Lines, Angles, and Triangles - GRE Quantitative Reasoning

They are congruent (equal in measure). Vertical angles are formed by intersecting lines and are equal because they are opposite each other.

They sum to $90^\circ$. Complementary angles are defined as two angles whose measures add up to $90^\circ$.

They sum to $180^\circ$. Supplementary angles are defined as two angles whose measures add up to $180^\circ$.

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

Alternate interior angles are congruent. Alternate interior angles are equal when lines are parallel, as they lie on opposite sides of the transversal inside the parallels.

They are supplementary: sum is $180^\circ$. Same-side interior angles add up to $180^\circ$ when lines are parallel, forming supplementary pairs.

$a+b>c$, $a+c>b$, and $b+c>a$. The theorem states that the sum of any two sides must exceed the third to form a triangle.

Longer side is opposite larger angle. In any triangle, the largest angle is opposite the longest side, and vice versa, by the law of sines.

$a^2+b^2=c^2$. The theorem relates the sides of a right triangle, where the square of the hypotenuse equals the sum of the squares of the legs.

$1:1:\sqrt{2}$. In a $45^\circ$-$45^\circ$-$90^\circ$ triangle, the legs are equal, and the hypotenuse is leg times $\sqrt{2}$.

$1:\sqrt{3}:2$. In a $30^\circ$-$60^\circ$-$90^\circ$ triangle, sides are in the ratio shortest to middle to hypotenuse as $1 : \sqrt{3} : 2$.

It has at least two equal side lengths. An isosceles triangle has at least two sides of equal length by definition.

Angles opposite equal sides are equal. In an isosceles triangle, the angles opposite the equal sides are congruent.

Exterior angle equals sum of the two remote interior angles. The exterior angle theorem states that an exterior angle equals the sum of the two non-adjacent interior angles.

$x=30$. Set up the equation $3x + 2x + 30 = 180$ and solve for $x$, as linear pairs sum to $180^\circ$.

$x=20$. Set $4x - 10 = 2x + 30$ and solve for $x$, since vertical angles are equal.

$x=70^\circ$. The sum of angles in a triangle is $180^\circ$, so $x = 180^\circ - 50^\circ - 60^\circ$.

$c=10$. Apply the Pythagorean theorem: $c = \sqrt{6^2 + 8^2} = \sqrt{100}$.

$180^\circ$. The sum of the interior angles in any triangle is always $180^\circ$ due to the properties of parallel lines and transversals.

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

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

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

They are supplementary: sum is $180^\circ$. Angles in a linear pair are adjacent and form a straight line, so their measures add up to $180^\circ$.