Worksheet on finding interior and exterior angles of polygons with four geometric figures.

Let's solve each of these problems step by step. The task is to find the value of \( x \) in each polygon, using the sum of interior angles of polygons.

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🔷 Key Concept: Sum of Interior Angles

For any n-sided polygon, the sum of the interior angles is:

\[
\text{Sum} = (n - 2) \times 180^\circ
\]

We'll use this formula for each shape.

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## 🔹 Problem 1: Quadrilateral

Given angles:
- 104°
- 130°
- 78°
- \( x \)

This is a quadrilateral → \( n = 4 \)

\[
\text{Sum} = (4 - 2) \times 180 = 2 \times 180 = 360^\circ
\]

Now add known angles:

\[
104 + 130 + 78 = 312^\circ
\]

So,

\[
x = 360 - 312 = 48^\circ
\]

✔ Answer: \( x = 48^\circ \)

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## 🔹 Problem 2: Pentagon

Given angles:
- 120°
- 135°
- 100°
- 99°
- \( x \)

This is a pentagon → \( n = 5 \)

\[
\text{Sum} = (5 - 2) \times 180 = 3 \times 180 = 540^\circ
\]

Add known angles:

\[
120 + 135 + 100 + 99 = 454^\circ
\]

So,

\[
x = 540 - 454 = 86^\circ
\]

✔ Answer: \( x = 86^\circ \)

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## 🔹 Problem 3: Pentagon

Given angles:
- 112°
- 112°
- 95°
- \( x \)
- \( x \)

Wait — two angles are labeled as \( x \), and others are given.

So, angles:
- 112°
- 112°
- 95°
- \( x \)
- \( x \)

Again, pentagon → sum = 540°

Add known values:

\[
112 + 112 + 95 = 319^\circ
\]

Remaining for two \( x \)'s:

\[
2x = 540 - 319 = 221^\circ
\]

\[
x = \frac{221}{2} = 110.5^\circ
\]

✔ Answer: \( x = 110.5^\circ \)

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## 🔹 Problem 4: Quadrilateral with a Right Angle

Given angles:
- 90° (right angle, marked with square)
- 120°
- 130°
- \( x \)

Quadrilateral → sum = 360°

Add known angles:

\[
90 + 120 + 130 = 340^\circ
\]

So,

\[
x = 360 - 340 = 20^\circ
\]

✔ Answer: \( x = 20^\circ \)

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✔ Final Answers:

1. \( x = 48^\circ \)
2. \( x = 86^\circ \)
3. \( x = 110.5^\circ \)
4. \( x = 20^\circ \)

Let me know if you'd like a visual explanation or help with drawing the shapes!