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MathChapter 16: Lines and Angles

Notes

Angles Notes

Name and classify angles, add adjacent parts, use angle bisectors, and connect vertical, complementary, and supplementary relationships.

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An angle is formed by two rays with a common endpoint. Most SAT angle problems are solved by identifying one relationship, writing its equation, and checking that the result fits the diagram.

Name an angle from its vertex

Anatomy and naming of angle ABCRays BA and BC share vertex B. The middle letter B names the vertex of angle ABC.
Anatomy and naming of angle ABCRays BA and BC share vertex B. The middle letter B names the vertex of angle ABC.CA∠ABCB
Vertex
The shared endpoint of the two rays. In \(\angle ABC\), the middle letter \(B\) must be the vertex.

Classify by exact measure

Four angle classificationsFour panels show a 35-degree acute angle, 90-degree right angle, 125-degree obtuse angle, and 180-degree straight angle.
Four angle classificationsFour panels show a 35-degree acute angle, 90-degree right angle, 125-degree obtuse angle, and 180-degree straight angle.Acute: 35°Right: 90°Obtuse: 125°Straight: 180°
Angle classification by measure
Angle typeMeasure rangeVisual cueExample
Acute\(0^\circ<\theta<90^\circ\)Narrower than a right angle\(35^\circ\)
Right\(\theta=90^\circ\)Square marker\(90^\circ\)
Obtuse\(90^\circ<\theta<180^\circ\)Wider than right, not straight\(125^\circ\)
Straight\(\theta=180^\circ\)Opposite rays\(180^\circ\)
Mini check

Classify exactly

How is a \(90^\circ\) angle classified?

  1. Acute
  2. Right
  3. Obtuse
Show answer and explanation

Answer: Right.

The acute interval stops before \(90^\circ\); equality belongs to the right-angle category.

Build a whole angle from adjacent parts

Angle additionInterior ray OC divides angle AOB into 42 degrees and 31 degrees, so the whole angle measures 73 degrees.
Angle additionInterior ray OC divides angle AOB into 42 degrees and 31 degrees, so the whole angle measures 73 degrees.BCA31°42°O
Angle Addition Postulate
\[m\angle AOB=m\angle AOC+m\angle COB\]

Ray \(OC\) must lie inside \(\angle AOB\).

Worked example

Add two interior angles

Ray \(OC\) lies inside \(\angle AOB\). If \(m\angle AOC=42^\circ\) and \(m\angle COB=31^\circ\), find \(m\angle AOB\).

  1. State the postulate

    \(m\angle AOB=m\angle AOC+m\angle COB\).

  2. Substitute

    \(42^\circ+31^\circ=73^\circ\).

\(m\angle AOB=73^\circ\).

Angle bisectors make equal angles

Angle bisector with matching arcsRay OC divides angle AOB into two congruent angles, shown with matching marked arcs.
Angle bisector with matching arcsRay OC divides angle AOB into two congruent angles, shown with matching marked arcs.BCAx°x°O
Angle-bisector relationships
\[m\angle AOC=m\angle COB=\frac12m\angle AOB\]

Matching arc marks visually encode the equality of the two smaller angles.

Worked example

Use a bisector with algebra

Ray \(OC\) bisects \(\angle AOB\). If \(m\angle AOC=4x+3\) and \(m\angle COB=6x-15\), find the whole angle.

  1. Set equal parts

    \(4x+3=6x-15\), so \(x=9\).

  2. Evaluate one half

    \(4(9)+3=39^\circ\).

  3. Double

    \(m\angle AOB=78^\circ\).

\(78^\circ\).

Vertical angles are opposite and congruent

Vertical angles from intersecting linesTwo lines intersect. Angles 1 and 3 are an opposite congruent pair; angles 2 and 4 are the other opposite congruent pair.
Vertical angles from intersecting linesTwo lines intersect. Angles 1 and 3 are an opposite congruent pair; angles 2 and 4 are the other opposite congruent pair.∠1∠2∠3∠4
Worked example

Solve a vertical-angle equation

Vertical angles measure \(3x+8\) degrees and \(5x-24\) degrees. Find each angle.

  1. Use congruence

    \(3x+8=5x-24\).

  2. Solve

    \(32=2x\), so \(x=16\).

  3. Evaluate

    \(3(16)+8=56^\circ\).

Each vertical angle measures \(56^\circ\).

Complementary and supplementary pairs

Complementary and supplementary angle pairsThe first panel shows 35 and 55 degrees forming 90 degrees. The second shows 118 and 62 degrees forming 180 degrees.
Complementary and supplementary angle pairsThe first panel shows 35 and 55 degrees forming 90 degrees. The second shows 118 and 62 degrees forming 180 degrees.Complementary: 35° + 55° = 90°35°55°Supplementary: 118° + 62° = 180°62°118°
Special angle-pair relationships
RelationshipRequired sumMust be adjacent?Example
Complementary\(90^\circ\)No\(35^\circ+55^\circ\)
Supplementary\(180^\circ\)No\(118^\circ+62^\circ\)
VerticalEqual measuresNo; they are opposite\(56^\circ\) and \(56^\circ\)
Angle-bisected partsEqual measuresYes, within the whole\(39^\circ\) and \(39^\circ\)
Worked example

Solve a supplementary equation

Supplementary angles measure \(2x+10\) and \(4x-4\) degrees. Find \(x\).

  1. Write the sum

    \((2x+10)+(4x-4)=180\).

  2. Solve

    \(6x+6=180\), so \(x=29\).

  3. Check

    The measures are \(68^\circ\) and \(112^\circ\), totaling \(180^\circ\).

\(x=29\).

Solve a multi-relation angle problem

  1. Mark the relationship

    Equal, complementary, supplementary, or additive?

  2. Write one equation

    Translate the named theorem before doing arithmetic.

  3. Solve and evaluate

    Substitute back for the requested angle, not only the variable.

  4. Check geometry

    Reject negative values or classifications inconsistent with the marks.

Key takeaways

Angles: key takeaways

  • The middle letter names the vertex.
  • Acute, right, obtuse, and straight intervals have exact boundaries.
  • Interior parts add to the whole angle.
  • A bisector makes equal halves; vertical angles make equal opposite pairs.
  • Complementary sums to \(90^\circ\); supplementary sums to \(180^\circ\).
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Put these notes into practice

Apply the ideas with SAT-style questions, then reinforce key details with flashcards.