MAAP High School Courses Geometry domains. Practice it free.

Same-side interior angles are supplementary: $(6x-5)+(4x+15)=180$, so $10x+10=180$ and $x=17$.

Set up the proportion $\dfrac{1{,}250\text{ people}}{1\text{ square mile}} = \dfrac{x\text{ people}}{48\text{ square miles}}\textsf{.}$ Cross-multiplying gives $x = 1{,}250 \times 48 = 60{,}000\text{ people}\textsf{.}$

When parallel lines are cut by a transversal, corresponding angles are congruent. The corresponding angle at the second intersection also measures $115^{\circ}$.

The hypotenuse is twice the short leg: $2 \times 10 = 20$.

$\cos\theta = \dfrac{\sin\theta}{\tan\theta} = \dfrac{\dfrac{12}{13}}{\dfrac{12}{5}} = \dfrac{12}{13} \cdot \dfrac{5}{12} = \dfrac{5}{13}$.

A reflection is a rigid transformation that maps the points from a preimage, over a line of reflection, to the points of an image that are equidistant to the line. In this example, that line is the perpendicular bisector of a line segment from a point to its image, so the segment maps onto itself.

Set up $\dfrac{4}{5} = \dfrac{h}{65}$. Cross-multiply: $5h = 260$, so $h = 52$ feet.

In a $30^{\circ}$-$60^{\circ}$-$90^{\circ}$ triangle, the hypotenuse is twice the short leg: $2 \times 10 = 20\textsf{.}$