Math 7 · 7.G.B.5 · G.CO.C.9

Adjacent angles and linear pairs

Two angles sit side by side at one corner. When is the angle they make together simply the two of them added, and what happens when their outer sides make a straight line?

You need

Complementary and supplementary anglesMath 7

Unlocks

Vertical anglesMath 7

Where you are 8 in Angle types and pairs

Geometry11

Points, lines and planes8

Angle types and pairs8

▴ 3 above Complementary and supplementary angles Perpendicular lines

Adjacent angles and linear pairs

Vertical angles Finding unknown angle measures

Why this exists

Measuring an angle means lining up a protractor, and a figure hands you most of its angles without a protractor anywhere near it. What makes that possible is a very ordinary situation: two angles sitting next to each other at one corner, like two slices cut side by side from the same pizza. When two angles sit like that, the bigger angle they make together is just the two of them added. Know two of the three and you get the third by adding or subtracting. But it only works when the angles really are side by side, so the arrangement needs a name and a careful definition. The name is adjacent.

One arrangement of adjacent angles comes up more than any other. A straight line is the one angle you never have to measure: its two halves point in opposite directions, so the turn from one to the other is half of a full turn, 180^\circ, every time. Let a single ray leave a point on that line. The straight angle is cut into two adjacent pieces, and you still know their total. So you only ever need to measure one of them; the other is whatever is left of 180^\circ. That pair, a linear pair, turns up wherever a line meets anything at all: under every crossing of two lines, beside every corner of a triangle when a side is extended, and at each point where a transversal cuts a line. Nearly every missing-angle problem, from a single crossing to a whole figure, is these two facts used again and again.

Definition

Adjacent angles

Two angles are adjacent when they lie in one plane and

they share a vertex,
they share a side, and
they do not overlap: no point lies inside both of them.

The side they share is called the common side. Each angle also has a side of its own, and these two are the outer sides. The common side sits between them.

Figure 1\angle 1 and \angle 2 share the vertex B and the side \overrightarrow{BD}, and they do not overlap. They are adjacent.

In Figure 1, \angle 1 is \angle ABD and \angle 2 is \angle DBC. They share the vertex B and the side \overrightarrow{BD}, and \angle 1 lies on one side of \overrightarrow{BD} while \angle 2 lies on the other, so they do not overlap. They are adjacent. When you name an angle by three points, an adjacent pair is easy to spot: the vertex letter is the same, and one of the outer letters is the same.

All three parts of the definition matter. Figure 2 shows a pair that fails each one.

Figure 2Three near misses. Each pair has one or two of the three features, never all three.

On the left the angles share a vertex but no side. In the middle they share a vertex and a side, but the small angle lies inside the large one, so they overlap. On the right they share a side, \overline{PQ}, but one angle has its vertex at P and the other at Q.

Linear pairs

Two angles form a linear pair when they are adjacent and their outer sides are opposite rays. Put plainly: the two angles sit side by side, sharing one side, and the two sides they do not share point in opposite directions along one straight line.

In Figure 3, A, B and C lie on one line with B between them, and D is off the line. The angles \angle ABD and \angle DBC share the vertex B and the side \overrightarrow{BD}, and their other sides, \overrightarrow{BA} and \overrightarrow{BC}, together make the line \overleftrightarrow{AC}. So \angle 1 and \angle 2 form a linear pair.

Figure 3\angle 1 and \angle 2 share the side \overrightarrow{BD}, and their other sides make the line \overleftrightarrow{AC}. They are a linear pair.

Both parts of this definition matter too. Figure 4 shows what happens when either one is missing.

Figure 4Neither is a linear pair. In (a) the angles share a side but the outer sides are not one line; in (b) the angles add to 180^\circ but share nothing.

In (a) the two angles are adjacent, but their outer sides do not line up: continue one of them through the vertex (the dashed ray) and the other misses it. In (b) the angles add to 180^\circ, so they are supplementary, but they sit apart and share no side. Neither is a linear pair.

Intuition

Stand at the vertex and look along one outer side. Now turn until you face along the common side: you have turned through the first angle. Keep turning in the same direction until you face along the other outer side: you have turned through the second angle as well. Two turns, one after the other, in the same direction, make one bigger turn, and its size is the two sizes added.

The “no overlap” rule is what keeps the turn going in one direction. If the second angle folded back over the first, you would turn forward and then partly back again, and the result would be a difference, not a sum. That is exactly what happens in the middle panel of Figure 2.

Now make the outer sides a straight line, as in Figure 3, and swing the common side \overrightarrow{BD} slowly to the right. \angle 2 closes and \angle 1 opens, and every degree that leaves one arrives in the other, because the two of them are always filling the same half of the picture above the line. Nothing leaks out and nothing comes in, so the total never changes from 180^\circ. When the ray stands straight up, the two angles are equal and each is 90^\circ. Push it almost flat and one angle is a sliver while the other is almost the whole 180^\circ. Between those, every split is possible, and every one of them adds to the same amount.

Results

Adjacent angles add. If \angle ABD and \angle DBC are adjacent and their measures add to 180^\circ or less, then together they make \angle ABC, and

m\angle ABC = m\angle ABD + m\angle DBC

Read the other way round, this is the angle addition postulate: a ray \overrightarrow{BD} drawn inside \angle ABC splits it into two adjacent angles whose measures add to the whole. Written as a subtraction, it finds one part from the whole and the other part:

m\angle DBC = m\angle ABC - m\angle ABD

Two special totals. When the outer sides form a right angle, the two adjacent angles add to 90^\circ and are complementary. When the outer sides form one straight line, the two add to 180^\circ; that is the next result.

Linear Pair Theorem. If two angles form a linear pair, they are supplementary.

m\angle 1 + m\angle 2 = 180^\circ

So if one angle of a linear pair measures t^\circ, the other measures (180 - t)^\circ.

The converse. If two adjacent angles are supplementary, their outer sides are opposite rays, so the two angles form a linear pair. This is how you prove that three points lie on one line: show that two adjacent angles at the middle point add to exactly 180^\circ.

Equal partners. If the two angles of a linear pair are equal, each measures 90^\circ, and the shared side is perpendicular to the line.

Bisectors. The bisectors of the two angles of a linear pair are always perpendicular. That result, its proof and its worked examples belong to angle bisectors, in Congruent angles, angle addition and bisectors. The same reasoning works for any two adjacent angles: each bisector takes half of its own angle, the two halves next to the common side are adjacent, so the angle between the bisectors is half the sum of the two angles.

A straight angle cut into several pieces. When two or more rays leave one point of a line on the same side, the angles they make side by side, each adjacent to the next, add to 180^\circ.

A ring of angles. Angles arranged around one point, each adjacent to the next, with no gaps and no overlaps, fill a whole turn, and their measures add to two straight angles:

m\angle 1 + m\angle 2 + \cdots + m\angle n = 360^\circ

where \angle 1, \angle 2, \ldots, \angle n are the angles in the ring.

Why it is true

Why adjacent angles add

Put a protractor’s center on B with its zero on \overrightarrow{BA}, facing the side where D lies. The protractor reads every ray on that side as a number from 0 to 180.

\overrightarrow{BD} crosses it at the reading m\angle ABD. Since \angle DBC is adjacent to \angle ABD, it opens on the far side of \overrightarrow{BD} from A, so \overrightarrow{BC} lies further round the scale, by exactly m\angle DBC. Its reading is

m\angle ABD + m\angle DBC

as long as that number has not gone past 180, the end of the scale. The reading of \overrightarrow{BC} is, by the way a protractor works, m\angle ABC. So m\angle ABC = m\angle ABD + m\angle DBC.

Two things in this argument are doing real work. The non-overlap condition is what puts \overrightarrow{BC} further round the scale and not back toward A. And the requirement that the sum stays at most 180 is what keeps \overrightarrow{BC} on the scale at all; the Boundaries section below shows what happens when it does not.

In a formal treatment of geometry this fact is not proved from anything simpler. It is taken as a postulate, the angle addition postulate, because it is the rule that says what “measure” means when angles are put together. The protractor picture is the reason the postulate is the right one to take.

Why a linear pair is supplementary

The whole argument is one fact about straight lines and one fact about adding angles.

Because \overrightarrow{BA} and \overrightarrow{BC} are opposite rays, \angle ABC is a straight angle, and a straight angle measures 180^\circ.

The ray \overrightarrow{BD} lies on one side of that line, inside the straight angle, so it splits \angle ABC into \angle ABD and \angle DBC. The angle addition postulate says the two parts add to the whole:

m\angle ABD + m\angle DBC = m\angle ABC = 180^\circ

That is the theorem. Some geometry courses take it as a postulate instead of proving it, under the name Linear Pair Postulate; either way it is the starting point for the angle facts that follow. It is Proposition 13 of Book I of Euclid’s Elements, which says that a line standing on a line makes two right angles or angles together equal to two right angles.

Why the pieces of a straight angle add to 180^\circ

Take two rays, \overrightarrow{BD} and \overrightarrow{BE}, leaving B on the same side of the line \overleftrightarrow{AC}, in the order A, D, E, C going round, so that \angle ABD, \angle DBE and \angle EBC sit side by side. The ray \overrightarrow{BE} lies inside the straight angle \angle ABC, so angle addition gives

m\angle ABE + m\angle EBC = 180^\circ

The ray \overrightarrow{BD} lies inside \angle ABE, so angle addition, used a second time, gives m\angle ABD + m\angle DBE = m\angle ABE. Put that in place of m\angle ABE:

m\angle ABD + m\angle DBE + m\angle EBC = 180^\circ

Each further ray costs one more use of the postulate in the same way, so any number of pieces that fill a straight angle side by side, each adjacent to the next, add to 180^\circ.

Why a ring of angles adds to 360^\circ

The only total the argument may use is the 180^\circ of a straight angle, so the 360^\circ of a whole turn has to be built from straight angles, not assumed. Let the ring’s angles go round the point O, each adjacent to the next, with no gaps and no overlaps. Each of them, like every angle named by three letters, is at most 180^\circ.

Pick one of the ring’s rays, \overrightarrow{OP}, and extend it backwards through O to the opposite ray \overrightarrow{OQ}. The line \overleftrightarrow{PQ} splits the picture into two sides, and on each side the angle from \overrightarrow{OP} round to \overrightarrow{OQ} is a straight angle.

Where does \overrightarrow{OQ} fall? Because the ring leaves no gaps, it lies either along one of the ring’s rays or inside one of the ring’s angles. In the first case, leave the ring as it is. In the second, \overrightarrow{OQ} splits that angle into two adjacent parts, and by angle addition the two parts add to exactly the angle they came from. So replacing the angle by its two parts does not change the ring’s total.

Now no piece crosses the line. A piece of at most 180^\circ with points on both sides of \overleftrightarrow{PQ} would have \overrightarrow{OP} or \overrightarrow{OQ} running through its inside. \overrightarrow{OQ} is now a side of two pieces, not inside any piece. And \overrightarrow{OP} is a side of two of the ring’s angles, so if it ran through the inside of a third, that third angle would overlap them, which the ring does not allow.

So the pieces fall into two groups. On one side of the line they run in order from \overrightarrow{OP} to \overrightarrow{OQ}, each adjacent to the next, with no gaps: they fill a straight angle and add to 180^\circ by the result above. On the other side they run from \overrightarrow{OQ} back to \overrightarrow{OP} and also add to 180^\circ. The whole ring therefore adds to

180^\circ + 180^\circ = 360^\circ

That is where the 360^\circ of a full turn comes from: it is two straight angles, and the proof used nothing but angle addition and the measure of a straight angle.

Why the converse holds

The converse, Euclid’s Proposition 14, runs the other way. Suppose \angle ABD and \angle DBC are adjacent and add to 180^\circ, but you do not yet know that A, B and C are on one line. Extend \overrightarrow{BA} backwards through B to a ray \overrightarrow{BC'}. Now \angle ABD and \angle DBC' are a linear pair, so m\angle DBC' = 180^\circ - m\angle ABD, which is exactly m\angle DBC. Both C and C' are on the side of \overleftrightarrow{BD} away from A: C' because it lies past B from A on one line, and C because the two angles are adjacent and do not overlap. On one side of a line there is only one ray making a given angle with \overrightarrow{BD}. So \overrightarrow{BC} and \overrightarrow{BC'} are the same ray, and C is on the line through A and B.

Worked examples

Splitting one angle into two parts and finding a part, with numbers or with expressions in x, is worked through in Congruent angles, angle addition and bisectors. The examples here are about recognizing adjacent angles and about the totals they make: a right angle, a straight angle, and a whole turn.

Example 1

Which pairs are adjacent

Use Figure 5.

Figure 5Four rays from one point give six angles. Only some pairs of them are adjacent. Also used in Exercises 1 to 4.

(a) For each pair, decide whether the two angles are adjacent: \angle AOB and \angle BOC; \angle AOC and \angle COD; \angle AOB and \angle COD; \angle AOC and \angle AOD.

(b) Find m\angle AOC, m\angle BOD and m\angle AOD.

Solution

(a) Test each pair against the three conditions.

\angle AOB and \angle BOC share the vertex O and the side \overrightarrow{OB}, and they lie on opposite sides of it. Adjacent.

\angle AOC and \angle COD share O and \overrightarrow{OC}, and lie on opposite sides of it. Adjacent, even though \angle AOC has another ray, \overrightarrow{OB}, running through its middle. What matters is how the two angles meet, not what is inside each one.

\angle AOB and \angle COD share the vertex but no side. Not adjacent.

\angle AOC and \angle AOD share O and \overrightarrow{OA}, but \angle AOC lies entirely inside \angle AOD. They overlap. Not adjacent.

(b) Each angle asked for is made of adjacent pieces whose total stays under 180^\circ, so add the pieces:

m\angle AOC = 30^\circ + 45^\circ = 75^\circ

m\angle BOD = 45^\circ + 60^\circ = 105^\circ

m\angle AOD = 30^\circ + 45^\circ + 60^\circ = 135^\circ

For \angle AOD you can also use the adjacent pair from part (a): m\angle AOC + m\angle COD = 75^\circ + 60^\circ = 135^\circ, which agrees.

Example 2

Two adjacent angles in a right angle

\angle ABC is a right angle and \overrightarrow{BD} lies inside it. m\angle ABD = (2y - 6)^\circ and m\angle DBC = (y + 15)^\circ. Find y and both angles.

Solution

The two angles are adjacent and together make the right angle, so they add to 90^\circ:

\begin{aligned} (2y - 6) + (y + 15) &= 90 \\ 3y + 9 &= 90 \\ 3y &= 81 \\ y &= 27 \end{aligned}

So m\angle ABD = 2(27) - 6 = 48^\circ and m\angle DBC = 27 + 15 = 42^\circ, and 48 + 42 = 90. A pair like this, adjacent and adding to 90^\circ, is an adjacent pair of complementary angles.

Example 3

Finding the partner in a linear pair

In Figure 6 a ray leaves a point on a straight line. One angle measures 58^\circ. Find x.

Figure 6One angle of a linear pair is given. Find the other.

Solution

The two angles share the ray, and their other sides run in opposite directions along the line, so they form a linear pair. A linear pair adds to 180^\circ:

\begin{aligned} x + 58 &= 180 \\ x &= 122 \end{aligned}

The angle marked x^\circ measures 122^\circ. It is obtuse, which matches the picture: the ray leans to the right, leaving the wider opening on the left.

Example 4

Linear pair or not?

Decide whether each pair of angles is a linear pair.

(a) \angle 1 measures 110^\circ and \angle 2 measures 70^\circ. \angle 1 is an angle of one triangle and \angle 2 an angle of another.

(b) \angle PQR and \angle RQS are adjacent, with m\angle PQR = 100^\circ and m\angle RQS = 70^\circ.

Solution

(a) No. The measures add to 110^\circ + 70^\circ = 180^\circ, so the angles are supplementary, but they have different vertices, so they are not adjacent. Adding to 180^\circ is not enough.

(b) No. They are adjacent, but 100^\circ + 70^\circ = 170^\circ. A linear pair always adds to 180^\circ, so the outer sides \overrightarrow{QP} and \overrightarrow{QS} cannot be opposite rays. They miss a straight line by 10^\circ.

(c) Yes. The angles are adjacent, and since K is between J and M on one line, the outer sides \overrightarrow{KJ} and \overrightarrow{KM} are opposite rays. Both parts of the definition hold, so m\angle JKL + m\angle LKM = 180^\circ without either measure being given.

Example 5

Solving for a variable in a linear pair

The angles of a linear pair measure (3x + 12)^\circ and (5x - 8)^\circ. Find x and both angles.

Solution

The angles form a linear pair, so their measures add to 180. Write that as an equation and solve:

\begin{aligned} (3x + 12) + (5x - 8) &= 180 \\ 8x + 4 &= 180 \\ 8x &= 176 \\ x &= 22 \end{aligned}

Substitute back into each expression: 3(22) + 12 = 78 and 5(22) - 8 = 102. The angles measure 78^\circ and 102^\circ.

Check. 78 + 102 = 180. Also check that both measures are positive; an expression that came out negative or zero would mean no such figure exists.

Example 6

Angles in a given ratio

The two angles of a linear pair are in the ratio 2 : 7. Find both.

Solution

A ratio of 2 : 7 means the angles are 2k and 7k degrees for some positive number k: the same unit, taken 2 times and 7 times. They form a linear pair, so

\begin{aligned} 2k + 7k &= 180 \\ 9k &= 180 \\ k &= 20 \end{aligned}

The angles are 2(20) = 40 degrees and 7(20) = 140 degrees, that is, 40^\circ and 140^\circ.

Check. 40 + 140 = 180, and 40 : 140 reduces to 2 : 7 on dividing both by 20.

Example 7

A straight angle cut into three

In Figure 7 two rays leave the same point on a straight line. Find x and all three angles.

Figure 7Two rays split a straight angle into three adjacent angles.

Solution

The three angles sit side by side, each adjacent to the next, and together fill the half above the line, which is a straight angle. Applying the angle addition postulate twice, the three measures add to 180^\circ, just as the two angles of a linear pair do:

\begin{aligned} x + (2x + 15) + (x + 25) &= 180 \\ 4x + 40 &= 180 \\ 4x &= 140 \\ x &= 35 \end{aligned}

The angles are x = 35^\circ, 2(35) + 15 = 85^\circ, and 35 + 25 = 60^\circ.

Check. 35 + 85 + 60 = 180. You can also see a linear pair hiding in the figure: the 35^\circ angle and the two angles to its left, taken together as one angle of 145^\circ, share a ray and fill the straight angle, and 35 + 145 = 180.

Example 8

Angles around a point

Find x in Figure 8, and the measure of every angle.

Figure 8Four angles, each adjacent to the next, closing up around a point.

Solution

The four angles go once around the point. Each is adjacent to the next, and together they leave no gap and never overlap, so they form a ring and, by the result proved above, add to 360^\circ:

\begin{aligned} 100 + 110 + x + 2x &= 360 \\ 210 + 3x &= 360 \\ 3x &= 150 \\ x &= 50 \end{aligned}

The angles are 100^\circ, 110^\circ, 50^\circ and 2(50) = 100^\circ.

Check. 100 + 110 + 50 + 100 = 360.

Boundaries

Adjacent is a fact about position, not about size. Being adjacent tells you that the measures add to the angle between the outer sides. It does not tell you that the two angles are equal, or complementary, or supplementary. For a number you need one more fact: the size of the whole, a right angle, or a straight line.

Past 180^\circ, the outer sides make a different angle. In Figure 9 the adjacent angles measure 140^\circ and 100^\circ. Their outer sides \overrightarrow{OA} and \overrightarrow{OC} do make an angle, but it is the one on the other side:

m\angle AOC = 360^\circ - (140^\circ + 100^\circ) = 120^\circ

Figure 9Adjacent angles of 140^\circ and 100^\circ. The angle between the outer sides is 120^\circ, not 240^\circ.

An angle named with three letters, such as \angle AOC, is the smaller opening between its two sides, never more than 180^\circ. When a sum comes to exactly 180^\circ the outer sides form a straight line, and the pair is a linear pair.

Overlapping angles subtract, they do not add. Two angles that share a vertex and a side but lie one inside the other, like \angle AOC and \angle AOD in Figure 5, are not adjacent. The measure between their other sides is the difference of the two.

Sharing a side is not enough. Two angles with a common side but different vertices, like those at the ends of \overline{PQ} in Figure 2, are not adjacent and their measures are unrelated.

Supplementary is not the same as a linear pair. Every linear pair is supplementary, but two supplementary angles form a linear pair only if they are also adjacent. Two angles of 120^\circ and 60^\circ drawn in different places add to 180^\circ and are not a linear pair.

The outer sides must really be one line. In a figure, a line that looks straight is not evidence. You need to be told that the three points are collinear, or to prove it (for example by the converse, from two adjacent angles adding to 180^\circ). A drawing where the outer sides miss by one degree looks identical and gives a sum of 179^\circ.

A linear pair is two angles, not more. When two or more rays split a straight angle, the pieces still add to 180^\circ by the same argument, but the name “linear pair” belongs only to two neighboring angles whose outer sides make the line.

Each angle of a linear pair is strictly between 0^\circ and 180^\circ. The shared ray is off the line, so neither angle is zero and neither is straight. In an algebra problem, a value of the variable that makes either angle zero or negative gives no figure and is rejected, even if it solves the equation.

One plane. At the corner of a box, the angle on the floor between two edges and the angle on a wall between two edges share a vertex and an edge, and each is 90^\circ. They do not lie in one plane, and the angle between their outer edges is 90^\circ, not 180^\circ. Adjacent angles, and the rule that they add, belong to a single plane.

The 180 is in degrees. Measured in radians, a straight angle is \pi, and the linear pair fact reads m\angle 1 + m\angle 2 = \pi.

Where this goes next

Adjacent pairs and linear pairs are the smallest pieces of angle chasing, and they are inside most of what comes after. Two linear pairs sharing an angle give the Vertical Angles Theorem: the angles opposite each other at a crossing are both supplementary to the same neighbor, so they are equal. The same move, used on any two angles with equal supplements, is the general fact that supplements of congruent angles are congruent. When the outer sides form a right angle instead, the adjacent pair is complementary.

Extend one side of a triangle and the new angle forms a linear pair with the interior angle beside it. Put that together with the sum of a triangle’s angles and you get the exterior angle theorem; the classic picture of the angle sum itself tears off the three corners and lays them side by side, each adjacent to the next, along one line. When a transversal crosses two parallel lines, linear pairs at each crossing are what carry one known angle to all eight, and they are why same side interior angles add to 180^\circ. Every chase through a whole figure, in finding unknown angle measures, is a sequence of adjacent pairs: a whole split into parts, or parts put together into a whole.

Exercises

Recognizing adjacent angles and linear pairs

Use Figure 5 for Exercises 1 to 4.

Are \angle BOC and \angle COD adjacent? Explain.
Are \angle AOB and \angle AOC adjacent? Explain.
Are \angle AOB and \angle BOD adjacent? Explain.
Name every angle in Figure 5 that is adjacent to \angle BOC.
True or false: any two angles that share a side are adjacent. Explain.

Use Figure 10 for Exercises 6 to 8. The points P, Q and R lie on one line.

Figure 10For Exercises 6 to 8, 14, 15 and 37.

Name two linear pairs in Figure 10.
Do \angle SQR and \angle RQT form a linear pair? Explain.
Name the angle that forms a linear pair with \angle PQT and has \overrightarrow{QT} as its shared side.

Finding a measure

Plain angle addition, a ray inside an angle with two of the three measures given, is practiced in Congruent angles, angle addition and bisectors. Here the whole is a right angle or a straight angle. In Exercises 9 and 10, \overrightarrow{BD} lies inside \angle ABC.

\angle ABC is a right angle and m\angle DBC = 27^\circ. Find m\angle ABD.
A, B and C lie on one straight line, and m\angle ABD = 73^\circ. Find m\angle DBC.

In Exercises 11 to 13, one angle of a linear pair has the measure given. Find the other.

47^\circ
90^\circ
138.5^\circ
In Figure 10, find m\angle PQS.
In Figure 10, find m\angle TQR.

Using algebra

In Exercises 16 and 17, \overrightarrow{BD} lies inside \angle ABC.

\angle ABC is a right angle, m\angle ABD = (x + 14)^\circ and m\angle DBC = (2x + 1)^\circ. Find x and both parts.
m\angle ABD and m\angle DBC are in the ratio 4 : 5, and m\angle ABC = 126^\circ. Find both parts.
The angles of a linear pair measure (2x + 30)^\circ and (4x)^\circ. Find x and both angles.
The angles of a linear pair measure (5x - 14)^\circ and (3x + 26)^\circ. Find x and both angles.
One angle of a linear pair is four times the other. Find both.
One angle of a linear pair is 24^\circ more than twice the other. Find both.
The angles of a linear pair measure (x^2)^\circ and (3x)^\circ. Find x and both angles.
The angles of a linear pair are in the ratio 5 : 4. Find both.

Several angles at one point

Two rays leave one point on a line and split the half on one side into three angles. Two of them measure 42^\circ and 73^\circ. Find the third.
Two rays split a straight angle into three angles in the ratio 1 : 2 : 3. Find all three.
Three angles fill a straight angle and measure (y + 10)^\circ, (2y)^\circ and (3y - 10)^\circ. Find y and all three angles.
Five angles go once around a point, each adjacent to the next, with no gaps. Four of them measure 70^\circ, 85^\circ, 40^\circ and 90^\circ. Find the fifth.
\angle AOB and \angle BOC are adjacent, with m\angle AOB = 150^\circ and m\angle BOC = 130^\circ. Find m\angle AOC.

Reasoning

The two angles of a linear pair are congruent. Show that each is a right angle, and say what that tells you about the shared side and the line.
Can both angles of a linear pair be acute? Explain.
Can two adjacent angles both be obtuse? If they can, what is the angle between their outer sides?
\angle 1 and \angle 2 form a linear pair, and \angle 2 and \angle 3 form a linear pair. What must be true about \angle 1 and \angle 3? Explain.
\angle A measures 125^\circ and \angle B measures 55^\circ. Must they form a linear pair? Explain.
Is the statement “adjacent angles are supplementary” always, sometimes or never true? Give an example to support your answer.
Explain why two vertical angles are never adjacent.
\angle ABD and \angle DBC are adjacent and together make a right angle. Show that the angle between their bisectors is 45^\circ.
In Figure 10, do S, Q and T lie on one line? Use the measures to decide, not the look of the drawing.

Answers

Yes. They share the vertex O and the side \overrightarrow{OC}, and they lie on opposite sides of it, so they do not overlap.
No. They share O and \overrightarrow{OA}, but \angle AOB lies inside \angle AOC, so they overlap.
Yes. They share O and \overrightarrow{OB}, with \angle AOB on one side of it and \angle BOD on the other.
\angle AOB and \angle COD. The only other angles with a side \overrightarrow{OB} or \overrightarrow{OC} are \angle BOD and \angle AOC, and each of those contains \angle BOC, so they overlap it.
False. The angles must also share a vertex and must not overlap. The pair at the ends of \overline{PQ} in Figure 2 shares a side and is not adjacent.
\angle PQS and \angle SQR, and \angle PQT and \angle TQR. Each pair shares a side, and its outer sides are \overrightarrow{QP} and \overrightarrow{QR}, which are opposite rays.
No. They are adjacent, sharing \overrightarrow{QR}, but their outer sides are \overrightarrow{QS} and \overrightarrow{QT}, and nothing says those are opposite rays. Exercise 37 shows they are not.
\angle TQR. It shares \overrightarrow{QT} with \angle PQT, and the outer sides \overrightarrow{QP} and \overrightarrow{QR} are opposite rays.
63^\circ. The parts add to 90^\circ, so m\angle ABD = 90^\circ - 27^\circ = 63^\circ.
107^\circ. The two angles form a linear pair, so m\angle DBC = 180^\circ - 73^\circ = 107^\circ.
133^\circ, since 180^\circ - 47^\circ = 133^\circ.
90^\circ, since 180^\circ - 90^\circ = 90^\circ. The ray is perpendicular to the line.
41.5^\circ, since 180^\circ - 138.5^\circ = 41.5^\circ.
m\angle PQS = 115^\circ. It forms a linear pair with the 65^\circ angle \angle SQR, so it is 180^\circ - 65^\circ.
m\angle TQR = 110^\circ. It forms a linear pair with the 70^\circ angle \angle PQT, so it is 180^\circ - 70^\circ.
x = 25, with parts 39^\circ and 51^\circ. From (x + 14) + (2x + 1) = 90, 3x + 15 = 90 and 3x = 75.
m\angle ABD = 56^\circ and m\angle DBC = 70^\circ. Write the parts as 4k and 5k; then 9k = 126, so k = 14.
x = 25, and the angles are 80^\circ and 100^\circ. From (2x + 30) + 4x = 180, 6x = 150. Then 2(25) + 30 = 80 and 4(25) = 100.
x = 21, and the angles are 91^\circ and 89^\circ. From (5x - 14) + (3x + 26) = 180, 8x + 12 = 180 and 8x = 168. Then 5(21) - 14 = 91 and 3(21) + 26 = 89.
36^\circ and 144^\circ. With angles t and 4t, 5t = 180, so t = 36 and 4t = 144.
52^\circ and 128^\circ. With angles t and 2t + 24, 3t + 24 = 180, so 3t = 156 and t = 52; then 2(52) + 24 = 128.
x = 12, and the angles are 144^\circ and 36^\circ. From x^2 + 3x = 180, x^2 + 3x - 180 = 0 and (x - 12)(x + 15) = 0, so x = 12 or x = -15. But x = -15 makes the second angle 3(-15) = -45, which is not an angle, so only x = 12 works: 12^2 = 144 and 3(12) = 36.
100^\circ and 80^\circ. With angles 5k and 4k, 9k = 180 and k = 20.
65^\circ. The three angles fill a straight angle, so the third is 180^\circ - 42^\circ - 73^\circ = 65^\circ.
30^\circ, 60^\circ and 90^\circ. With angles k, 2k and 3k, 6k = 180 and k = 30.
y = 30, and the angles are 40^\circ, 60^\circ and 80^\circ. The measures add to 180: (y + 10) + 2y + (3y - 10) = 6y = 180. Then 30 + 10 = 40, 2(30) = 60 and 3(30) - 10 = 80.
75^\circ. The five fill a whole turn: 360^\circ - (70^\circ + 85^\circ + 40^\circ + 90^\circ) = 75^\circ.
80^\circ. The sum 150^\circ + 130^\circ = 280^\circ is past 180^\circ, so the outer sides make the angle on the other side: 360^\circ - 280^\circ = 80^\circ.
Call each angle t. They form a linear pair, so t + t = 180^\circ, and 2t = 180^\circ gives t = 90^\circ. Each is a right angle, so the shared side is perpendicular to the line.
No. An acute angle is less than 90^\circ, so two acute angles add to less than 90^\circ + 90^\circ = 180^\circ. A linear pair always adds to exactly 180^\circ.
Yes, for example 100^\circ and 120^\circ. Two obtuse angles always add to more than 180^\circ, so the angle between the outer sides is 360^\circ minus their sum, here 140^\circ.
m\angle 1 = m\angle 3. Both linear pairs add to 180^\circ, so m\angle 1 = 180^\circ - m\angle 2 and m\angle 3 = 180^\circ - m\angle 2. Both equal the same thing, so they equal each other. When \angle 1 and \angle 3 sit opposite each other at a crossing, this is the Vertical Angles Theorem.
No. They are supplementary, since 125^\circ + 55^\circ = 180^\circ, but nothing says they are adjacent. Only if they shared a vertex and a side, with the outer sides forming a line, would they be a linear pair.
Sometimes. Adjacent angles are supplementary exactly when their outer sides form a straight line. Adjacent angles of 30^\circ and 40^\circ are not supplementary; adjacent angles of 70^\circ and 110^\circ with outer sides along one line are.
Vertical angles share only their vertex. Each side of one is the continuation of a side of the other, pointing the opposite way, so they never share a side, and sharing a side is part of being adjacent.
Call the angles a and b, with a + b = 90. The bisectors make angles of \displaystyle \frac{a}{2} and \displaystyle \frac{b}{2} with the common side, on opposite sides of it, so they are adjacent and the angle between the bisectors is \displaystyle \frac{a}{2} + \frac{b}{2} = \frac{90}{2} = 45^\circ.
No. m\angle RQT = 180^\circ - 70^\circ = 110^\circ, so the adjacent angles \angle SQR and \angle RQT add to 65^\circ + 110^\circ = 175^\circ. If S, Q and T were on one line, those two angles would form a linear pair and add to exactly 180^\circ. They fall 5^\circ short.