1. **Problem Statement:** We have a tessellation figure and a heptagon with interior angles expressed in terms of $x$. Also, a straight line $AOB$ with angles labeled in terms of $x$. 2. **Tessellation Questions:** 2.1.1. Name any TWO shapes in the tessellation figure. 2.1.2. Give ONE real life example of tessellation. 3. **Heptagon Interior Angles:** Given angles: $x^\circ$, $(x - 20)^\circ$, $3x^\circ$, $4x^\circ$, $(x + 75)^\circ$, $(2x + 10)^\circ$, $(2x - 5)^\circ$. 3.1. Find $x$. 3.2. Determine if the polygon is regular or irregular. 3.3. Provide reason. 4. **Angles on Straight Line $AOB$:** Angles given: $4x$, $7x$, $2x$, $5x$. 4.1. Find $x$. 4.2. Find complement of $x$. 4.3. Find all angles. --- ### Step-by-step solution: 1. **Tessellation shapes:** - Common shapes in tessellations include squares, triangles, hexagons, or other polygons. - From the figure, two shapes could be: **hexagon** and **triangle** (or any two visible shapes). 2. **Real life example of tessellation:** - Examples include **floor tiles**, **honeycomb**, or **brick walls**. --- 3. **Heptagon interior angles sum:** - Sum of interior angles of an $n$-sided polygon is given by: $$\text{Sum} = (n - 2) \times 180^\circ$$ - For a heptagon, $n=7$: $$\text{Sum} = (7 - 2) \times 180 = 5 \times 180 = 900^\circ$$ 4. **Set up equation for $x$:** $$x + (x - 20) + 3x + 4x + (x + 75) + (2x + 10) + (2x - 5) = 900$$ 5. **Simplify left side:** $$x + x - 20 + 3x + 4x + x + 75 + 2x + 10 + 2x - 5 = 900$$ Combine like terms: $$ (x + x + 3x + 4x + x + 2x + 2x) + (-20 + 75 + 10 - 5) = 900$$ $$ (14x) + 60 = 900$$ 6. **Solve for $x$:** $$14x + 60 = 900$$ Subtract 60 from both sides: $$14x + \cancel{60} - \cancel{60} = 900 - 60$$ $$14x = 840$$ Divide both sides by 14: $$\frac{14x}{\cancel{14}} = \frac{840}{\cancel{14}}$$ $$x = 60$$ 7. **Determine if polygon is regular or irregular:** - A regular polygon has all interior angles equal. - Check angles with $x=60$: $$x = 60^\circ$$ $$(x - 20) = 40^\circ$$ $$3x = 180^\circ$$ $$4x = 240^\circ$$ $$(x + 75) = 135^\circ$$ $$(2x + 10) = 130^\circ$$ $$(2x - 5) = 115^\circ$$ - Angles are not equal, so polygon is **irregular**. 8. **Reason:** - Because the interior angles are not all equal, the polygon is irregular. --- 9. **Angles on straight line $AOB$:** - Given angles: $4x$, $7x$, $2x$, $5x$. - Since $AOB$ is a straight line, sum of adjacent angles on the line is $180^\circ$. 10. **Find $x$:** - Sum of all angles on the straight line: $$4x + 7x + 2x + 5x = 180$$ $$18x = 180$$ Divide both sides by 18: $$\frac{18x}{\cancel{18}} = \frac{180}{\cancel{18}}$$ $$x = 10$$ 11. **Complement of $x$:** - Complement of an angle $x$ is $90^\circ - x$: $$90 - 10 = 80^\circ$$ 12. **Find all angles:** - Substitute $x=10$: $$4x = 40^\circ$$ $$7x = 70^\circ$$ $$2x = 20^\circ$$ $$5x = 50^\circ$$ --- **Final answers:** - 2.1.1: Hexagon and triangle (or any two shapes visible). - 2.1.2: Floor tiles (or honeycomb). - 2.2.1: $x = 60$. - 2.2.2: Irregular polygon. - 2.2.3: Angles are not equal. - 2.3.1: $x = 10$. - 2.3.2: Complement of $x$ is $80^\circ$. - 2.3.3: Angles are $40^\circ$, $70^\circ$, $20^\circ$, and $50^\circ$ respectively.