Angles Built from Two Rays
An angle is a figure formed by two rays that share a common endpoint. Think of it as a “turn” from one ray to the other, with the shared endpoint acting like the hinge.
Key Parts of an Angle
- Vertex: the common endpoint of the two rays.
- Sides (or arms): the two rays that form the angle.
- Interior: the set of points “between” the two sides (the region you would shade if you were coloring the inside of the angle).
- Exterior: the set of points not in the interior and not on the sides (everything outside the angle).
When you see an angle drawn, the vertex is usually where the two rays meet, and the interior is the smaller region between them (unless stated otherwise).
Diagram 1: A Basic Angle (Opening to the Right)
C• (ray BC goes up-right) / / B•----------------> A• (ray BA goes right) In this diagram, the two rays share endpoint B, so B is the vertex. The sides are ray BA and ray BC. The interior is the region between those rays (the wedge-shaped region above the horizontal ray).
Diagram 2: Same Angle Concept, Different Orientation (Opening Up)
D• ^ | | (ray ED goes up) | E•------> F• (ray EF goes right) Angles can face any direction. Here, E is the vertex, and the sides are ray ED and ray EF. The interior is the region between the upward ray and the rightward ray.
Diagram 3: An Obtuse-Looking Opening (Wider Interior)
G•<-------------------H• \ \ \ I• (ray HI goes down-right) The vertex is H. The sides are ray HG (pointing left) and ray HI (slanting down-right). The interior is the wide region between those rays.
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Interior vs. Exterior (How to Tell)
A practical way to identify the interior: imagine standing at the vertex and looking along one side, then turning the shorter way to face the other side. The region swept out by that turn is the interior. Everything else (except the sides themselves) is the exterior.
| Part | What to look for in a diagram |
|---|---|
| Vertex | The shared endpoint where both rays start |
| Sides | The two rays starting at the vertex |
| Interior | The region between the sides (often the smaller “wedge”) |
| Exterior | All points not in the interior and not on the sides |
Naming Angles (Three Common Methods)
You will see angles named in three main ways. The correct choice depends on what information is shown and whether there could be confusion.
1) Naming an Angle by Three Points: ∠ABC
When an angle is named with three letters, the middle letter is always the vertex. The other two letters lie on the sides of the angle.
Example using Diagram 1: the angle formed by rays BA and BC can be named ∠ABC or ∠CBA.
∠ABCmeans: start on side BA, vertex at B, end on side BC.∠CBAmeans the same angle, but the order of the side points is reversed.
Important: ∠ABC is not the same as ∠BAC. In ∠BAC, the vertex would be A (middle letter), which describes a different angle.
Order Matters: Vertex Must Be in the Middle
Use this quick check: if you read ∠ABC, the vertex is B. If the picture shows the rays meeting at B, then the name is consistent.
Correct: ∠ABC (vertex B) Incorrect for Diagram 1: ∠BAC (vertex A) Incorrect for Diagram 1: ∠ACB (vertex C) 2) Naming an Angle by Its Vertex Only: ∠B
If there is only one angle at a vertex in the diagram (or the context makes it clear which angle is meant), you may name it by the vertex alone.
In Diagram 1, if only one angle is drawn at point B, you can write ∠B.
When not to use it: if multiple angles share the same vertex (several rays meet at the same point), then ∠B is ambiguous and should be avoided.
3) Naming an Angle by a Number or Label: ∠1
Sometimes an angle is marked with a number or label inside the interior region. Then you can name it using that label.
K• / / (1) / J•----------> L• The angle at vertex J is labeled 1, so you can write ∠1. You could also name it with three points if points are provided on both sides (for example, ∠KJL if K and L lie on the sides and J is the vertex).
Step-by-Step: Identify Vertex, Sides, and Write a Correct Name
Procedure
- Find the common endpoint where the two rays start. That is the vertex.
- Pick one point on each ray (not the vertex). Those points help you name the angle with three letters.
- Write the angle name with the vertex in the middle:
∠(point on first side)(vertex)(point on second side). - If a number/label is shown inside the angle, you may also use
∠label.
Worked Example
P• / / O•-----------> Q• Step 1: The rays share endpoint O, so the vertex is O.
Step 2: Choose points on the sides: P is on one ray, Q is on the other ray.
Step 3: Valid three-point names include ∠POQ and ∠QOP. (Both have vertex O in the middle.)
Practice: Identify Vertex and Sides
Practice Set A
For each diagram: (1) state the vertex, (2) name the two sides as rays, (3) write two correct three-point names for the angle.
A1
S• ^ | | R•------> T• A2
U•<-----------V•-----------> W• \ \ X• A3
Z• / (2) / Y•----------------> A• In A3, also write the name using the label inside the angle.
Answer Check (Reveal After You Try)
- A1: Vertex R; sides are rays RS and RT; angle names:
∠SRT,∠TRS. - A2: Vertex V; sides are rays VU and VX (or VW and VX depending on which angle is indicated—use the two rays that actually form the highlighted angle); example names:
∠UVX,∠XVU(or∠WVX,∠XVW). - A3: Vertex Y; sides are rays YZ and YA; angle names:
∠ZYA,∠AYZ; label name:∠2.
Quick Checks: Match Angle Names to Highlighted Angles
In each figure, match the given angle name to the correct highlighted angle. (Assume the highlighted interior is the intended angle.)
Quick Check 1
M• / (shaded) / N•-----------> O• - i)
∠MNO - ii)
∠ONM - iii)
∠N
Write which names are valid for the shaded angle.
Quick Check 2 (Multiple Angles at One Vertex)
D• / / C•------> E• \ \ F• - i)
∠DCE - ii)
∠ECF - iii)
∠C
Decide which names are unambiguous and which are ambiguous, based on the diagram.
Quick Check 3 (Label Inside)
H• / / (1) / G•----------> I• - i)
∠HGI - ii)
∠1 - iii)
∠G
List all correct names for the marked angle.
