Geometry

Angles

Big Picture

Angles are formed when two rays share an endpoint. Angles are classified by their measure. Acute angles measure less than 90°, right angles measure exactly 90°, obtuse angles measure between 90 to 180°, and straight angles are 180°. Angles also have relationships with each other. For example, two angles sharing a side are adjacent angles, and two adjacent angles that add up to 180° are linear pairs

Key Terms

Angle: Figure formed when two rays share a common endpoint.

Vertex: Common endpoint of an angle.

Degree: The measure of an angle; one degree is equivalent to 1⁄360 of a circle.

Acute Angle: An angle that measures between 0 to 90°.

Right Angle:  An angle that measures exactly 90°.

Obtuse Angle: An angle that measures between 90 to 180°.

Straight Angle:  An angle that measures exactly 180°.

Congruent Angles: Angles that have the same measure.

Angle Bisector: A ray that divides an angle into two congruent angles, each with a measure equal to exactly half of the original angle.

Linear Pair: A pair of adjacent angles whose non-common sides are opposite rays.

Adjacent Angles: Two angles that have the same vertex and share one side but do not overlap (no common interior points).

Vertical Angles: Two non-adjacent angles formed by intersecting lines.

Theorem: A statement that can be proven using postulates, definitions, logic, etc.

Angles

An angle is formed when two rays (\(\overrightarrow{AB} \text{ and }\overrightarrow{AT}\)) share a vertex (point A).

  • The two rays are the sides of the angle.
  • The interior of an angle is the set of all points between the sides of the angle.
  • The exterior is the set of all points outside the angle.

Naming Angles

There are three ways to name an angle symbol .

  • By number: 1
  • By the single letter at the vertex:  A.
  • We can use the vertex to name the angle if there is only one angle at point A.
  • By three points that form the angle:  BAT or  TAB
  • If named with three letters, the middle letter will be the vertex
  • If two or more angles share same vertex, we MUST use three points to name the angle!
    Angles

    In the diagram below, we cannot name the angle  ∠U because it is not clear which angle we are referring to. The figure has three angles:

    • XUZ or ZUX
    • XUY or YUX
    • YUZ or ZUY
    Naming Angles

    Geometry

    Angles cont.

    Measuring Angles

    Angles are measured in degrees, a unit measuring the amount of rotation from one side to another.

    • The symbol for degree is °.
    • A full circle has 360°, so a degree is 1 ⁄ 360 of a circle.
    • The measure of an angle is denoted by m

    An angle’s degrees can be measured with a protractor. A protractor is a half-circle measuring device with angle measures marked for each degree. To use a protractor, line up the vertex of the angle with the center of the protractor.

    • To use a protractor, line up the vertex of the angle with the center of the protractor.

    Protractor Postulate

    Protractor Postulate: For every angle, there is a number between 0 and 180 that is the measure of the angle in degrees. The angle’s measure on a protractor is the absolute value of the difference of the numbers shown on the protractor.

    • You do not need to measure the angle from 0°:
    Protractor Postulate

    Angle Addition Postulate

    Angle Addition Postulate: The measure of any angle can be found by adding the measures of the smaller angles that comprise it.

           If R is in the interior of  BAT, then m∠BAR + mRAT = mBAT

    Angle Addition Postulate

    Classifying Angles

    Angles can be classified by their measure:

    Acute Angle

    Obtuse Angle

    Right Angle

    Straight Angle

    Acute Angle
    Obtuse Angle
    Right Angle
    Straight Angle

    Measures between 0 to 90°

    Measures between 90 to 180°

    Measures 90°
    Marked with a small square
    When two lines intersect at a right angle, they are perpendicular

    Measures 180°
    Formed by two opposite rays, looks just like a straight line

    Congruent Angles

    Congruent angles are angles with the same measure. Arc marks are used to show that the angles are congruent (use symbol   to indicate congruence).

    not equal different angle marks

    equal angles same arc marks

    equal angles same arc marks

    not equal different angle marks

    Right Angle Theorem: If two angles are right angles, then the angles are congruent.

    Angle Bisectors

    An angle bisector will divide an angle into two congruent angles.

    If \(\overline{BD}\) is the angle bisector of  ABC, then:

    • ABD DBC
    • m ABD = \(\frac{1}{2}\)mABC
    Angle Bisectors

    Angle Bisector Postulate: Every angle has exactly one angle bisector.

    Geometry

    Angles cont.

    Angle Pairs

    Complimentary Angles

    A pair of angles are complementary if the sum of their measures is 90°.

    • The angles do not need to be congruent or touching.
    Complimentary Angles

    Same Angle Complements Theorem: If two angles are complementary to the same angle (or to congruent angles), then the angles are congruent.

    Supplementary Angles

    A pair of angles are supplementary if their measures sum to 180°.

    • The angles do not need to be congruent or touching.
    Supplementary Angles

    Same Angle Supplements Theorem: If two angles are supplementary to the same angle (or to congruent angles), then the angles are congruent.

    Tip: c comes before s, just like 90° comes before 180°, so complementary angles add up to 90° and suplementary angles add up to 180°

    Linear Pairs

    A linear pair is a pair of adjacent angles whose non-common sides form a line.

    \(\overrightarrow{PN}\) is the common side. ∠MNP and ∠PNO are adjacent angles.

    Linear Pairs

    Linear Pair Postulate: If two angles are a linear pair, then they are supplementary.

    Vertical Angles

    Vertical angles are two non-adjacent angles formed by intersecting lines.

    • 1 and  3 are vertical angles
    • 2 and   4 are vertical angle
    Vertical Angles

    Vertical Angles Theorem: Vertical angles are congruent.

    • 1 3 and 2 4

    Notes