SOLUTION: 2 the angle addition postulate - Studypool - Free Printable

Let's solve each of these geometry problems step by step. These are all angle-related problems involving algebraic expressions, and we'll use angle addition or subtraction based on the diagrams (which are described in text). We’ll assume standard geometric configurations (e.g., angles on a straight line, adjacent angles, etc.).

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Problem 11

Given:
- $ m\angle HGF = 16x + 4 $
- $ m\angle EGF = 110^\circ $
- $ m\angle HGE = 3x + 11 $

From the diagram (points G, F, E, H), it appears that point G is the vertex, and rays go to H, F, and E. The angle $ \angle HGF $ is composed of $ \angle HGE $ and $ \angle EGF $, since E lies between H and F.

So:

$$
\angle HGF = \angle HGE + \angle EGF
$$

Substitute values:

$$
16x + 4 = (3x + 11) + 110
$$

Simplify:

$$
16x + 4 = 3x + 121
$$

Subtract $3x$ from both sides:

$$
13x + 4 = 121
$$

Subtract 4:

$$
13x = 117
$$

Divide:

$$
x = 9
$$

✔ Answer: $ x = 9 $

---

Problem 12

Given:
- $ m\angle VUT = 175^\circ $
- $ m\angle VUJ = 17x - 3 $
- $ m\angle JUT = 17x + 8 $

From the diagram, point U is the vertex, and ray UJ lies between UV and UT, so:

$$
\angle VUT = \angle VUJ + \angle JUT
$$

$$
175 = (17x - 3) + (17x + 8)
$$

Simplify:

$$
175 = 34x + 5
$$

Subtract 5:

$$
170 = 34x
$$

Divide:

$$
x = 5
$$

✔ Answer: $ x = 5 $

---

Problem 13

Given:
- $ m\angle FCD = x + 41 $
- $ m\angle BCF = x + 78 $
- $ m\angle BCD = 95^\circ $

From the diagram, point C is the vertex, and ray CF lies between CB and CD, so:

$$
\angle BCD = \angle BCF + \angle FCD
$$

$$
95 = (x + 78) + (x + 41)
$$

$$
95 = 2x + 119
$$

Subtract 119:

$$
-24 = 2x
$$

$$
x = -12
$$

Wait — negative angle? Let’s double-check.

But $ \angle BCF = x + 78 = -12 + 78 = 66^\circ $,
$ \angle FCD = x + 41 = -12 + 41 = 29^\circ $,
Sum: $ 66 + 29 = 95^\circ $. So mathematically correct.

But angles can't be negative — but here, x is just a variable, and the measures are positive. So even if $ x = -12 $, the actual angles are valid.

✔ Answer: $ x = -12 $

(Note: This is acceptable as long as resulting angle measures are positive.)

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Problem 14

Given:
- $ m\angle BJK = 146 + 2x $
- $ m\angle IJK = 172^\circ $
- $ m\angle IJB = 2x + 26 $

From the diagram, point J is the vertex. Rays go to B, I, and K. Since $ \angle IJK $ is larger than $ \angle BJK $, likely ray BJ lies inside $ \angle IJK $, so:

$$
\angle IJK = \angle IJB + \angle BJK
$$

$$
172 = (2x + 26) + (146 + 2x)
$$

$$
172 = 4x + 172
$$

Subtract 172:

$$
0 = 4x \Rightarrow x = 0
$$

Check:
- $ \angle IJB = 2(0) + 26 = 26^\circ $
- $ \angle BJK = 146 + 0 = 146^\circ $
- Sum: $ 26 + 146 = 172^\circ $ ✔

✔ Answer: $ x = 0 $

---

Problem 15

Given:
- $ m\angle GFZ = 38^\circ $
- $ m\angle ZFE = 2x + 125 $
- $ m\angle GFE = x + 163 $

From the diagram, point F is the vertex. Ray FZ lies between FG and FE, so:

$$
\angle GFE = \angle GFZ + \angle ZFE
$$

$$
x + 163 = 38 + (2x + 125)
$$

$$
x + 163 = 2x + 163
$$

Subtract $ x $ and 163:

$$
0 = x
$$

So $ x = 0 $

Check:
- $ \angle ZFE = 2(0) + 125 = 125^\circ $
- $ \angle GFZ = 38^\circ $
- $ \angle GFE = 38 + 125 = 163^\circ $
- Given $ \angle GFE = x + 163 = 0 + 163 = 163^\circ $ ✔

✔ Answer: $ x = 0 $

---

Problem 16

Given:
- $ m\angle LMN = 135^\circ $
- $ m\angle LMV = -1 + 45x $
- $ m\angle VMN = 23x $

From the diagram, point M is the vertex. Ray MV lies between ML and MN, so:

$$
\angle LMN = \angle LMV + \angle VMN
$$

$$
135 = (-1 + 45x) + 23x
$$

$$
135 = 68x - 1
$$

Add 1:

$$
136 = 68x
$$

$$
x = 2
$$

Check:
- $ \angle LMV = -1 + 45(2) = -1 + 90 = 89^\circ $
- $ \angle VMN = 23(2) = 46^\circ $
- Sum: $ 89 + 46 = 135^\circ $ ✔

✔ Answer: $ x = 2 $

---

Problem 17

Given:
- $ m\angle WOY = 10x $
- $ m\angle XOY = 145^\circ $
- $ m\angle WOX = 2x + 13 $

From the diagram, ray OX lies between OW and OY, so:

$$
\angle WOY = \angle WOX + \angle XOY
$$

$$
10x = (2x + 13) + 145
$$

$$
10x = 2x + 158
$$

Subtract $2x$:

$$
8x = 158
$$

$$
x = 19.75
$$

Wait — let's check:

$ x = 158 / 8 = 19.75 $ → $ x = 79/4 $

But let’s verify:

- $ \angle WOY = 10x = 197.5^\circ $
- $ \angle WOX = 2(19.75) + 13 = 39.5 + 13 = 52.5^\circ $
- $ \angle XOY = 145^\circ $
- Sum: $ 52.5 + 145 = 197.5^\circ $ ✔

So it's correct.

✔ Answer: $ x = 19.75 $ or $ \frac{79}{4} $

---

Problem 18

Given:
- $ m\angle ABC = 17x + 4 $
- $ m\angle ABD = 42^\circ $
- $ m\angle DBC = 12x + 4 $
- Find $ m\angle ABC $

From the diagram, ray BD lies between BA and BC, so:

$$
\angle ABC = \angle ABD + \angle DBC
$$

$$
17x + 4 = 42 + (12x + 4)
$$

$$
17x + 4 = 12x + 46
$$

Subtract $12x$:

$$
5x + 4 = 46
$$

Subtract 4:

$$
5x = 42 \Rightarrow x = 8.4
$$

Now find $ m\angle ABC = 17x + 4 = 17(8.4) + 4 = 142.8 + 4 = 146.8^\circ $

Or as fraction:

$ x = 42/5 = 8.4 $

$ 17x = 17 × 42 / 5 = 714 / 5 = 142.8 $

+4 = 146.8°

✔ Answer: $ m\angle ABC = 146.8^\circ $ or $ \frac{734}{5}^\circ $

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Problem 19

Given:
- $ m\angle ZBG = 11x - 1 $
- $ m\angle BBG = 24^\circ $ ← Wait! "BBG"? That must be typo.

Likely: $ m\angle ABG = 24^\circ $? Or $ m\angle DBG $? But labeled as $ m\angle BBG $ — impossible.

Looking at the diagram: probably points Z, B, G, and maybe D or something.

Wait: Likely typo — should be $ m\angle DBG = 24^\circ $? But not clear.

Alternatively, perhaps it's $ m\angle ZBD = 24^\circ $? But the problem says:

> $ m\angle ZBG = 11x - 1 $, $ m\angle BBG = 24^\circ $, $ m\angle DBG = 12x + 13 $. Find $ m\angle DBG $

"BBG" is invalid — probably meant $ m\angle ZBD = 24^\circ $? Or $ m\angle DBZ = 24^\circ $? Let's assume:

Maybe the angles are around point B, with rays going to Z, D, G.

Suppose:
- $ \angle ZBG = 11x - 1 $
- $ \angle DBZ = 24^\circ $
- $ \angle DBG = 12x + 13 $

And ray BD lies between BZ and BG?

Then:

$$
\angle ZBG = \angle ZBD + \angle DBG
$$

$$
11x - 1 = 24 + (12x + 13)
$$

$$
11x - 1 = 12x + 37
$$

Subtract $11x$:

$$
-1 = x + 37 \Rightarrow x = -38
$$

Then $ \angle DBG = 12(-38) + 13 = -456 + 13 = -443^\circ $ → Invalid!

So assumption wrong.

Alternative: Maybe $ \angle DBG = 12x + 13 $, and $ \angle ZBG = 11x - 1 $, and $ \angle ZBD = 24^\circ $, but now suppose ray ZB is outside.

Wait — perhaps $ \angle ZBG $ includes $ \angle ZBD $ and $ \angle DBG $, so:

$$
\angle ZBG = \angle ZBD + \angle DBG
$$

But then:

$$
11x - 1 = 24 + (12x + 13) = 12x + 37
$$

Same as before → $ x = -38 $ → invalid.

Wait — maybe $ \angle ZBG $ is not the sum?

Perhaps ray BD is between Z and G, so:

$$
\angle ZBG = \angle ZBD + \angle DBG
$$

But again, same equation.

Unless the labeling is different.

Wait — perhaps “$ m\angle BBG $” is a typo for $ m\angle DBG = 24^\circ $? But then $ m\angle DBG $ is given twice?

No — problem says:
> $ m\angle ZBG = 11x - 1 $, $ m\angle BBG = 24^\circ $, $ m\angle DBG = 12x + 13 $. Find $ m\angle DBG $

“BBG” is nonsense — likely typo.

Possibility: It's $ m\angle ZBD = 24^\circ $, and $ m\angle DBG = 12x + 13 $, and $ m\angle ZBG = 11x - 1 $

Then:

$$
\angle ZBG = \angle ZBD + \angle DBG
\Rightarrow 11x - 1 = 24 + (12x + 13)
\Rightarrow 11x - 1 = 12x + 37
\Rightarrow -1 - 37 = x \Rightarrow x = -38
$$

Still invalid.

Wait — maybe the order is reversed.

Suppose ray BG is between BZ and BD? Then:

$$
\angle ZBD = \angle ZBG + \angle GB D
\Rightarrow 24 = (11x - 1) + (12x + 13)
\Rightarrow 24 = 23x + 12
\Rightarrow 12 = 23x \Rightarrow x = 12/23
$$

Then $ \angle DBG = 12x + 13 = 12(12/23) + 13 = 144/23 + 299/23 = 443/23 ≈ 19.26^\circ $

But is this consistent?

Let’s try:

- $ x = 12/23 $
- $ \angle ZBG = 11x - 1 = 132/23 - 23/23 = 109/23 ≈ 4.74^\circ $
- $ \angle DBG = 12x + 13 = 144/23 + 299/23 = 443/23 ≈ 19.26^\circ $
- Sum: $ 4.74 + 19.26 = 24^\circ $ ✔

So possible.

But the problem says $ m\angle BBG = 24^\circ $ — which doesn’t make sense.

But if we interpret $ m\angle BBG $ as a typo for $ m\angle ZBD = 24^\circ $, then yes.

So assuming:

- $ \angle ZBD = 24^\circ $
- $ \angle ZBG = 11x - 1 $
- $ \angle DBG = 12x + 13 $
- And $ \angle ZBD = \angle ZBG + \angle DBG $

Then:

$$
24 = (11x - 1) + (12x + 13) = 23x + 12
\Rightarrow 23x = 12 \Rightarrow x = \frac{12}{23}
$$

Then $ m\angle DBG = 12x + 13 = 12(\frac{12}{23}) + 13 = \frac{144}{23} + \frac{299}{23} = \frac{443}{23} \approx 19.26^\circ $

But the question asks to find $ m\angle DBG $, so:

$$
m\angle DBG = 12x + 13 = 12 \cdot \frac{12}{23} + 13 = \frac{144}{23} + \frac{299}{23} = \frac{443}{23}^\circ
$$

✔ Answer: $ \frac{443}{23}^\circ $ or approximately $ 19.26^\circ $

But due to ambiguity in label "BBG", this is speculative.

Alternatively, if "BBG" is meant to be "DBZ" or "ZBD", then this works.

We'll go with this interpretation.

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Problem 20

Given:
- $ m\angle GPE = 4x + 10 $
- $ m\angle NPE = 14x + 3 $
- $ m\angle GPN = 157^\circ $
- Find $ m\angle NPE $

From the diagram, point P is vertex, rays go to G, N, E.

Assuming ray PN lies between PG and PE, then:

$$
\angle GPE = \angle GPN + \angle NPE
$$

$$
4x + 10 = 157 + (14x + 3)
$$

$$
4x + 10 = 14x + 160
$$

Subtract $4x$:

$$
10 = 10x + 160
\Rightarrow -150 = 10x \Rightarrow x = -15
$$

Then $ m\angle NPE = 14(-15) + 3 = -210 + 3 = -207^\circ $ → Invalid.

So assumption wrong.

Try reverse: ray PG lies between PN and PE?

Then:

$$
\angle NPE = \angle NPG + \angle GPE
$$

But $ \angle NPG = \angle GPN = 157^\circ $, so:

$$
14x + 3 = 157 + (4x + 10)
\Rightarrow 14x + 3 = 4x + 167
\Rightarrow 10x = 164 \Rightarrow x = 16.4
$$

Then $ m\angle NPE = 14(16.4) + 3 = 229.6 + 3 = 232.6^\circ $

But $ \angle GPE = 4(16.4) + 10 = 65.6 + 10 = 75.6^\circ $

Then $ \angle NPE = \angle NPG + \angle GPE = 157 + 75.6 = 232.6^\circ $ ✔

So it works.

Thus, $ m\angle NPE = 14x + 3 = 14(16.4) + 3 = 232.6^\circ $

Or as fraction:

$ x = 16.4 = 82/5 $

$ 14x = 14 × 82 / 5 = 1148 / 5 = 229.6 $

+3 = 232.6°

✔ Answer: $ m\angle NPE = 232.6^\circ $ or $ \frac{1163}{5}^\circ $

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✔ Final Answers Summary:

| Problem | Answer |
|--------|--------|
| 11 | $ x = 9 $ |
| 12 | $ x = 5 $ |
| 13 | $ x = -12 $ |
| 14 | $ x = 0 $ |
| 15 | $ x = 0 $ |
| 16 | $ x = 2 $ |
| 17 | $ x = 19.75 $ |
| 18 | $ m\angle ABC = 146.8^\circ $ |
| 19 | $ m\angle DBG = \frac{443}{23}^\circ \approx 19.26^\circ $ |
| 20 | $ m\angle NPE = 232.6^\circ $ |

Let me know if you'd like explanations for any specific one or clarification on the ambiguous ones!