, using three letters.
- Name the angle of \triangle ABD opposite \overline{AB}, and the angle of \triangle CBD opposite \overline{CB}.
Counting
- Five points lie on one line, and a sixth point A is off the line. A is joined to each of the five. How many triangles does the figure contain?
- Seven points are placed so that no three of them lie on one line, and every pair is joined by a segment. How many triangles does the figure contain?
Reasoning
- Explain why every side of a triangle has exactly one opposite angle, and never two or none.
- Explain why the side opposite \angle A is the same side as the one included between \angle B and \angle C.
- Explain why \triangle ABC and \triangle CAB are the same triangle, and say how many ways there are to write a triangle’s name.
- Explain, using the letters A, B, C, why a triangle has exactly three pairs of sides and exactly three pairs of angles.
- In Example 3, a side that was not between the two given angles became a side between two known angles. Explain why this always works once two angles are known, and why it does not work when only one angle is known.
Answers
- The vertices are U, V, W. The sides are \overline{UV}, \overline{VW}, \overline{WU}. The angles are \angle U, \angle V, \angle W, which can also be written \angle VUW, \angle UVW, \angle VWU. There is one side for each pair of vertices and one angle at each vertex.
- \overline{UW}. It is the only side that does not have V as an endpoint.
- \angle W. The endpoints of \overline{UV} are U and V, so W is the vertex left out.
- \overline{VW} is included between \angle V and \angle W, because its endpoints are the two vertices. \angle U is included between \overline{UV} and \overline{UW}, because U is the vertex the two sides share.
- \triangle WUV, \triangle UVW and \triangle WVU are names for it. \triangle VUU is not: it uses U twice and never mentions W. The order of the letters is free, but each of the three vertices appears exactly once.
- The side opposite \angle E is \overline{DF}, the side without E. The angle opposite \overline{DE} is \angle F, the vertex left out.
- The included side is \overline{GJ}, which joins the two vertices. The included angle is \angle H, since \overline{HG} and \overline{HJ} meet at H.
- (a) \overline{BC}, because a is the side opposite A and so has endpoints B and C. (b) \angle B, since b is the side \overline{CA} and the vertex left out is B. (c) \angle A. The side of length 6 is \overline{CA} and the side of length 7 is \overline{AB}, and they share A.
- m\angle C = 65^\circ and the opposite side is \overline{AB}. From 180^\circ - 35^\circ - 80^\circ = 65^\circ; the side opposite \angle C is the one without C.
- m\angle M = 48^\circ and the included side is \overline{LM}. From 180^\circ - 38^\circ - 94^\circ = 48^\circ; the side between \angle L and \angle M joins L to M.
- No: the included side of \angle D and \angle E is \overline{DE}, and \overline{EF} does not have an end at D. m\angle F = 180^\circ - 48^\circ - 67^\circ = 65^\circ. Then \overline{EF} is included between \angle E and \angle F, which measure 67^\circ and 65^\circ.
- x = 20, and the angles are 40^\circ, 70^\circ and 70^\circ. The angles add to 180^\circ. The sum of the three is 10x - 20, and 10x - 20 = 180 gives x = 20. Then 2(20) = 40, 3(20) + 10 = 70 and 5(20) - 30 = 70.
- x = 5, with JK = 17, JL = 17 and KL = 11. The legs are equal: 3x + 2 = 5x - 8 gives 2x = 10. The base is \overline{KL}, the side that is not a leg; the vertex angle is \angle J, between the legs; the base angles are \angle K and \angle L, at the ends of the base.
- The hypotenuse is \overline{TU}, the side opposite the right angle at S. The legs are \overline{ST} and \overline{SU}. m\angle U = 180^\circ - 90^\circ - 41^\circ = 49^\circ.
- \triangle ABD and \triangle CBD. Each uses B, D and one end of \overline{AC}.
- \overline{BD}. It has endpoints B and D, and both triangles contain both points.
- \angle ABD, \angle DBC and \angle ABC. The third is the whole angle of the large triangle, and it is made of the other two, which sit side by side.
- In \triangle ABD the vertex not on \overline{AB} is D, so the angle is \angle ADB. In \triangle CBD the vertex not on \overline{CB} is D, so the angle is \angle CDB. Both are at D, on opposite sides of \overline{BD}.
- 10. Every triangle uses A and two of the five points on the line, and 5 \cdot 4 \div 2 = 10.
- 35. Any three of the seven points are not on one line, and all their sides are drawn, so every choice of three points is a triangle. There are 7 \cdot 6 \cdot 5 = 210 ordered choices, and each triangle is counted six times, once for each of its names, so 210 \div 6 = 35.
- A side has two endpoints, which are two of the triangle’s three vertices, so exactly one vertex is left over. That vertex has exactly one angle of the triangle. So the side has one opposite angle: not none, because a vertex is always left over, and not two, because only one is.
- The side opposite \angle A is the one that does not have A as an endpoint. Its endpoints are therefore B and C. The side included between \angle B and \angle C is the side joining B to C. Both descriptions give \overline{BC}.
- \triangle ABC and \triangle CAB both name the triangle with vertices A, B, C and the three sides between them. A triangle is those three points and segments, and the order the letters are written in is not part of it. Three letters can be put in order in 6 ways (three choices for the first, two for the second, one for the third), so there are six ways to write the name.
- A pair of sides is a pair chosen from the three sides \overline{AB}, \overline{BC}, \overline{CA}, and there are three choices, because choosing a pair is the same as choosing the one side to leave out. The angles are at A, B and C, and a pair of angles is a pair of vertices, which is again three choices, one for each vertex left out.
- Once two angles are known, the triangle angle sum gives the third, since it is 180^\circ minus the two. Every side joins two vertices, and all three angles are now known, so every side is included between two known angles. With only one angle known, say \angle A, the side opposite it, \overline{BC}, is included between \angle B and \angle C, which are both unknown. The angle sum fixes only their total, 180^\circ - m\angle A, and not either one.