Angle Relationships - ISEE Upper Level: Quantitative Reasoning

$30^\circ$. Each exterior angle of a regular 12-gon is found by dividing $360^\circ$ by 12.

$x=75^\circ$. The exterior angle equals the sum of remote interiors, so subtract $55^\circ$ from $130^\circ$ to find the other remote angle.

$x=70^\circ$. Quadrilateral angles sum to $360^\circ$, so subtract the sum of $95^\circ$, $110^\circ$, and $85^\circ$ from $360^\circ$.

$x=69^\circ$. Triangle angles sum to $180^\circ$, so subtract the sum of $48^\circ$ and $63^\circ$ from $180^\circ$.

$x=79^\circ$. Vertical angles are congruent, so the unknown angle equals the given measure of $79^\circ$.

$x=68^\circ$. Supplementary angles sum to $180^\circ$, so subtract $112^\circ$ from $180^\circ$ to find the unknown angle.

$x=133^\circ$. Linear pairs sum to $180^\circ$, so subtract $47^\circ$ from $180^\circ$ to find the unknown angle.

They are supplementary (a linear pair). Adjacent angles on a straight line form a linear pair and are supplementary, summing to $180^\circ$.

Exterior angle $=$ sum of the two remote interior angles. By the exterior angle theorem, the exterior angle of a triangle equals the sum of the two non-adjacent interior angles.

They are supplementary (sum to $180^\circ$). Same-side interior angles are supplementary when parallel lines are cut by a transversal, summing to $180^\circ$.

Alternate exterior angles are congruent. Alternate exterior angles are congruent when parallel lines are intersected by a transversal, based on the alternate exterior angles theorem.

Alternate interior angles are congruent. Alternate interior angles are congruent when formed by parallel lines and a transversal, per the alternate interior angles theorem.

Corresponding angles are congruent. When parallel lines are cut by a transversal, corresponding angles are equal due to the corresponding angles postulate.

$\frac{360^\circ}{n}$. In a regular $n$-gon, each exterior angle is equal, found by dividing the total sum of $360^\circ$ by $n$.

$360^\circ$. The sum of exterior angles, one at each vertex, equals $360^\circ$ for any polygon, representing a full turn.

$\frac{(n-2)\cdot 180^\circ}{n}$. In a regular $n$-gon, all interior angles are equal, so the total sum is divided evenly by $n$.

$(n-2)\cdot 180^\circ$. The formula derives from dividing an $n$-gon into $(n-2)$ triangles, each contributing $180^\circ$.

$360^\circ$. A quadrilateral's interior angles sum to $360^\circ$ as it can be divided into two triangles.

$180^\circ$. The interior angles of any triangle sum to $180^\circ$ based on the triangle angle sum theorem.

Their measures sum to $180^\circ$. Supplementary angles add up to a straight angle, measuring $180^\circ$.

Their measures sum to $90^\circ$. Complementary angles add up to a right angle, which measures $90^\circ$.

Vertical angles are congruent (equal). Vertical angles, formed opposite each other by intersecting lines, are always congruent due to the properties of intersecting lines.

$360^\circ$. Angles around a point form a complete circle, totaling $360^\circ$ as a full rotation.

$180^\circ$. A linear pair consists of adjacent angles on a straight line, which always sum to $180^\circ$ due to the straight angle property.