Geometry Worksheets | Angles Worksheets - Free Printable

Here's the complete solution to the worksheet, with explanations for each section.

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## Section 1: Name the vertex and sides of each angle.

An angle is formed by two rays (sides) that share a common endpoint (vertex).

- The vertex is the point where the two rays meet.
- The sides are the two rays forming the angle.

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1)

Vertex: O
Sides: Ray OF and Ray OR
*(The angle opens between points F and R, with O at the corner.)*

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2)

Vertex: O
Sides: Ray OE and Ray OF
*(Rays OE and OF meet at O.)*

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3)

Vertex: K
Sides: Ray KJ and Ray KL
*(Rays from K go to J and L.)*

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4)

Vertex: H
Sides: Ray HO and Ray HJ
*(Rays from H go to O and J.)*

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## Section 2: Name each angle in four ways.

There are 4 standard ways to name an angle:

1. By the vertex letter alone — *only if there’s only one angle at that vertex.*
2. By three letters, with the vertex in the middle: ∠ABC or ∠CBA
3. By a number inside the angle (if labeled)
4. By a letter/variable assigned to the angle (if labeled)

*(Note: In this worksheet, angles are labeled with letters at endpoints and sometimes numbers or variables. We’ll use the three-letter format and number/variable if present.)*

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5)

Angle has vertex at R, with rays to Q and S. Number “1” is inside.

✔ Four ways:
- ∠QRS (or ∠SRQ)
- ∠R (since only one angle at R)
- ∠1
- (No variable given, so skip if not labeled)

> Answer: ∠QRS, ∠SRQ, ∠R, ∠1

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6)

Vertex at M, rays to L and N. Angle labeled “2”.

✔ Four ways:
- ∠LMN (or ∠NML)
- ∠M
- ∠2
- (No variable)

> Answer: ∠LMN, ∠NML, ∠M, ∠2

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7)

Vertex at N, rays to M and O. Angle labeled “3”.

✔ Four ways:
- ∠MNO (or ∠ONM)
- ∠N
- ∠3
- (No variable)

> Answer: ∠MNO, ∠ONM, ∠N, ∠3

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8)

Vertex at L, rays to K and M. Angle labeled “4”.

✔ Four ways:
- ∠KLM (or ∠MLK)
- ∠L
- ∠4
- (No variable)

> Answer: ∠KLM, ∠MLK, ∠L, ∠4

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## Section 3: Name all the angles that have V as a vertex.

We must list every angle that has point V as its vertex — meaning both rays of the angle start at V.

We’ll list them using three-letter notation (vertex in the middle) and/or number if labeled.

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9)

Points around V: D, E, F (with rays to them). Angles labeled 1, 2, 3.

Angles at V:
- ∠DVE (angle 1)
- ∠EVF (angle 2)
- ∠DVF (angle 3 — the whole angle from D to F)

> Answer: ∠DVE, ∠EVF, ∠DVF, ∠1, ∠2, ∠3
*(You can also write ∠EVD, ∠FVE, etc., but standard is vertex in middle.)*

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10)

Points around V: F, G, H, J (rays to them). Angles labeled 1, 2, 3, 4.

Angles at V:
- ∠FVG (angle 1)
- ∠GVH (angle 2)
- ∠HVJ (angle 3)
- ∠FVJ (angle 4 — big angle from F to J)
- Also: ∠FVH (combines 1+2), ∠GVJ (combines 2+3)

But typically, worksheets expect only the *individual* and *obvious combined* angles labeled or easily seen.

Since 4 angles are labeled, we list those + any obvious combinations.

> Answer: ∠FVG, ∠GVH, ∠HVJ, ∠FVJ, ∠1, ∠2, ∠3, ∠4
*(Also acceptable: ∠FVH, ∠GVJ if asked for all possible — but based on labeling, stick to labeled ones.)*

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11)

Points around V: E, F, G. Angles labeled 1, 2, 3.

Angles at V:
- ∠EVF (angle 1)
- ∠FVG (angle 2)
- ∠EVG (angle 3 — full angle from E to G)

> Answer: ∠EVF, ∠FVG, ∠EVG, ∠1, ∠2, ∠3

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12)

Points around V: J, K, L, M. Angles labeled 1, 2, 3, 4.

Angles at V:
- ∠JVK (angle 1)
- ∠KVL (angle 2)
- ∠LVM (angle 3)
- ∠JVM (angle 4 — full angle from J to M)
- Also: ∠JVL (1+2), ∠KVM (2+3)

But again, based on labeling and typical expectations:

> Answer: ∠JVK, ∠KVL, ∠LVM, ∠JVM, ∠1, ∠2, ∠3, ∠4

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## ✔ Final Answers Summary:

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1) Vertex: O; Sides: OF, OR

2) Vertex: O; Sides: OE, OF

3) Vertex: K; Sides: KJ, KL

4) Vertex: H; Sides: HO, HJ

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5) ∠QRS, ∠SRQ, ∠R, ∠1

6) ∠LMN, ∠NML, ∠M, ∠2

7) ∠MNO, ∠ONM, ∠N, ∠3

8) ∠KLM, ∠MLK, ∠L, ∠4

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9) ∠DVE, ∠EVF, ∠DVF, ∠1, ∠2, ∠3

10) ∠FVG, ∠GVH, ∠HVJ, ∠FVJ, ∠1, ∠2, ∠3, ∠4

11) ∠EVF, ∠FVG, ∠EVG, ∠1, ∠2, ∠3

12) ∠JVK, ∠KVL, ∠LVM, ∠JVM, ∠1, ∠2, ∠3, ∠4

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✔ Tips for Students:
- Always put the vertex letter in the middle when naming with 3 letters.
- If there’s only one angle at the vertex, you can name it with just the vertex letter.
- Use numbers or variables if they’re clearly marked inside the angle.
- For multiple angles at one vertex, list each distinct angle formed by adjacent rays.

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