Unit 1: Basics of Geometry - Angles and Angle Addition Postulate Study Guide Notes | Knowt

What is an angle in geometry?

A figure formed by two rays, called the sides of the angle, sharing a common endpoint called the vertex.

What is the vertex of an angle?

The common endpoint where two rays meet to form an angle.

What are the sides of an angle?

The two rays that share a common endpoint (vertex) to form the angle.

What is an acute angle?

An angle with a measure strictly greater than $0^\circ$ and less than $90^\circ$.

What is a right angle?

An angle whose measure is exactly $90^\circ$.

What is an obtuse angle?

An angle with a measure strictly greater than $90^\circ$ and less than $180^\circ$.

What is a straight angle?

An angle whose measure is exactly $180^\circ$, forming a straight line.

What is a reflex angle?

An angle with a measure strictly greater than $180^\circ$ and less than $360^\circ$.

What is the Angle Addition Postulate?

If point $B$ lies in the interior of $\angle AEC$, then $m\angle AEB + m\angle BEC = m\angle AEC$.

What are adjacent angles?

Two angles in a plane that share a common vertex and a common side, but have no common interior points.

What are complementary angles?

Two angles whose measures have a sum of $90^\circ$.

What are supplementary angles?

Two angles whose measures have a sum of $180^\circ$.

What is a linear pair of angles?

A pair of adjacent angles whose non-common sides are opposite rays, forming a straight line and summing to $180^\circ$.

What are vertical angles?

Two non-adjacent angles formed by two intersecting lines; vertical angles are congruent.

What is an angle bisector?

A ray that divides an angle into two congruent adjacent angles.

If ray $BD$ bisects $\angle ABC$, what is the relationship between $m\angle ABD$ and $m\angle ABC$?

$m\angle ABD = \frac{1}{2} m\angle ABC$ (or $m\angle ABC = 2 \times m\angle ABD$).

What does it mean for two angles to be congruent ($\cong$)?

It means the two angles have equal measures ($m\angle A = m\angle B$).

What tool is standardly used to measure an angle in degrees?

A protractor.

What is the unit of measure commonly used for angles in basic geometry?

Degrees ($^\circ$).

If $m\angle 1 = (2x + 10)^\circ$ and $m\angle 2 = (3x - 15)^\circ$, and $\angle 1$ and $\angle 2$ are complementary, what is the value of $x$?

$x = 19$ (since $(2x + 10) + (3x - 15) = 90 \rightarrow 5x - 5 = 90 \rightarrow 5x = 95$).

If $m\angle ABD = 35^\circ$ and $m\angle DBC = 45^\circ$, where point $D$ lies in the interior of $\angle ABC$, what is $m\angle ABC$?

$m\angle ABC = 80^\circ$ (since $35^\circ + 45^\circ = 80^\circ$).

What are interior points of an angle?

All points lying between the two rays that form the angle.

What are exterior points of an angle?

All points in the plane that do not lie on the angle or in its interior.

If two angles form a linear pair and one angle measures $115^\circ$, what is the measure of the second angle?

$65^\circ$ (since $180^\circ - 115^\circ = 65^\circ$).

According to the Angle Addition Postulate, if $m\angle PQR = 120^\circ$ and ray $QS$ is inside $\angle PQR$ such that $m\angle PQS = 2x^\circ$ and $m\angle SQR = 4x^\circ$, what is $x$?

$x = 20$ (since $2x + 4x = 120 \rightarrow 6x = 120$).