Angles

Study Sheet

Angles

Naming, classifying, and solving for angle measures

Naming Angles and Their Parts

Concept
What is an angle?
55°

An angle is formed by two rays that share a common endpoint. That shared endpoint is the vertex, and the two rays are the sides of the angle. We measure the “opening” between the sides in degrees (∘^\circ).

There are three ways to name an angle:

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  • Three points: ∠AOB\angle AOB or ∠BOA\angle BOA --- the vertex letter always goes in the middle.
  • The vertex alone: ∠O\angle O --- only when there is no confusion about which angle you mean.
  • A number: ∠1\angle 1 --- when a small number is written inside the angle.

An angle measures a turn.

Example
Naming the angle below

The rays OA→\overrightarrow{OA} and OB→\overrightarrow{OB} meet at the vertex OO. We can call this angle ∠AOB\angle AOB, ∠BOA\angle BOA, or simply ∠O\angle O.

Tip

Protractor tip. To measure an angle, place the protractor's center hole on the vertex and line up one side with the 0∘0^\circ mark. Read where the other side crosses the scale. Use the scale that starts at 0∘0^\circ on the side you lined up --- most protractors have two rows of numbers!

Classifying Angles

Concept
Five kinds of angles
55°
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  • Acute: greater than 0∘0^\circ and less than 90∘90^\circ.
  • Right: exactly 90∘90^\circ (marked with a small square).
  • Obtuse: greater than 90∘90^\circ and less than 180∘180^\circ.
  • Straight: exactly 180∘180^\circ (a straight line).
  • Reflex: greater than 180∘180^\circ and less than 360∘360^\circ.

An angle measures a turn.

Tip

Remember. A right angle is your reference. If an angle looks “smaller than the corner of a page,” it is acute; if it looks “more open than a corner,” it is obtuse.

Adjacent Angles & the Angle Addition Postulate

Concept
Adjacent angles
55°

Two angles are adjacent when they share a common vertex and a common side, but do not overlap (no interior points in common). Think of two angles sitting side by side.

An angle measures a turn.

Concept
Angle Addition Postulate

If point BB is in the interior of ∠AOC\angle AOC, then the two small angles add up to the big angle:

m∠AOB+m∠BOC=m∠AOC.m\angle AOB + m\angle BOC = m\angle AOC .
Example
Adding adjacent angles

Ray OB→\overrightarrow{OB} lies inside ∠AOC\angle AOC. If m∠AOB=35∘m\angle AOB = 35^\circ and m∠BOC=40∘m\angle BOC = 40^\circ, then

m∠AOC=35∘+40∘=75∘.m\angle AOC = 35^\circ + 40^\circ = 75^\circ .
Tip

Tip. The Angle Addition Postulate works both ways: you can add two pieces to get the whole, or subtract a known piece from the whole to find the missing piece.

Angle Bisectors

Concept
What is a bisector?
55°

A ray that bisects an angle divides it into two congruent (equal) angles. If OB→\overrightarrow{OB} bisects ∠AOC\angle AOC, then

m∠AOB=m∠BOC=12 m∠AOC.m\angle AOB = m\angle BOC = \tfrac{1}{2}\, m\angle AOC .

An angle measures a turn.

Example
Using a bisector

OB→\overrightarrow{OB} bisects ∠AOC\angle AOC, and m∠AOC=80∘m\angle AOC = 80^\circ. Then each half is

m∠AOB=m∠BOC=12(80∘)=40∘.m\angle AOB = m\angle BOC = \tfrac{1}{2}(80^\circ) = 40^\circ .
Tip

Remember. “Bisect” means “cut into two equal parts.” The two halves are always congruent, so you can set them equal to each other when solving with algebra.

Complementary & Supplementary Angles

Concept
Two special sums
55°
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  • Complementary angles have measures that add to 90∘\mathbf{90^\circ}.
  • Supplementary angles have measures that add to 180∘\mathbf{180^\circ}.

The angles do not have to be next to each other --- only their measures matter.

An angle measures a turn.

Example
Finding a complement and a supplement

The complement of 30∘30^\circ is 90∘−30∘=60∘90^\circ - 30^\circ = 60^\circ.

The supplement of 30∘30^\circ is 180∘−30∘=150∘180^\circ - 30^\circ = 150^\circ.

Together the two angles above form a right angle, so they are complementary.

Tip

Memory trick. C comes before S in the alphabet, and 9090 comes before 180180: Complementary =90∘= 90^\circ, Supplementary =180∘= 180^\circ. (Or: “Corner” is 90∘90^\circ, “Straight” is 180∘180^\circ.)

Linear Pairs & Vertical Angles

Concept
Linear pairs
55°

When two angles are adjacent and their outer sides form a straight line, they make a linear pair. The angles of a linear pair are always supplementary (they add to 180∘180^\circ).

An angle measures a turn.

Example
A linear pair

∠1\angle 1 and ∠2\angle 2 below form a linear pair, so m∠1+m∠2=180∘m\angle 1 + m\angle 2 = 180^\circ. If m∠1=110∘m\angle 1 = 110^\circ, then m∠2=180∘−110∘=70∘m\angle 2 = 180^\circ - 110^\circ = 70^\circ.

Concept
Vertical angles

When two straight lines cross, they form two pairs of vertical angles --- the angles “across” from each other. Vertical angles are always congruent (equal).

Example
Vertical angles are equal

Two lines cross at OO. Angles ∠1\angle 1 and ∠3\angle 3 are vertical (so m∠1=m∠3m\angle 1 = m\angle 3), and ∠2\angle 2 and ∠4\angle 4 are vertical (so m∠2=m∠4m\angle 2 = m\angle 4). If m∠1=65∘m\angle 1 = 65^\circ, then m∠3=65∘m\angle 3 = 65^\circ, while ∠2\angle 2 and ∠4\angle 4 each measure 180∘−65∘=115∘180^\circ - 65^\circ = 115^\circ.

Tip

Key facts. Linear pair ⇒\Rightarrow angles are supplementary (180∘180^\circ). Vertical angles ⇒\Rightarrow angles are congruent (equal). These two rules solve most crossing-line puzzles!

Finding Unknown Angles with Algebra

Concept
Set up an equation
55°

When an angle relationship gives you a total (90∘90^\circ, 180∘180^\circ, or a bisected angle), write an equation, then solve for the variable. Finally, substitute back to find each actual angle measure.

An angle measures a turn.

Example
Complementary angles with algebra

Two complementary angles measure xx and 2x2x. Find each angle.

x+2x=903x=90x=30\begin{aligned} x + 2x &= 90 \\ 3x &= 90 \\ x &= 30 \end{aligned}

So the angles are x=30∘x = 30^\circ and 2x=60∘2x = 60^\circ. Check: 30∘+60∘=90∘30^\circ + 60^\circ = 90^\circ. ✓

Example
Vertical angles with algebra

Two vertical angles measure (3x+10)∘(3x+10)^\circ and (x+50)∘(x+50)^\circ. Because vertical angles are congruent, set them equal:

3x+10=x+502x=40x=20\begin{aligned} 3x + 10 &= x + 50 \\ 2x &= 40 \\ x &= 20 \end{aligned}

Each angle measures 3(20)+10=70∘3(20)+10 = 70^\circ (and 20+50=70∘20 + 50 = 70^\circ). ✓

Tip

Always check! After solving, plug your answer back in. Complementary angles should total 90∘90^\circ, supplementary and linear pairs should total 180∘180^\circ, and vertical angles should come out equal.

Going Deeper: Advanced Angle Ideas

Concept
Angle chasing: work one step at a time
55°

Angle chasing means finding an unknown angle by filling in other angles one at a time, using the rules you already know, until the one you want is forced. At every step, ask which single rule turns something you know into something new:

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  • angles on a straight line add to 180∘180^\circ (linear pair);
  • angles around a point add to 360∘360^\circ;
  • vertical angles are equal;
  • a bisector splits an angle into two equal halves.

Mark each angle on the figure as soon as you find it --- one known angle usually unlocks the next.

An angle measures a turn.

Concept
Two counting rules that act as constraints

These two facts are the “budget” every angle problem must respect:

along a line:   ∑angles=180∘,around a point:   ∑angles=360∘.\text{along a line: } \; \sum \text{angles} = 180^\circ, \qquad \text{around a point: } \; \sum \text{angles} = 360^\circ .

If three rays leave a single point OO and go all the way around, the three angles between neighboring rays must total 360∘360^\circ. If several angles sit in a row on one side of a straight line, they must total 180∘180^\circ. Turning a picture into one of these equations is the whole game.

Example
Angle chasing around a point

Four rays leave point OO. Going around, the angles between neighbors are 90∘90^\circ, xx, 130∘130^\circ, and 2x2x. Find xx and the two unknown angles.

Everything around a point must add to 360∘360^\circ:

90+x+130+2x=3603x+220=3603x=140x=1403≈46.7.\begin{aligned} 90 + x + 130 + 2x &= 360 \\ 3x + 220 &= 360 \\ 3x &= 140 \\ x &= \tfrac{140}{3} \approx 46.7 . \end{aligned}

So x≈46.7∘x \approx 46.7^\circ and 2x≈93.3∘2x \approx 93.3^\circ. Check: 90+46.7+130+93.3=360∘90 + 46.7 + 130 + 93.3 = 360^\circ. ✓

Concept
The Exterior Angle Theorem

Extend one side of a triangle to make an exterior angle. That exterior angle equals the sum of the two remote (non-adjacent) interior angles:

m∠ext=m∠A+m∠B.m\angle \text{ext} = m\angle A + m\angle B .

This follows from two facts working together: the three interior angles add to 180∘180^\circ, and the exterior angle forms a linear pair (180∘180^\circ) with the interior angle beside it. Subtracting the shared interior angle from both leaves exactly the two remote angles.

Example
Using the exterior angle theorem

In the triangle below, the two remote interior angles are 50∘50^\circ and 65∘65^\circ, and ∠4\angle 4 is an exterior angle. Instead of finding the third interior angle first, add the remote angles directly:

m∠4=50∘+65∘=115∘.m\angle 4 = 50^\circ + 65^\circ = 115^\circ .

Check the long way: the interior angle beside ∠4\angle 4 is 180−115=65∘180 - 115 = 65^\circ, and 50+65+65=180∘50 + 65 + 65 = 180^\circ. ✓

Concept
Bisectors and the angle between them

Angle bisectors set up equal pieces you can chain together. A useful pattern: if two rays bisect neighboring angles that sit on a straight line, the angle between the two bisectors is always 90∘90^\circ. Reason: the four half-angles come in two equal pairs, and all four together fill the 180∘180^\circ line, so the middle two halves total exactly half of 180∘180^\circ. Whenever a problem mentions a bisector, replace the whole angle by “two equal halves” and the algebra usually falls out.

Example
A bisector condition with algebra

OB→\overrightarrow{OB} bisects ∠AOC\angle AOC. The two halves are given as m∠AOB=(4x−5)∘m\angle AOB = (4x - 5)^\circ and m∠BOC=(2x+15)∘m\angle BOC = (2x + 15)^\circ. Because a bisector makes the halves equal, set them equal:

4x−5=2x+152x=20x=10.\begin{aligned} 4x - 5 &= 2x + 15 \\ 2x &= 20 \\ x &= 10 . \end{aligned}

Each half is 4(10)−5=35∘4(10) - 5 = 35^\circ, so the whole angle is m∠AOC=70∘m\angle AOC = 70^\circ. ✓

Concept
Clock hands as angles

A clock face is a circle of 360∘360^\circ split into 1212 hours, so each hour mark is 360/12=30∘360 / 12 = 30^\circ apart. The two hands move at different, steady speeds:

minute hand: 360∘60 min=6∘ per minute,hour hand: 30∘60 min=0.5∘ per minute.\text{minute hand: } \tfrac{360^\circ}{60\text{ min}} = 6^\circ \text{ per minute}, \qquad \text{hour hand: } \tfrac{30^\circ}{60\text{ min}} = 0.5^\circ \text{ per minute}.

The minute hand gains on the hour hand at 6−0.5=5.5∘6 - 0.5 = 5.5^\circ per minute. To find the angle between the hands at a given time, compute each hand's position from the top (1212) and subtract. If the gap is more than 180∘180^\circ, use 360∘360^\circ minus that gap for the smaller angle.

Example
The angle at 3:40, and when the hands overlap

Part 1 --- angle at 3:40. Measure each hand clockwise from the 1212:

minute hand=40×6=240∘,hour hand=3×30+40×0.5=90+20=110∘.\begin{aligned} \text{minute hand} &= 40 \times 6 = 240^\circ, \\ \text{hour hand} &= 3 \times 30 + 40 \times 0.5 = 90 + 20 = 110^\circ. \end{aligned}

The gap is 240−110=130∘240 - 110 = 130^\circ, which is already less than 180∘180^\circ, so the angle between the hands is 130∘\mathbf{130^\circ}.

Part 2 --- when do the hands overlap after 12:00? At 12:0012{:}00 both hands sit at 0∘0^\circ. The minute hand closes the gap at 5.5∘5.5^\circ per minute, and it must make up a full 360∘360^\circ to lap the hour hand and line up again:

t=3605.5=36011/2=72011≈65.45 minutes.t = \frac{360}{5.5} = \frac{360}{11/2} = \frac{720}{11} \approx 65.45 \text{ minutes}.

So the hands first overlap again at about 1:055111{:}05\tfrac{5}{11}. In general, consecutive overlaps are exactly 72011=3605.5\dfrac{720}{11} = \dfrac{360}{5.5} minutes apart, which is why the hands line up only 1111 times every 1212 hours --- not 1212.

Example
An angle condition that leads to a quadratic

An angle measures xx degrees. Its supplement is numerically equal to the square of its complement divided by 1010. Find xx.

The complement is (90−x)(90 - x) and the supplement is (180−x)(180 - x), so

180−x=(90−x)210.180 - x = \frac{(90 - x)^2}{10}.

Multiply through by 1010 and expand:

1800−10x=(90−x)21800−10x=8100−180x+x20=x2−170x+6300.\begin{aligned} 1800 - 10x &= (90 - x)^2 \\ 1800 - 10x &= 8100 - 180x + x^2 \\ 0 &= x^2 - 170x + 6300 . \end{aligned}

Factor (or use the quadratic formula): x2−170x+6300=(x−50)(x−120)x^2 - 170x + 6300 = (x - 50)(x - 120), so

x=50orx=120.x = 50 \qquad \text{or} \qquad x = 120 .

Reject the extra root. A complement only exists for an angle less than 90∘90^\circ, and x=120x = 120 has no complement (and a negative 90−x90 - x). So we discard x=120x = 120 and keep x=50∘\boxed{x = 50^\circ}. Check: complement =40= 40, supplement =130= 130, and 402/10=1600/10=13040^2 / 10 = 1600/10 = 130. ✓

Reminder — The quadratic formula:  x=−b±b2−4ac2a(a≠0)x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\qquad (a\neq 0)
Tip

Advanced checklist. (1) Chase angles one rule at a time and mark each result on the figure. (2) Treat “on a line =180∘= 180^\circ” and “around a point =360∘= 360^\circ” as equations. (3) An exterior angle equals the two remote interior angles. (4) Replace any bisected angle by two equal halves. (5) For clocks, use 6∘6^\circ/min and 0.5∘0.5^\circ/min, with overlaps every 72011\tfrac{720}{11} minutes. (6) When algebra gives two roots, keep only the one that makes every angle valid (complements need angles below 90∘90^\circ; all measures must be positive).

Formulas, Proofs & Tips

Tip
Triangle angle sum and the exterior angle
A+B+C=180∘,exterior=sum of the two remote interior anglesA+B+C=180^\circ, \qquad \text{exterior} = \text{sum of the two remote interior angles}

What it means. Every triangle's angles total a straight angle; an exterior angle equals the two angles it is not next to.

Example. Angles 50∘50^\circ and 60∘60^\circ give a third of 70∘70^\circ; the exterior angle there is 110∘110^\circ.

Why it works. Draw a line through one vertex parallel to the opposite side. The two alternate interior angles equal the other two triangle angles, and together with the third they form a straight line — 180∘180^\circ. The exterior result follows since exterior =180∘−adjacent=180^\circ-\text{adjacent}, and the other two also total 180∘−adjacent180^\circ-\text{adjacent}.

Tip. The exterior angle shortcut saves a step: no need to find the third angle first.

Tip
Angle pair relationships
vertical angles are equal;linear pair sums to 180∘;complements sum to 90∘\text{vertical angles are equal};\quad \text{linear pair sums to }180^\circ;\quad \text{complements sum to }90^\circ

What it means. Angles formed by crossing lines come in predictable pairs.

Example. If two angles are supplementary and one is 110∘110^\circ, the other is 70∘70^\circ.

Why it works. Two angles on a straight line total 180∘180^\circ. If ∠1\angle 1 and ∠2\angle 2 form a line, and ∠2\angle 2 and ∠3\angle 3 also form a line, then both equal 180∘−∠2180^\circ-\angle 2 — so the vertical angles ∠1\angle 1 and ∠3\angle 3 are equal.

Tip. With parallel lines cut by a transversal, corresponding and alternate angles are equal, while same-side interior angles are supplementary.