Geometry: Angles and Angle Pairs Flashcards

Vocabulary and theorems covering angle measurement, types of angles (acute, right, obtuse), angle pairs (adjacent, linear pair, vertical, complementary, supplementary), and properties of congruence.

Postulate 11 (The Angle Measurement Postulate)

To every angle there corresponds a real number between $0$ and $180$.

Postulate 12 (The Angle Construction Postulate)

Let $AB$ be a ray on the edge of the half-plane $H$. For every number $r$ between $0$ and $180$ there is exactly one ray $AP$, with $P$ in $H$, such that $m \angle PAB = r$.

Postulate 13 (The Angle Addition Postulate)

If $D$ is in the interior of $\angle BAC$, then $m \angle BAC = m \angle BAD + m \angle DAC$.

Interior of $\angle BAC$

The set of all points $P$ in the plane of $\angle BAC$ such that (1) $P$ and $B$ are on the same side of line $AC$ and (2) $P$ and $C$ are on the same side of line $AB$.

Exterior of $\angle BAC$

The set of all points of the plane of $\angle BAC$ that lie neither on the angle nor in its interior.

Degree

The unit of measure used for angles.

Protractor

The instrument used to measure angles.

Ray

In this context, equivalent to a 'zero angle'.

Line

In this context, equivalent to a 'straight angle'.

Adjacent Angles

Two angles that have a common side, a common endpoint, and no common interior points.

Linear Pair

Formed if $AB$ and $AD$ are opposite rays and $AC$ is any other ray, resulting in $\angle BAC$ and $\angle CAD$.

Supplementary Angles

Two angles whose measures sum to $180$; each is called a supplement of the other.

Postulate 14 (The Supplement Postulate)

If two angles form a linear pair, then they are supplementary.

Theorem 4-1

Every right angle has measure $90$, and every angle with measure $90$ is a right angle.

Right Angle

An angle formed if the angles in a linear pair have the same measure, or an angle with measure exactly $90$.

Perpendicular

Rays, lines, or segments that intersect to form a right angle; denoted by the symbol $\perp$.

Complementary Angles

Two angles whose measures sum to $90$; each is a complement of the other.

Acute Angle

An angle with a measure less than $90$.

Obtuse Angle

An angle with a measure greater than $90$, but less than $180$, given that $0 < r < 180$.

Congruent Angles

Two angles with the same measure; denoted by the symbol $\cong$.

Theorem 4-2

Congruence between angles is an equivalence relation (reflexive, symmetric, and transitive).

Theorem 4-3

If two angles are complementary, then both are acute.

Theorem 4-4

Any two right angles are congruent.

Theorem 4-5

If two angles are both congruent and supplementary, then each is a right angle.

Theorem 4-6 (The Supplement Theorem)

Supplements of congruent angles are congruent.

Theorem 4-7 (The Complement Theorem)

Complements of congruent angles are congruent.

Theorem 4-8 (Vertical Angle Theorem)

Vertical angles are congruent.

Theorem 4-9

If two intersecting lines form one right angle, then they form four right angles.

Reflexive Property of Congruence

$\angle A \cong \angle A$ for every $\angle A$.

Symmetric Property of Congruence

If $\angle A \cong \angle B$, then $\angle B \cong \angle A$.

Transitive Property of Congruence

If $\angle A \cong \angle B$ and $\angle B \cong \angle C$, then $\angle A \cong \angle C$.

Vertical Angles

Two angles whose sides form two pairs of opposite rays.