Find the missing angles in various polygons using the sum of interior angles formula.

Let's solve each problem step by step in the "Finding Missing Angles in a Polygon Worksheet".

We'll use key geometry principles:
- The sum of interior angles of an *n*-gon is:
$$
(n - 2) \times 180^\circ
$$
- In parallelograms, opposite angles are equal and adjacent angles are supplementary (add to 180°).
- In rectangles, all angles are 90°.
- In isosceles triangles, two sides are equal → two angles are equal.
- In quadrilaterals, sum of angles = 360°.
- In triangles, sum of angles = 180°.

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Problem 1

Quadrilateral with angles: 135°, 60°, 50°, and x°

Sum of interior angles of a quadrilateral = 360°

$$
x = 360 - (135 + 60 + 50) = 360 - 245 = 115^\circ
$$

✔ Answer: x = 115°

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Problem 2

Quadrilateral with two right angles (90°), one angle 50°, find x

Sum = 360°

$$
x = 360 - (90 + 90 + 50) = 360 - 230 = 130^\circ
$$

✔ Answer: x = 130°

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Problem 3

Triangle with two equal sides (marked with ticks), so it’s isosceles. One angle is 60°, find x.

The two equal sides mean the angles opposite them are equal.

But we don't know which angles are equal yet.

Given: one angle is 60°, and two sides are marked equal → the angles opposite those sides are equal.

So, suppose the 60° angle is not between the two equal sides.

Let’s analyze:

- If the triangle has two equal sides, then two angles are equal.
- We’re given one angle: 60°, and need to find x.
- But since the triangle has two tick marks on sides, the two angles opposite those sides are equal.

Let’s assume:
- The side with tick mark is opposite the unknown angle x.
- Another side with tick mark is opposite the 60° angle? Wait — actually, the diagram shows:
- Two sides have tick marks → so two sides are equal → two angles are equal.
- One angle is labeled 60°, and another is labeled x°.
- The third angle is not labeled.

Wait — but only one angle is labeled: 60°, and x° is at the top vertex.

From the diagram:
- The triangle has two equal sides: left and right sides (each with a tick mark).
- So base angles are equal.
- The bottom-left angle is 60°.
- Therefore, the bottom-right angle must also be 60° (since triangle is isosceles).
- Then, the top angle x = 180 - 60 - 60 = 60°.

So this is an equilateral triangle! All angles are 60°.

✔ Answer: x = 60°

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Problem 4

Heptagon (7 sides) with several angles given:
- 130°, 122°, 130°, 130°, 138°, 128°, and x°

Sum of interior angles of a heptagon:
$$
(7 - 2) \times 180 = 5 \times 180 = 900^\circ
$$

Add known angles:
$$
130 + 122 + 130 + 130 + 138 + 128 = ?
$$
Step-by-step:
- 130 + 122 = 252
- 252 + 130 = 382
- 382 + 130 = 512
- 512 + 138 = 650
- 650 + 128 = 778

So:
$$
x = 900 - 778 = 122^\circ
$$

✔ Answer: x = 122°

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Problem 5

Quadrilateral with angles: 120°, 107°, 90° (right angle), and x°

Sum = 360°

$$
x = 360 - (120 + 107 + 90) = 360 - 317 = 43^\circ
$$

✔ Answer: x = 43°

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Problem 6

Quadrilateral with angles: 50°, 140°, 70°, and x°

Sum = 360°

$$
x = 360 - (50 + 140 + 70) = 360 - 260 = 100^\circ
$$

✔ Answer: x = 100°

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Problem 7

Parallelogram — opposite angles equal, adjacent angles supplementary.

Given:
- One angle is 60°
- Opposite angle is also 60°
- Adjacent angles add to 180° → so other two angles are 120° each

We're asked for x°, which is at the top-left corner.

Top-left is adjacent to 60°, so:
$$
x = 180 - 60 = 120^\circ
$$

Alternatively, opposite to 120° (bottom-right), so x = 120°

✔ Answer: x = 120°

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Problem 8

Rhombus or kite-like shape, but looks like a kite or rhombus with two pairs of equal angles.

Angles given:
- Top: 140°
- Left: 40°
- Bottom: 140°
- Right: x°

This appears symmetric.

Sum of angles in a quadrilateral = 360°

$$
x = 360 - (140 + 40 + 140) = 360 - 320 = 40^\circ
$$

Also, notice that 140° and 140° are opposite, and 40° and x° are opposite → so x should be 40°.

✔ Answer: x = 40°

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✔ Final Answers:

| Problem | Answer |
|--------|--------|
| 1 | 115° |
| 2 | 130° |
| 3 | 60° |
| 4 | 122° |
| 5 | 43° |
| 6 | 100° |
| 7 | 120° |
| 8 | 40° |

Let me know if you'd like these explained further or want a printed version!