Comprehensive geometry worksheet focusing on the interior angles of polygons, featuring eight distinct sections ranging from basic angle calculations to complex proofs and tessellation problems.

Problem Analysis and Solution

The image contains a series of tasks related to the interior angles of polygons. Below, I will solve each task step by step.

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#### 1. Factual Recall: Find the missing angles
Task: Find the missing angles in the given diagrams.

##### a)
- The triangle has one right angle (90°) and another angle of 65°.
- The sum of the interior angles of a triangle is always 180°.
- Let the missing angle be \( s \).
- Using the angle sum property:
\[
65^\circ + 90^\circ + s = 180^\circ
\]
\[
155^\circ + s = 180^\circ
\]
\[
s = 25^\circ
\]

##### b)
- The triangle has angles of 31°, 51°, and a missing angle \( t \).
- Using the angle sum property:
\[
31^\circ + 51^\circ + t = 180^\circ
\]
\[
82^\circ + t = 180^\circ
\]
\[
t = 98^\circ
\]

##### c)
- The polygon is a quadrilateral with angles 125°, 85°, and 204°.
- The sum of the interior angles of a quadrilateral is:
\[
(n-2) \times 180^\circ = (4-2) \times 180^\circ = 360^\circ
\]
- Let the missing angle be \( u \).
- Using the angle sum property:
\[
125^\circ + 85^\circ + 204^\circ + u = 360^\circ
\]
\[
414^\circ + u = 360^\circ
\]
\[
u = -54^\circ
\]
This result is incorrect because an angle cannot be negative. There might be a mistake in the problem setup or interpretation. Assuming the problem is correct as stated, we would need clarification.

---

#### 2. Carry out a routine procedure: Find the missing angles
Task: Find the missing angles in the given diagrams.

##### a)
- The polygon is a pentagon with angles 135°, 110°, 130°, and 121°.
- The sum of the interior angles of a pentagon is:
\[
(n-2) \times 180^\circ = (5-2) \times 180^\circ = 540^\circ
\]
- Let the missing angle be \( p \).
- Using the angle sum property:
\[
135^\circ + 110^\circ + 130^\circ + 121^\circ + p = 540^\circ
\]
\[
596^\circ + p = 540^\circ
\]
\[
p = -56^\circ
\]
This result is incorrect because an angle cannot be negative. There might be a mistake in the problem setup or interpretation. Assuming the problem is correct as stated, we would need clarification.

##### b)
- The polygon is a quadrilateral with angles 130°, 7°, and 125°.
- The sum of the interior angles of a quadrilateral is:
\[
(n-2) \times 180^\circ = (4-2) \times 180^\circ = 360^\circ
\]
- Let the missing angle be \( q \).
- Using the angle sum property:
\[
130^\circ + 7^\circ + 125^\circ + q = 360^\circ
\]
\[
262^\circ + q = 360^\circ
\]
\[
q = 98^\circ
\]

##### c)
- The polygon is a hexagon with angles 147°, 116°, 151°, 140°, and 124°.
- The sum of the interior angles of a hexagon is:
\[
(n-2) \times 180^\circ = (6-2) \times 180^\circ = 720^\circ
\]
- Let the missing angle be \( r \).
- Using the angle sum property:
\[
147^\circ + 116^\circ + 151^\circ + 140^\circ + 124^\circ + r = 720^\circ
\]
\[
678^\circ + r = 720^\circ
\]
\[
r = 42^\circ
\]

---

#### 3. Classify some mathematical object: Which question is the odd one out? Why?
Task: Identify the odd one out among the given polygons.

##### a)
- The polygon has angles 121°, 62°, and \( x \).
- The sum of the interior angles of a triangle is 180°.
- Using the angle sum property:
\[
121^\circ + 62^\circ + x = 180^\circ
\]
\[
183^\circ + x = 180^\circ
\]
\[
x = -3^\circ
\]
This result is incorrect because an angle cannot be negative. There might be a mistake in the problem setup or interpretation.

##### b)
- The polygon has angles 121°, 37°, and \( x \).
- The sum of the interior angles of a triangle is 180°.
- Using the angle sum property:
\[
121^\circ + 37^\circ + x = 180^\circ
\]
\[
158^\circ + x = 180^\circ
\]
\[
x = 22^\circ
\]

##### c)
- The polygon has angles 60°, 60°, and \( x \).
- The sum of the interior angles of a triangle is 180°.
- Using the angle sum property:
\[
60^\circ + 60^\circ + x = 180^\circ
\]
\[
120^\circ + x = 180^\circ
\]
\[
x = 60^\circ
\]

Odd One Out: The first polygon (a) is the odd one out because it results in a negative angle, which is not possible.

---

#### 4. Interpret a situation or answer: Tiling a bathroom
Task: Determine if Jimmy's choice of a regular pentagon tile is correct for tiling his bathroom without gaps.

##### Solution:
- A regular pentagon has interior angles of:
\[
\text{Interior angle} = \frac{(n-2) \times 180^\circ}{n} = \frac{(5-2) \times 180^\circ}{5} = \frac{3 \times 180^\circ}{5} = 108^\circ
\]
- For a polygon to tessellate (fit together without gaps), its interior angle must divide evenly into 360°.
- Checking divisibility:
\[
360^\circ \div 108^\circ = \frac{360}{108} = \frac{10}{3}
\]
Since 10/3 is not an integer, a regular pentagon cannot tessellate.

Conclusion: Jimmy has not chosen correctly. Regular pentagons do not tessellate.

---

#### 5. Prove, show, justify: Interior angle of a regular octagon
Task: Prove that an interior angle of a regular octagon is 135°.

##### Solution:
- The formula for the interior angle of a regular polygon is:
\[
\text{Interior angle} = \frac{(n-2) \times 180^\circ}{n}
\]
- For an octagon (\( n = 8 \)):
\[
\text{Interior angle} = \frac{(8-2) \times 180^\circ}{8} = \frac{6 \times 180^\circ}{8} = \frac{1080^\circ}{8} = 135^\circ
\]

Conclusion: The interior angle of a regular octagon is indeed 135°.

---

#### 6. Extend a concept: Tessellation with squares, hexagons, and a third polygon
Task: Determine the third polygon that can tessellate with squares and hexagons.

##### Solution:
- Squares have interior angles of 90°.
- Hexagons have interior angles of 120°.
- For tessellation, the sum of the angles around a point must be 360°.
- Possible combinations:
- 3 squares: \( 3 \times 90^\circ = 270^\circ \) (remaining: \( 360^\circ - 270^\circ = 90^\circ \))
- 2 hexagons: \( 2 \times 120^\circ = 240^\circ \) (remaining: \( 360^\circ - 240^\circ = 120^\circ \))
- 1 square and 2 hexagons: \( 90^\circ + 2 \times 120^\circ = 330^\circ \) (remaining: \( 360^\circ - 330^\circ = 30^\circ \))

The only polygon that fits is a triangle (with an interior angle of 60°), but it does not work here. The correct third polygon is a dodecagon (12-sided polygon) with an interior angle of 150°, which can fit with squares and hexagons.

---

#### 7. Construct an instance: Sketch a polygon with specific angle types
Task: Sketch a polygon with at least 2 reflex angles, 2 obtuse angles, and 2 acute angles.

##### Solution:
- A reflex angle is greater than 180°.
- An obtuse angle is between 90° and 180°.
- An acute angle is less than 90°.
- A suitable polygon could be a heptagon (7-sided polygon) with angles:
- Reflex angles: 200°, 220°
- Obtuse angles: 120°, 150°
- Acute angles: 30°, 40°
- Remaining angle: calculated to satisfy the angle sum property.

---

#### 8. Criticise a fallacy: Sum of interior angles of a nonagon
Task: Critique the student's calculation of the sum of interior angles of a nonagon.

##### Solution:
- The student calculates the sum of the interior angles of a nonagon as:
\[
180^\circ \times 9 = 1620^\circ
\]
- The correct formula for the sum of the interior angles of a polygon is:
\[
(n-2) \times 180^\circ
\]
- For a nonagon (\( n = 9 \)):
\[
(9-2) \times 180^\circ = 7 \times 180^\circ = 1260^\circ
\]

Conclusion: The student's calculation is incorrect. The correct sum is 1260°.

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Final Answers:

1. \( s = 25^\circ \), \( t = 98^\circ \), \( u = \text{invalid} \)
2. \( p = \text{invalid} \), \( q = 98^\circ \), \( r = 42^\circ \)
3. Odd one out: (a)
4. Jimmy has not chosen correctly.
5. Proven: Interior angle of a regular octagon is 135°.
6. Third polygon: Dodecagon.
7. Constructed: Heptagon with specified angles.
8. Critique: Student's calculation is incorrect; correct sum is 1260°.

\boxed{25^\circ, 98^\circ, \text{invalid}, \text{invalid}, 98^\circ, 42^\circ, \text{odd one out: (a)}, \text{not correct}, \text{proven}, \text{dodecagon}, \text{constructed}, \text{critique: 1260^\circ}}