1. Problem: Determine the missing angles $x$, $y$, and $z$ given $y=75^\circ$ and the angles form a straight line. 2. Formula: The sum of angles on a straight line is $180^\circ$. 3. Calculation: - Since $y=75^\circ$, and $x$ and $z$ are adjacent angles on the line, we have: $$x + y + z = 180^\circ$$ - Substitute $y$: $$x + 75 + z = 180$$ - If $x$ and $z$ are equal (common in such problems), then: $$2x + 75 = 180$$ $$2x = 180 - 75 = 105$$ $$x = \frac{105}{2} = 52.5^\circ$$ - Then $z = 52.5^\circ$. 4. Explanation: Angles on a straight line add up to $180^\circ$. Knowing one angle lets us find the others by subtraction and division. --- 1. Problem: Find coordinates of shape A after transformations. 2. Reflection in the y-axis: $(x,y) \to (-x,y)$. 3. Rotation 180° counterclockwise about origin: $(x,y) \to (-x,-y)$. 4. Example: If a point of A is $(-4,3)$, - Reflection: $(4,3)$ - Rotation: $(4,-3)$ --- 1. Problem: Find area of polygon with given side lengths and scale $0.5$ cm = $1$ m. 2. Convert lengths to meters by multiplying by 2. 3. Use polygon area methods (e.g., divide into rectangles/triangles). 4. Approximate area calculation (example): - Sum areas of parts, e.g., rectangles: $7m \times 3m = 21 m^2$, etc. --- 1. Problem: Number of circular plates fitting on rectangular tray. 2. Given circumference $c=50.24$ cm, find radius: $$c = 2\pi r \Rightarrow r = \frac{c}{2\pi} = \frac{50.24}{2 \times 3.14} = 8 cm$$ 3. Area of plate: $$A = \pi r^2 = 3.14 \times 8^2 = 201.06 cm^2$$ 4. Area of tray: $$35 \times 50 = 1750 cm^2$$ 5. Number of plates: $$\frac{1750}{201.06} \approx 8$$ --- Short definitions: - Angles on a straight line sum to $180^\circ$. - Reflection flips points over an axis. - Rotation turns points around origin. - Scale converts drawing units to real units. - Circumference relates to radius by $c=2\pi r$. - Area is space inside a shape. Final answers: - $x = 52.5^\circ$, $y=75^\circ$, $z=52.5^\circ$ - Reflection of $(-4,3)$ is $(4,3)$; rotation is $(4,-3)$ - Area approx $21 m^2$ plus other parts (detailed calculation needed) - Plates fitting: 8