Here are the solutions to the problems: 11. Define the term triangle. A triangle is a polygon with three straight sides and three vertices. It is a closed two-dimensional shape whose interior angles sum to $180^\circ$. 12. Classify the following triangles with angles: a) $60^\circ, 60^\circ, 60^\circ$: Equilateral triangle (all angles are equal, implying all sides are equal). b) $70^\circ, 40^\circ, 70^\circ$: Isosceles triangle (two angles are equal, implying two sides are equal). c) $40^\circ, 72^\circ, 68^\circ$: Scalene triangle (all angles are different, implying all sides are different). 13. Calculate both the perimeter and area for the following: a) Step 1: Calculate the perimeter. The sides of the triangle are $9 \text{ cm}$, $12 \text{ cm}$, and $15 \text{ cm}$. $$ \text{Perimeter} = 9 \text{ cm} + 12 \text{ cm} + 15 \text{ cm} $$ $$ \text{Perimeter} = \boxed{\text{}36 \text{ cm}\text{}} $$ Step 2: Calculate the area. The base is $15 \text{ cm}$ and the height is $6 \text{ cm}$. $$ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} $$ $$ \text{Area} = \frac{1}{2} \times 15 \text{ cm} \times 6 \text{ cm} $$ $$ \text{Area} = \frac{1}{2} \times 90 \text{ cm}^2 $$ $$ \text{Area} = \boxed{\text{}45 \text{ cm}^2\text{}} $$ b) Step 1: Calculate the area. The base is $7 \text{ cm}$ and the height is $4 \text{ cm}$. $$ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} $$ $$ \text{Area} = \frac{1}{2} \times 7 \text{ cm} \times 4 \text{ cm} $$ $$ \text{Area} = \frac{1}{2} \times 28 \text{ cm}^2 $$ $$ \text{Area} = \boxed{\text{}14 \text{ cm}^2\text{}} $$ Note: The perimeter cannot be calculated without the length of the third side. The given side $2.5 \text{ cm}$ is one of the non-base sides. c) The given side lengths are $3 \text{ cm}$, $3 \text{ cm}$, and $8 \text{ cm}$. According to the triangle inequality theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Here, $3 \text{ cm} + 3 \text{ cm} = 6 \text{ cm}$. Since $6 \text{ cm} < 8 \text{ cm}$, a triangle cannot be formed with these dimensions. 14. In the triangle below, angles at A and B are $55^\circ$ and $115^\circ$ respectively. Find the value of $x$ at C. Step 1: Find the interior angle at B. The exterior angle at B is $115^\circ$. The interior angle at B and the exterior angle at B are supplementary (they add up to $180^\circ$). $$ \text{Interior Angle at B} = 180^\circ - 115^\circ $$ $$ \text{Interior Angle at B} = 65^\circ $$ Step 2: Use the sum of angles in a triangle. The sum of the interior angles of a triangle is $180^\circ$. $$ \text{Angle A} + \text{Interior Angle B} + \text{Angle C} = 180^\circ $$ $$ 55^\circ + 65^\circ + x = 180^\circ $$ $$ 120^\circ + x = 180^\circ $$ Step 3: Solve for $x$. $$ x = 180^\circ - 120^\circ $$ $$ x = \boxed{\text{}60^\circ\text{}} $$