This geometry handout explains the Angle Addition Postulate and provides practice problems for finding missing angle measures using algebra.

It looks like you've uploaded a worksheet titled "ANGLE ADDITION POSTULATE" from a Geometry class. The worksheet includes definitions, diagrams, and several problems related to the Angle Addition Postulate, which states:

> If point $ D $ lies in the interior of $ \angle ABC $, then
> $$
> m\angle ABD + m\angle DBC = m\angle ABC
> $$

This postulate allows us to break down an angle into smaller adjacent angles whose measures add up to the total measure of the larger angle.

Let’s go through each section and solve the problems step by step.

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Part 1: Definitions & Concepts

#### How to Name an Angle
- An angle is named using three points: the vertex in the middle.
- Example: $ \angle ABC $, where $ B $ is the vertex.

#### Congruent Angles
- Angles with equal measures.
- Marked with arcs (same number of arcs for congruent angles).

#### Types of Angles
| Type | Description |
|------------|------------------------------|
| Acute | Less than 90° |
| Right | Exactly 90° |
| Obtuse | Greater than 90° but less than 180° |
| Straight | Exactly 180° |

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Problem 1:

> If $ m\angle ABC = 65^\circ $ and $ m\angle ABD = 16^\circ $, find $ m\angle DBC $.

We use the Angle Addition Postulate:
$$
m\angle ABD + m\angle DBC = m\angle ABC
$$
Substitute known values:
$$
16^\circ + m\angle DBC = 65^\circ
$$
Solve:
$$
m\angle DBC = 65^\circ - 16^\circ = 49^\circ
$$

✔ Answer: $ m\angle DBC = 49^\circ $

---

Problem 2:

> If $ m\angle ABC = 120^\circ $, find the value of $ x $.

From the diagram:
- $ m\angle ABD = 3x $
- $ m\angle DBC = 2x $
- Together: $ m\angle ABC = 3x + 2x = 5x $

Set equal to given measure:
$$
5x = 120^\circ
$$
$$
x = \frac{120}{5} = 24^\circ
$$

✔ Answer: $ x = 24^\circ $

---

Problem 3:

> $ m\angle COW = 27^\circ $, $ m\angle WOP = 15^\circ $, and $ m\angle COE = (2x + 19)^\circ $. Find the value of $ x $.

From the diagram:
- $ \angle COE = \angle COW + \angle WOP + \angle POE $? Wait — need to interpret carefully.

But looking at the labels:
- Points are: $ C $, $ O $, $ W $, $ P $, $ E $
- $ \angle COW = 27^\circ $
- $ \angle WOP = 15^\circ $
- So $ \angle COE = \angle COW + \angle WOP + \angle POE $? But no info about $ \angle POE $.

Wait — actually, it says $ m\angle COE = (2x + 19)^\circ $, and we’re told $ m\angle COW = 27^\circ $, $ m\angle WOP = 15^\circ $. But unless $ P $ is on $ OE $, this may not be additive directly.

Looking at the diagram (as described): likely, $ \angle COE $ is composed of $ \angle COW $ and $ \angle WOE $, but $ WOP $ is part of that?

Wait — perhaps $ \angle COE = \angle COW + \angle WOP $? But $ P $ might be between $ W $ and $ E $? Let's assume the ray order is $ OC $, $ OW $, $ OP $, $ OE $, so:

Then:
$$
m\angle COE = m\angle COW + m\angle WOP + m\angle POE
$$

But we don’t know $ m\angle POE $. Hmm.

Wait — maybe $ \angle COE $ is just $ \angle COW + \angle WOP $? That would only be true if $ P $ is on $ OE $ and $ W $ is between $ C $ and $ P $, etc.

Alternatively, perhaps $ \angle COE $ is made up of $ \angle COW $ and $ \angle WOE $, but we're told $ m\angle WOP = 15^\circ $, which suggests $ P $ is on $ OE $, and $ \angle WOP = 15^\circ $, so $ \angle WOE = 15^\circ $? Or is $ \angle WOP $ part of $ \angle WOE $?

Wait — perhaps the figure shows:
- Ray $ OC $
- Ray $ OW $
- Ray $ OP $
- Ray $ OE $

With $ \angle COW = 27^\circ $, $ \angle WOP = 15^\circ $, and $ \angle COE = (2x + 19)^\circ $

So if $ \angle COE = \angle COW + \angle WOP + \angle POE $? But again, no info about $ \angle POE $.

Wait — unless $ \angle COE = \angle COW + \angle WOP $, meaning $ P $ is on $ OE $, and $ W $ is between $ C $ and $ P $, and $ P $ is between $ W $ and $ E $? Then $ \angle COE = \angle COW + \angle WOP + \angle POE $? Still missing info.

Wait — maybe there's a typo or misreading.

Actually, let's re-read:
> "If $ m\angle COW = 27^\circ $, $ m\angle WOP = 15^\circ $, and $ m\angle COE = (2x + 19)^\circ $, find the value of $ x $."

And the diagram shows rays from point $ O $: $ OC $, $ OW $, $ OP $, $ OE $, in that order.

So:
$$
\angle COE = \angle COW + \angle WOP + \angle POE
$$

But we don't have $ \angle POE $. Unless... wait — maybe $ \angle COE $ is just $ \angle COW + \angle WOP $? But that would mean $ P $ is on $ OE $, and $ \angle POE = 0 $? No.

Wait — perhaps $ \angle COE = \angle COW + \angle WOE $, and $ \angle WOE = \angle WOP $? But unless $ P $ is on $ OE $, we can't assume that.

Wait — maybe $ \angle COE = \angle COW + \angle WOP $, and $ \angle WOP $ is part of $ \angle COE $, and $ P $ is between $ W $ and $ E $, so $ \angle COE = \angle COW + \angle WOP $, and no more?

But $ \angle COE = (2x + 19)^\circ $, and $ \angle COW = 27^\circ $, $ \angle WOP = 15^\circ $, so:

$$
\angle COE = 27^\circ + 15^\circ = 42^\circ
$$

So:
$$
2x + 19 = 42
$$
$$
2x = 23
$$
$$
x = 11.5
$$

But that seems odd — usually these problems use integers.

Wait — maybe I misread the diagram.

Let me double-check: Is $ \angle COE $ composed of $ \angle COW $ and $ \angle WOP $? Only if $ P $ is on $ OE $, and $ W $ is between $ C $ and $ P $, and $ P $ is between $ W $ and $ E $? Then yes, $ \angle COE = \angle COW + \angle WOP + \angle POE $? But still missing $ \angle POE $.

Wait — perhaps the diagram shows that $ \angle COE = \angle COW + \angle WOP $, meaning $ P $ is on $ OE $, and $ \angle WOP $ is the rest? But $ \angle COE $ should include everything from $ C $ to $ E $.

Alternatively, maybe $ \angle COE $ is split into $ \angle COW $ and $ \angle WOE $, and $ \angle WOE = \angle WOP $? But that doesn’t make sense unless $ P = E $.

Wait — perhaps $ \angle COE = \angle COW + \angle WOP $, and $ P $ is on $ OE $, so $ \angle WOP $ is part of $ \angle WOE $, but we don't know $ \angle POE $.

Hmm — unless the problem assumes $ \angle COE = \angle COW + \angle WOP $, even though $ P $ is not necessarily $ E $, but that would be incorrect.

Wait — look again at the problem statement:

> $ m\angle COW = 27^\circ $, $ m\angle WOP = 15^\circ $, and $ m\angle COE = (2x + 19)^\circ $

And the diagram likely shows that $ \angle COE = \angle COW + \angle WOP + \angle POE $? But we don’t have $ \angle POE $.

Unless $ P $ is $ E $? Then $ \angle WOP = \angle WOE $, and $ \angle COE = \angle COW + \angle WOP = 27 + 15 = 42^\circ $

Then:
$$
2x + 19 = 42
\Rightarrow 2x = 23 \Rightarrow x = 11.5
$$

But $ x = 11.5 $ is acceptable unless specified otherwise.

But let’s check if there's another interpretation.

Wait — maybe $ \angle COE $ is not the sum of those two, but rather $ \angle COE = \angle COW + \angle WOE $, and $ \angle WOE = \angle WOP $? But again, unless $ P = E $, no.

Alternatively, perhaps $ \angle COE $ is made up of $ \angle COW $ and $ \angle WOE $, and $ \angle WOE = \angle WOP $, so $ \angle COE = 27 + 15 = 42^\circ $, same as before.

So:
$$
2x + 19 = 42 \Rightarrow x = 11.5
$$

✔ Answer: $ x = 11.5 $

But let's see if that makes sense.

Wait — maybe the diagram shows that $ \angle COE = \angle COW + \angle WOP $, and $ \angle WOP $ is given, so total is 42°, and set equal to $ 2x + 19 $, so $ x = 11.5 $

Yes, that must be it.

So:
$$
2x + 19 = 27 + 15 = 42 \\
2x = 23 \\
x = 11.5
$$

✔ Answer: $ x = 11.5 $

---

Problem 4:

> $ m\angle A = 45^\circ $, $ m\angle C = 23^\circ $, find the value of $ x $, if $ m\angle B = (x + 2)^\circ $

Wait — this is a triangle?

The diagram shows triangle $ ABC $, with angles at $ A $, $ B $, $ C $.

Sum of angles in a triangle is $ 180^\circ $.

So:
$$
m\angle A + m\angle B + m\angle C = 180^\circ
$$
$$
45^\circ + (x + 2)^\circ + 23^\circ = 180^\circ
$$
$$
(45 + 23 + x + 2) = 180
\Rightarrow 70 + x = 180
\Rightarrow x = 110
$$

Wait — $ x = 110 $? Then $ m\angle B = 112^\circ $? Let's check:
$$
45 + 112 + 23 = 180? \quad 45+23=68, 68+112=180 ✔
$$

But $ m\angle B = x + 2 = 110 + 2 = 112^\circ $, correct.

So $ x = 110 $

✔ Answer: $ x = 110 $

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Summary of Answers

1. $ m\angle DBC = 49^\circ $
2. $ x = 24^\circ $
3. $ x = 11.5 $
4. $ x = 110 $

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Final Answers:

| Problem | Answer |
|--------|--------|
| 1 | $ m\angle DBC = 49^\circ $ |
| 2 | $ x = 24 $ |
| 3 | $ x = 11.5 $ |
| 4 | $ x = 110 $ |

Let me know if you'd like a visual explanation or clarification on any problem!