Angle Addition Postulate Color by Number - Free Printable

Let's solve each problem step by step using the Angle Addition Postulate, which states:

> If point $ I $ lies in the interior of $ \angle FGH $, then
> $$
> m\angle FGH = m\angle FGI + m\angle IGH
> $$

We’ll apply this to each question.

---

1.

Given:
- $ m\angle FGI = 22^\circ $
- $ m\angle IGH = 110^\circ $

Find: $ m\angle FGH $

Using the Angle Addition Postulate:
$$
m\angle FGH = m\angle FGI + m\angle IGH = 22^\circ + 110^\circ = 132^\circ
$$

✔ Answer: $ \boxed{132} $ (maroon)

---

2.

Given:
- $ m\angle LKJ = 160^\circ $
- $ m\angle LKG = 45^\circ $

Find: $ m\angle GKJ $

Note: $ \angle LKJ $ is made up of $ \angle LKG $ and $ \angle GKJ $. So,
$$
m\angle LKJ = m\angle LKG + m\angle GKJ
$$
$$
160^\circ = 45^\circ + m\angle GKJ
$$
$$
m\angle GKJ = 160^\circ - 45^\circ = 115^\circ
$$

✔ Answer: $ \boxed{115} $ (sky blue)

---

3.

Given:
- $ m\angle IJE = x + 40 $
- $ m\angle EJK = x + 100 $
- $ m\angle IJK = 120^\circ $

Note: $ \angle IJK $ is composed of $ \angle IJE $ and $ \angle EJK $, so:
$$
m\angle IJK = m\angle IJE + m\angle EJK
$$
$$
120 = (x + 40) + (x + 100)
$$
$$
120 = 2x + 140
$$
$$
2x = 120 - 140 = -20
$$
$$
x = -10
$$

Wait — a negative angle measure? That seems odd, but let’s check.

But angles can't be negative. Let’s double-check the diagram.

Looking at the figure: Point $ J $ has rays going to $ I $, $ E $, and $ K $. The angle from $ I $ to $ K $ via $ E $ should add up.

But if $ m\angle IJE = x+40 $, $ m\angle EJK = x+100 $, and total $ m\angle IJK = 120^\circ $, then:
$$
(x+40) + (x+100) = 120 \\
2x + 140 = 120 \\
2x = -20 \Rightarrow x = -10
$$

This implies $ m\angle IJE = -10 + 40 = 30^\circ $, and $ m\angle EJK = -10 + 100 = 90^\circ $, sum = 120° — mathematically consistent.

But geometrically, it’s unusual for an expression like $ x+100 $ to represent an angle when $ x $ is negative. However, since the algebra checks out, we accept it.

✔ Answer: $ \boxed{-10} $ (black)

⚠️ But wait — maybe the diagram shows that $ \angle IJK $ is split into $ \angle IJE $ and $ \angle EJK $, but perhaps the order is different?

Alternatively, could $ \angle IJE $ and $ \angle EJK $ be adjacent forming $ \angle IJK $? Yes — and they do.

So even though $ x = -10 $, it satisfies the equation.

But let’s see if there's a misinterpretation.

Wait — maybe $ \angle IJE $ and $ \angle EJK $ are not both parts of $ \angle IJK $? But the diagram shows ray $ JE $ between $ JI $ and $ JK $, so yes, it *is* additive.

So unless there’s a typo, $ x = -10 $ is correct.

But let’s keep going — maybe others will make more sense.

---

4.

Given:
- $ m\angle KGH = x + 100 $
- $ m\angle FGH = 115^\circ $
- $ m\angle FGK = x + 29 $

Note: From the diagram, point $ K $ lies between $ F $ and $ H $ on the same side of $ G $, so:
$$
m\angle FGH = m\angle FGK + m\angle KGH
$$
$$
115 = (x + 29) + (x + 100)
$$
$$
115 = 2x + 129
$$
$$
2x = 115 - 129 = -14
$$
$$
x = -7
$$

Again, negative value. But let’s check:
- $ m\angle FGK = -7 + 29 = 22^\circ $
- $ m\angle KGH = -7 + 100 = 93^\circ $
- Sum: $ 22 + 93 = 115^\circ $ → Correct.

So again, algebraically valid.

✔ Answer: $ \boxed{-7} $ (yellow)

---

5.

Given:
- $ m\angle MLR = 88^\circ $
- $ m\angle RLK = 3x + 6 $
- $ m\angle MLK = 11x + 14 $

From the diagram: $ \angle MLK $ is made up of $ \angle MLR $ and $ \angle RLK $

So:
$$
m\angle MLK = m\angle MLR + m\angle RLK
$$
$$
11x + 14 = 88 + (3x + 6)
$$
$$
11x + 14 = 3x + 94
$$
$$
11x - 3x = 94 - 14
$$
$$
8x = 80 \Rightarrow x = 10
$$

✔ Answer: $ \boxed{10} $ (purple)

---

6.

Given:
- $ m\angle EFG = 113^\circ $
- $ m\angle AFG = 8x - 3 $
- $ m\angle EFA = 8x + 4 $

Note: From the diagram, ray $ FA $ is between $ FE $ and $ FG $, so:
$$
m\angle EFG = m\angle EFA + m\angle AFG
$$
$$
113 = (8x + 4) + (8x - 3)
$$
$$
113 = 16x + 1
$$
$$
16x = 112 \Rightarrow x = 7
$$

Check:
- $ m\angle EFA = 8(7)+4 = 56+4 = 60^\circ $
- $ m\angle AFG = 8(7)-3 = 56-3 = 53^\circ $
- Sum: $ 60 + 53 = 113^\circ $ → Correct.

✔ Answer: $ \boxed{7} $ (maroon)

---

Final Answers Summary:

| Problem | Answer | Color |
|--------|--------|-------|
| 1 | $ 132 $ | maroon |
| 2 | $ 115 $ | sky blue |
| 3 | $ -10 $ | black |
| 4 | $ -7 $ | yellow |
| 5 | $ 10 $ | purple |
| 6 | $ 7 $ | maroon |

---

Notes:

- Problems 3 and 4 give negative values for $ x $, but the angle measures are still positive and satisfy the equations.
- This suggests either:
- The expressions were set up with variables that allow negative values (perhaps for teaching purposes),
- Or there might be a typo in the problem (e.g., wrong signs), but based on given info, the math is correct.

---

✔ Final Answers:

1. $ \boxed{132} $
2. $ \boxed{115} $
3. $ \boxed{-10} $
4. $ \boxed{-7} $
5. $ \boxed{10} $
6. $ \boxed{7} $

Now you can color the picture accordingly!