Naming a triangle

The idea

A triangle has three vertices and three sides joining pairs of vertices; it is named by its vertices in any order, such as ∆ABC, and its angles ∠CAB, ∠ABC, ∠BCA are written ∠A, ∠B, ∠C.

One triangle, two names

A triangle has three corner points and three sides that join them. Two friends look at the same triangle. One calls it ∆ABC. The other calls it ∆CAB.

Who is right?

• Only ∆ABC: the letters must go A, B, C
• Both are right
• Neither: a triangle has one name only

Parts and names

Tap the parts. Then name the triangle with letters. Then name its angles.

Any order, and the middle letter is the corner

A triangle has 3 vertices (its corner points) and 3 sides. Each side joins a pair of vertices.

To name the triangle, write its vertices in any order. ∆ABC, ∆BCA and ∆CAB are the same triangle.

For an angle, the middle letter is the corner. ∠CAB is the angle at A. Its short name is ∠A.

A triangle has three vertices and three sides joining pairs of vertices. It is named by its vertices in any order, such as ∆ABC. Its angles ∠CAB, ∠ABC, ∠BCA are written ∠A, ∠B, ∠C.

• We write | It means
• ∆ABC | the triangle with vertices A, B, C
• AB | the side that joins A and B
• ∠CAB or ∠A | the angle at the vertex A

Notes

A triangle has three vertices and three sides joining pairs of vertices. It is named by its vertices in any order (∆ABC = ∆BCA = ∆CAB). In an angle name the middle letter is the corner: ∠CAB = ∠A.

Check yourself

Which of these is not a name for ∆ABC?

In ∆ABC, what is AC?

In ∆PQR, the angle at P can be written with three letters, the corner in the middle: ∠QPR. How many different three-letter names does the angle at P have?

Answer: 2

∠QPR and ∠RPQ: the corner P stays in the middle and Q, R can swap ends. That is 2 names.

A student says: “∠ABC is the angle at A, because A comes first.” What is wrong?

• ∆CBA. This uses A, B and C in another order. That is still ∆ABC.
• ∆BCA. This uses A, B and C in another order. That is still ∆ABC.
• ∆ABD — correct. Yes! D is not a vertex of this triangle. A name uses its three vertices, and only those.
• ∆CAB. This uses A, B and C in another order. That is still ∆ABC.
• A vertex: a corner point.. A vertex has one letter, like A. AC has two letters: it is the segment that joins A and C.
• An angle.. An angle is written with ∠, like ∠A. AC is the side that joins A and C.
• A side: it joins the vertices A and C. — correct. Yes! Two letters name a side: the line segment between two vertices.
• Nothing is wrong: the first letter is the corner.. The first letter is only one end of the angle. The corner is the middle letter.
• ∠ABC names the whole triangle, not an angle.. The sign ∠ means an angle. A triangle is written with ∆.
• The corner is the middle letter, so ∠ABC is the angle at B. — correct. Yes! In ∠ABC the middle letter B is the corner.