Math 7 · 7.G.B.5

Complementary and supplementary angles

A square corner or a straight line is cut in two. If you know one piece, what is the other?

You need

Acute, right, obtuse and straight anglesGrade 4

Unlocks

Adjacent angles and linear pairsMath 7 Finding unknown angle measuresMath 7

Where you are 8 in Angle types and pairs

Geometry11

Points, lines and planes8

Angle types and pairs8

▴ 1 above Acute, right, obtuse and straight angles Congruent angles, angle addition and bisectors

Complementary and supplementary angles

Perpendicular lines Adjacent angles and linear pairs ▾ Show all 8 (2 more)

Why this exists

Two shapes turn up in almost every figure in geometry. One is the square corner: the corner of a page, a wall meeting a floor, the two lines of a plus sign. Each is a right angle, exactly 90^\circ. The other is the straight line, which as an angle measures 180^\circ.

Put a ray through either one and you cut it into two pieces. You do not need to measure both. The pieces have to fill the whole between them, so once you know one, the other is whatever is left of 90^\circ, or of 180^\circ. That trade happens so often, in rectangles, in right triangles, on every pair of perpendicular lines, every time a ray stands on a line or a side of a triangle is extended, that each kind of pair gets a name of its own.

The names are worth having even when the two angles are nowhere near each other. Two angles can be the two pieces of one corner or one line in spirit without ever touching on the page, and the names let you say so in one word.

Definition

Complementary angles

Two angles are complementary when their measures add to 90^\circ. Each angle is then the complement of the other. In symbols, \angle A and \angle B are complementary when

m\angle A + m\angle B = 90^\circ

Figure 1Side by side or apart, 35^\circ and 55^\circ are complementary, because 35 + 55 = 90.

The two angles do not have to touch. In Figure 1 the 35^\circ and 55^\circ angles on the left share a vertex and a side, so they are adjacent and together they fill a right angle. The pair on the right sit apart and point different ways, and they are just as complementary. The word is about the two numbers, not about where the angles are drawn.

The word itself means “the part that completes”. The complement of an angle is what it needs to become a right angle.

Supplementary angles

Two angles are supplementary when their measures add to 180^\circ. Each angle is then the supplement of the other:

m\angle A + m\angle B = 180^\circ

Again the definition is about the two measures and nothing else: the angles may share a vertex or sit on opposite sides of the page, and they may point any way at all. In Figure 2 the 130^\circ angle and the 50^\circ angle are supplementary on the left, drawn apart, and still supplementary on the right, where they have been slid together.

Figure 2Two angles need not touch to be supplementary. Slid together, they fill a straight line.

When two supplementary angles do sit side by side, sharing a vertex and a side, their outer sides make one straight line. That special arrangement is a linear pair. Every linear pair is supplementary, but supplementary angles need not be a linear pair.

The word is from the Latin supplere, to fill up. The supplement is what fills an angle up to a straight line, as the complement fills it up to a right angle.

Intuition

Stand a ray inside a right angle and swing it. As the ray turns, the angle on one side of it opens and the angle on the other side closes, and by exactly the same amount, because the corner they share never changes size. Opening one piece by 10^\circ closes the other by 10^\circ. Push the ray right over to one side and one piece shrinks to nothing while the other becomes the whole corner. Stop the ray exactly in the middle and the two pieces are equal, 45^\circ each. Every position in between gives a complementary pair, and each pair is two acute angles.

Now stand the ray on a straight line instead. It cuts the line’s 180^\circ into two parts. Call the part on the right x^\circ; the part on the left is simply everything that is left over.

Figure 3The supplement of x^\circ is whatever is left of the straight angle, (180 - x)^\circ.

So the supplement of an angle of x^\circ is always (180 - x)^\circ, just as the complement is always (90 - x)^\circ. Tilt the ray to the right and the right-hand angle shrinks while the left-hand one grows by exactly the same amount. The two are on a see-saw with a fixed total. When the ray stands straight up, the see-saw balances and both are 90^\circ.

That picture also shows which angles can be supplementary partners. A small angle on one side leaves a large one on the other: one acute, one obtuse. Only the balanced case gives two equal angles, and they are both right angles.

Results

The complement and the supplement of an angle. An angle of x^\circ with 0 < x < 90 has exactly one complement, and an angle of x^\circ with 0 < x < 180 has exactly one supplement. They measure

(90 - x)^\circ \qquad\text{and}\qquad (180 - x)^\circ

Complementary angles are both acute. If two angles are complementary, each measures more than 0^\circ and less than 90^\circ. An angle of 90^\circ or more has no complement, and an angle of 180^\circ or more has no supplement.

What kinds of angle make a supplementary pair. In a supplementary pair, either both angles are right angles, or one is acute and the other is obtuse. Two acute angles, or two obtuse angles, are never supplementary.

Complements of the same angle are equal, and so are supplements. If \angle 1 and \angle 3 are both complementary to \angle 2, or both supplementary to \angle 2, then

\angle 1 \cong \angle 3

The same holds for complements, or supplements, of two congruent angles: if \angle 2 \cong \angle 4, an angle complementary to \angle 2 and an angle complementary to \angle 4 are congruent, and likewise for supplements.

The acute angles of a right triangle are complementary. In a triangle with a right angle, the other two angles add to 90^\circ.

Supplement and complement differ by 90^\circ. For an acute angle of x^\circ, the supplement is always 90^\circ larger than the complement:

(180 - x) - (90 - x) = 90

Why it is true

The complement and the supplement. If m\angle A + m\angle B = 90^\circ and m\angle A = x^\circ, subtracting x from both sides gives m\angle B = (90 - x)^\circ, and that is the only number that works. The supplement is the same with 180 in place of 90. Every angle has a measure greater than 0^\circ, so each of a complementary pair is less than the whole 90^\circ: both are acute. And if x \ge 90, then 90 - x is zero or negative, which no angle can measure, so there is no complement at all. In the same way an angle of 180^\circ or more would need a supplement of 0^\circ or less.

The kinds of angle in a supplementary pair. If both angles were acute, each would be less than 90^\circ and the sum would be less than 180^\circ. If both were obtuse, the sum would be more than 180^\circ. So if neither angle is right, one must be acute and the other obtuse. If one angle is right, the other is 180^\circ - 90^\circ = 90^\circ, right as well.

Complements of the same angle. Look at Figure 4.

Figure 4Two right angles that overlap in \angle 2. What is left over on each side must be the same size.

\angle 1 is complementary to \angle 2, so

m\angle 1 + m\angle 2 = 90^\circ

\angle 3 is complementary to \angle 2 as well, so

m\angle 2 + m\angle 3 = 90^\circ

Both sums equal 90^\circ, so they equal each other:

m\angle 1 + m\angle 2 = m\angle 2 + m\angle 3

Subtract m\angle 2 from both sides, and m\angle 1 = m\angle 3. Nothing in this used the picture: the argument works whether or not the angles touch, and for any size of \angle 2. If the two starting angles are merely equal rather than the same angle, replace m\angle 2 in the second line with the equal measure m\angle 4 and the subtraction goes through unchanged.

Supplements of the same angle. Write 180^\circ in place of 90^\circ in every line above and nothing else changes: if \angle 1 and \angle 3 are each supplementary to \angle 2, then m\angle 1 + m\angle 2 = m\angle 3 + m\angle 2, and subtracting m\angle 2 leaves m\angle 1 = m\angle 3. Written as a formal two-column proof this is one of the first proofs about angle pairs, and it is exactly the argument that makes vertical angles equal: two vertical angles are supplements of the same neighbor.

The right triangle. The three angles of any triangle add to 180^\circ, which is the triangle angle sum. The right angle uses 90^\circ of that, so the other two share the remaining 180^\circ - 90^\circ = 90^\circ.

Supplement and complement. Subtract: (180 - x) - (90 - x) = 180 - 90 = 90. The x cancels, so the gap is the same for every acute angle.

Worked examples

Example 1

Finding complements and supplements

Find (a) the complement of 28^\circ, (b) the complement of 63.5^\circ, (c) the supplement of 58^\circ, and then the supplement of that answer, and (d) the complement and the supplement of an angle of x^\circ.

Solution

The complement is whatever is left when the angle is taken away from 90^\circ, and the supplement is whatever is left when it is taken away from 180^\circ.

(a) 90^\circ - 28^\circ = 62^\circ.

(b) 90^\circ - 63.5^\circ = 26.5^\circ. Decimal measures are fine; the rule does not care.

(c) 180^\circ - 58^\circ = 122^\circ. The supplement of 122^\circ is found the same way: 180^\circ - 122^\circ = 58^\circ. Taking the supplement twice returns the angle you started with, because the relationship is symmetric: if 58^\circ and 122^\circ add to 180^\circ, each is the supplement of the other. One is acute and the other obtuse, as the results say it must be.

(d) The same subtractions, done with a letter: the complement is (90 - x)^\circ and the supplement is (180 - x)^\circ. The first only makes sense while 0 < x < 90 and the second while 0 < x < 180, since outside those ranges the answer is not the measure of any angle.

Check. 28 + 62 = 90, 63.5 + 26.5 = 90 and 58 + 122 = 180.

Example 2

Two angles described in words

(a) Two angles are complementary. The larger is 18^\circ more than twice the smaller. Find both angles.

(b) The supplement of an angle is four times the angle. Find the angle and its supplement.

Solution

(a) Name the angle that everything else is described from. Let the smaller angle be x^\circ. Then the larger is (2x + 18)^\circ. Complementary means they add to 90^\circ:

\begin{aligned} x + (2x + 18) &= 90 \\ 3x + 18 &= 90 \\ 3x &= 72 \\ x &= 24 \end{aligned}

The smaller angle is 24^\circ and the larger is 2(24) + 18 = 66 degrees.

(b) Let the angle measure y^\circ. Its supplement is four times as large, 4y^\circ. The two are supplementary, so their measures add to 180^\circ:

\begin{aligned} y + 4y &= 180 \\ 5y &= 180 \\ y &= 36 \end{aligned}

The angle measures 36^\circ, and its supplement measures 4 \times 36^\circ = 144^\circ.

Check. 24 + 66 = 90, and 66 is 18 more than 48, which is twice 24. Also 36 + 144 = 180, and 144 = 4 \times 36.

Example 3

A right angle split into two expressions

In Figure 5, \angle ABC is a right angle and ray BD lies inside it. Find x, then find m\angle ABD and m\angle DBC.

Figure 5A right angle split into two parts written in x.

Solution

The ray splits the right angle into two parts, so the parts add to the whole, by the angle addition postulate. The whole is 90^\circ, so the two parts are complementary:

\begin{aligned} (2x + 10) + (3x - 5) &= 90 \\ 5x + 5 &= 90 \\ 5x &= 85 \\ x &= 17 \end{aligned}

Substitute back into each expression: 2(17) + 10 = 44 and 3(17) - 5 = 46. So m\angle ABD = 44^\circ and m\angle DBC = 46^\circ.

Check. 44 + 46 = 90.

Example 4

Two angles on a straight line

In Figure 6, O lies on line AB. Find x, then the measures of \angle AOC and \angle COB.

Figure 6A ray standing on a straight line splits it into two supplementary angles.

Solution

\angle AOC and \angle COB sit side by side and together fill the straight angle \angle AOB, so they form a linear pair and are supplementary:

\begin{aligned} (3x + 12) + (5x - 16) &= 180 \\ 8x - 4 &= 180 \\ 8x &= 184 \\ x &= 23 \end{aligned}

Substitute back into each expression:

\begin{aligned} m\angle AOC &= 3(23) + 12 = 81^\circ \\ m\angle COB &= 5(23) - 16 = 99^\circ \end{aligned}

Check. 81 + 99 = 180. The ray leans slightly left, so the left-hand angle should be the smaller one, and it is.

Example 5

Perpendicular lines and one more ray

In Figure 7, lines AC and BD are perpendicular at O, and ray OE lies between OA and OB with m\angle AOE = 27^\circ. Find m\angle EOB, m\angle EOC and m\angle EOD.

Figure 7Two perpendicular lines and one more ray. One measure gives the rest.

Solution

The lines are perpendicular, so \angle AOB is a right angle, and ray OE splits it into \angle AOE and \angle EOB. Those two are complementary:

m\angle EOB = 90^\circ - 27^\circ = 63^\circ

A, O and C lie on one straight line, so \angle AOE and \angle EOC sit side by side along it and form a linear pair. They are supplementary:

m\angle EOC = 180^\circ - 27^\circ = 153^\circ

\angle EOD is made of \angle EOA and the right angle \angle AOD, so

m\angle EOD = 27^\circ + 90^\circ = 117^\circ

Check. Going the other way round, \angle EOC is \angle EOB plus the right angle \angle BOC, and 63 + 90 = 153, which agrees.

Example 6

Supplement and complement in one problem

The supplement of an angle is 20^\circ more than twice its complement. Find the angle.

Solution

Let the angle measure x^\circ. Its supplement measures (180 - x)^\circ and its complement (90 - x)^\circ. Translate the sentence directly: the supplement equals twice the complement, plus 20.

\begin{aligned} 180 - x &= 2(90 - x) + 20 \\ 180 - x &= 200 - 2x \\ x &= 20 \end{aligned}

The angle measures 20^\circ.

Check. Its supplement is 160^\circ and its complement is 70^\circ, and 2(70) + 20 = 160. The supplement is also 90^\circ more than the complement, as it always is.

Example 7

Two complements, and two supplements, of one angle

(a) \angle P and \angle Q are complementary, and \angle Q and \angle R are complementary. m\angle P = (3t + 7)^\circ and m\angle R = (5t - 13)^\circ. Find t and all three angles.

(b) \angle 1 and \angle 2 are supplementary, and \angle 2 and \angle 3 are supplementary. m\angle 1 = (4y - 10)^\circ and m\angle 3 = (2y + 30)^\circ. Find y and m\angle 2.

Solution

(a) There is no number for \angle Q, so do not start there. Both \angle P and \angle R are complements of the same angle, \angle Q, and complements of the same angle are equal:

\begin{aligned} 3t + 7 &= 5t - 13 \\ 20 &= 2t \\ t &= 10 \end{aligned}

Then m\angle P = 3(10) + 7 = 37 degrees and m\angle R = 5(10) - 13 = 37 degrees, which agree as they must. \angle Q is the complement of 37^\circ, so m\angle Q = 53^\circ.

(b) The same idea with 180^\circ. \angle 1 and \angle 3 are both supplements of \angle 2, so their measures are equal, and \angle 2 is not needed to find y:

\begin{aligned} 4y - 10 &= 2y + 30 \\ 2y &= 40 \\ y &= 20 \end{aligned}

Then m\angle 1 = 4(20) - 10 = 70^\circ, and m\angle 3 = 2(20) + 30 = 70^\circ as well. \angle 2 is the supplement of a 70^\circ angle, so

m\angle 2 = 180^\circ - 70^\circ = 110^\circ

Check. 37 + 53 = 90 for both pairs in (a), and 70 + 110 = 180 for both pairs in (b). Nothing in either problem said where the angles are. The equal measures follow from the sums alone.

Example 8

The acute angles of a right triangle

Triangle ABC in Figure 8 has a right angle at C. Find x and the measures of the angles at A and B.

Figure 8The two acute angles of a right triangle, written in x.

Solution

The two acute angles of a right triangle are complementary, because the three angles add to 180^\circ and the right angle takes 90^\circ of it:

\begin{aligned} (4x + 2) + (5x - 2) &= 90 \\ 9x &= 90 \\ x &= 10 \end{aligned}

So m\angle A = 4(10) + 2 = 42 degrees and m\angle B = 5(10) - 2 = 48 degrees.

Check. 42 + 48 + 90 = 180, as the angles of a triangle must.

This is the fact that makes right triangles so useful: pick either acute angle and you already know the other one.

Example 9

The exterior angle of a triangle

In triangle ABC (Figure 9), side \overline{BC} is extended past C to a point D. The exterior angle \angle ACD measures (4x + 3)^\circ and the interior angle \angle ACB measures (3x - 5)^\circ. Find x and both angles.

Figure 9Extending a side of a triangle makes an exterior angle, the supplement of the interior angle beside it.

Solution

B, C and D lie on one straight line, and the ray \overrightarrow{CA} stands on it. So \angle ACB and \angle ACD fill the straight angle \angle BCD between them and are supplementary:

\begin{aligned} (4x + 3) + (3x - 5) &= 180 \\ 7x - 2 &= 180 \\ 7x &= 182 \\ x &= 26 \end{aligned}

So m\angle ACD = 4(26) + 3 = 107^\circ and m\angle ACB = 3(26) - 5 = 73^\circ.

Check. 107 + 73 = 180.

There is a second exterior angle at C: extend \overline{AC} past C instead and it makes an angle with \overline{CB}. That angle is also a supplement of \angle ACB, so by the result on supplements of the same angle it measures 107^\circ too. An exterior angle at a vertex has the same measure whichever side you extend, and that is what lets the exterior angle theorem speak of one number.

Boundaries

A pair, and only a pair. Three angles of 30^\circ add to 90^\circ, and the three angles of a triangle add to 180^\circ, but neither triple is called complementary or supplementary. Both words describe two angles, and each angle’s complement or supplement is one angle.

Both angles must be real angles. Each angle of a complementary pair is acute, so an angle of 90^\circ or more has no complement. Only an angle strictly between 0^\circ and 180^\circ has a supplement: a straight angle would need a partner of 0^\circ, which is not an angle, and an angle larger than 180^\circ would need a negative one. The formulas 90 - x and 180 - x still produce numbers for those; the numbers are not angles. In trigonometry the expression 90^\circ - x is used for every x, including negative and very large angles, but that is an identity between numbers, not a pair of angles drawn at a point.

Only the measures matter, not the position. Two complementary or supplementary angles may share a side, may sit far apart, or may even be in different figures. When they are adjacent, complementary angles make a right angle together and supplementary angles make a straight line. If you need the two to form a straight line, you need a linear pair, which asks for more: a shared vertex and a shared side as well. The converse for right angles needs care. Two adjacent angles whose outer sides are perpendicular are complementary only when their shared side lies inside the right angle those outer sides make. Rays pointing at 0^\circ, 135^\circ and 270^\circ give two adjacent 135^\circ angles whose outer sides are perpendicular, and 135 + 135 = 270.

Supplementary does not mean congruent. The two angles of a supplementary pair are equal only in the one case where both are 90^\circ, and a complementary pair only when both are 45^\circ. The congruence in the results is between two complements, or two supplements, of the same angle, not between an angle and its partner.

The two words never overlap. Complementary angles add to 90^\circ and supplementary angles add to 180^\circ, so no pair is both. A linear pair is always supplementary, and so never complementary.

The numbers 90 and 180 belong to degrees. Measured in radians, a right angle is \displaystyle \frac{\pi}{2} and a straight angle is \pi, and two angles are complementary when their measures add to \displaystyle \frac{\pi}{2} and supplementary when they add to \pi.

Where this goes next

Complementary and supplementary angles, with vertical angles, are the basic moves for finding an unknown angle in a figure: see a square corner, and every pair of pieces inside it is complementary; see a straight line, and every pair of pieces along it is supplementary. A linear pair is the arrangement that guarantees a supplementary pair, and vertical angles are equal because they are supplements of the same neighbor. Put these together and a whole figure’s angles can be chased from one given measure.

The results that complements of equal angles are equal, and supplements of equal angles are equal, are among the first short proofs in proofs about angle pairs.

When a line crosses two parallel lines, the two interior angles on the same side are supplementary: that is part of alternate and same side angles, and it is the fact behind the consecutive angles of a parallelogram adding to 180^\circ. In a cyclic quadrilateral the opposite angles are supplementary.

The biggest payoff is in trigonometry. Because the two acute angles of a right triangle are complementary, the leg opposite one of them is the leg next to the other, and that is why sine and cosine swap. The name records it: cosine is short for the sine of the complement, and the cofunction identities say exactly that for every angle. An angle and its supplement share the same sine, which is why the ambiguous case can give two different triangles.

Exercises

Complementary, supplementary or neither

Decide whether each pair of angles is complementary, supplementary or neither.

37^\circ and 53^\circ
45^\circ and 55^\circ
112^\circ and 68^\circ
95^\circ and 95^\circ

Finding a complement or a supplement

Find the complement of 18^\circ.
Find the supplement of 124^\circ.
Find the complement and the supplement of 47.5^\circ.
Find the supplement of 90^\circ. Then find the angle that is its own complement.
Find the complement of an angle of (2x + 15)^\circ.
Find the supplement of an angle of (2y + 10)^\circ.

Using algebra

Two complementary angles measure (3x)^\circ and (x + 10)^\circ. Find x and both angles.
Two supplementary angles measure (5x + 8)^\circ and (3x + 12)^\circ. Find x and both angles.
The supplement of an angle is five times the angle. Find the angle.
An angle is 24^\circ less than its supplement. Find the angle and its supplement.
Two complementary angles are in the ratio 7 : 8. Find both angles.
The supplement of an angle is five times its complement. Find the angle.
The complement of an angle is 10^\circ more than one third of the angle. Find the angle.
\angle P and \angle Q are supplementary, and \angle Q and \angle R are supplementary. If m\angle P = (6y - 20)^\circ and m\angle R = (4y + 14)^\circ, find y and m\angle Q.
Two supplementary angles measure (x^2 + 20)^\circ and (8x - 20)^\circ. Find every value of x that gives two genuine angles, and the two measures.

In figures

\angle PQR is a right angle and ray QS lies inside it, with m\angle PQS = (x + 14)^\circ and m\angle SQR = (2x - 5)^\circ. Find x and both angles.
Lines \ell and m cross at O with \ell \perp m. A ray from O lies inside one of the right angles and makes a 38^\circ angle with one half of \ell. Find the angle it makes with the neighboring half of m, and the angle it makes with the other half of \ell.
The acute angles of a right triangle measure (2x)^\circ and (3x - 10)^\circ. Find x and both angles.
One side of a triangle is extended, and the exterior angle formed measures 119^\circ. Find the interior angle of the triangle at that vertex.
O is a point on line AB. Rays \overrightarrow{OC} and \overrightarrow{OD} are drawn on the same side of the line, in the order A, C, D, B, with m\angle AOC = 40^\circ, m\angle COD = x^\circ and m\angle DOB = (3x)^\circ. Find x, and name the supplement of \angle AOC and give its measure.
Two vertical angles are also complementary. Find their measures. Then find them if instead the two vertical angles are supplementary.

Reasoning

Explain why an angle of 90^\circ or more has no complement, and why a straight angle has no supplement.
Three angles measure 30^\circ, 30^\circ and 30^\circ, so they add to 90^\circ. Three others measure 45^\circ, 45^\circ and 90^\circ, so they add to 180^\circ. Is the first set complementary, or the second supplementary? Explain.
Can two obtuse angles be supplementary? Can two acute angles? Explain.
\angle A and \angle B are complementary, and \angle B and \angle C are complementary. Show that \angle A \cong \angle C.
\angle A and \angle B are supplementary and congruent. Show that both are right angles.
Explain why the two angles of a linear pair can never be complementary.
Show that for any acute angle, its supplement is exactly 90^\circ more than its complement.

Answers

Complementary. 37 + 53 = 90.
Neither. 45 + 55 = 100, which is neither 90 nor 180.
Supplementary. 112 + 68 = 180.
Neither. 95 + 95 = 190, which is more than 180. Both are obtuse, and two obtuse angles never make a supplementary pair.
72^\circ, since 90 - 18 = 72.
56^\circ, since 180 - 124 = 56.
The complement is 42.5^\circ and the supplement is 132.5^\circ, since 90 - 47.5 = 42.5 and 180 - 47.5 = 132.5.
90^\circ, since 180^\circ - 90^\circ = 90^\circ. A right angle is its own supplement (and has no complement). An angle that is its own complement measures 45^\circ: if x + x = 90, then x = 45.
(75 - 2x)^\circ, since 90 - (2x + 15) = 75 - 2x. This is only an angle while 2x + 15 lies strictly between 0 and 90.
(170 - 2y)^\circ, since 180 - (2y + 10) = 170 - 2y.
x = 20, and the angles are 60^\circ and 30^\circ. From 3x + (x + 10) = 90, 4x = 80.
x = 20, and the angles are 108^\circ and 72^\circ. From (5x + 8) + (3x + 12) = 180, 8x + 20 = 180, so 8x = 160. Then 5(20) + 8 = 108 and 3(20) + 12 = 72.
30^\circ. With the angle x^\circ, x + 5x = 180, so 6x = 180. Its supplement is 150^\circ.
The angle is 78^\circ and its supplement 102^\circ. With the angle x^\circ, x = (180 - x) - 24, so 2x = 156.
42^\circ and 48^\circ. Write them as 7k and 8k: 15k = 90, so k = 6.
67.5^\circ. From 180 - x = 5(90 - x), 180 - x = 450 - 5x, so 4x = 270. Check: the complement is 22.5^\circ, the supplement is 112.5^\circ, and 5 \times 22.5 = 112.5.
60^\circ. From \displaystyle 90 - x = \frac{x}{3} + 10, \displaystyle 80 = \frac{4x}{3}, so x = 60. Check: the complement is 30^\circ, and one third of 60 is 20, plus 10 is 30.
y = 17 and m\angle Q = 98^\circ. \angle P and \angle R are supplements of the same angle, so 6y - 20 = 4y + 14 and 2y = 34. Then m\angle P = 6(17) - 20 = 82^\circ and m\angle Q = 180^\circ - 82^\circ.
x = 10 only, giving 120^\circ and 60^\circ. The sum gives x^2 + 8x = 180, so x^2 + 8x - 180 = 0 and (x - 10)(x + 18) = 0. x = 10 gives 10^2 + 20 = 120 and 8(10) - 20 = 60. x = -18 gives 344 and -164: a negative number is not an angle measure, and 344^\circ has no supplement, so x = -18 is rejected.
x = 27, m\angle PQS = 41^\circ and m\angle SQR = 49^\circ. The parts fill the right angle: (x + 14) + (2x - 5) = 90, so 3x + 9 = 90 and 3x = 81.
52^\circ and 142^\circ. The ray lies in a right angle, so its angle with the half of m beside it is the complement, 90^\circ - 38^\circ = 52^\circ. The two halves of \ell make a straight line, so the other angle forms a linear pair with the 38^\circ angle: 180^\circ - 38^\circ = 142^\circ.
x = 20, and the angles are 40^\circ and 50^\circ. The acute angles of a right triangle are complementary, so 2x + (3x - 10) = 90 and 5x = 100.
61^\circ. The exterior and interior angles at a vertex are supplementary: 180^\circ - 119^\circ = 61^\circ.
x = 35, and the supplement of \angle AOC is \angle COB, which measures 140^\circ. The three angles fill the straight angle, so 40 + x + 3x = 180, 4x = 140. \angle COB is made of \angle COD and \angle DOB, 35^\circ + 105^\circ = 140^\circ, and it forms a linear pair with \angle AOC.
45^\circ each if complementary, and 90^\circ each if supplementary. Vertical angles are equal, so call each x^\circ. Complementary gives x + x = 90, so x = 45; supplementary gives x + x = 180, so x = 90, and the two lines are perpendicular.
A complement of x^\circ would measure (90 - x)^\circ. When x \ge 90 that is zero or negative, and no angle has a measure of zero or less. A straight angle measures 180^\circ, so its supplement would have to measure 0^\circ, and that is not an angle either.
No to both. Each word describes a pair of angles whose two measures add to 90^\circ or 180^\circ, and these are sets of three. Put the two 45^\circ angles together into one 90^\circ angle, though, and that angle is supplementary to the third.
No to both. Each obtuse angle is more than 90^\circ, so two of them add to more than 180^\circ. Each acute angle is less than 90^\circ, so two of them add to less than 180^\circ. A supplementary pair is either two right angles or one acute and one obtuse.
m\angle A + m\angle B = 90^\circ and m\angle B + m\angle C = 90^\circ, so m\angle A + m\angle B = m\angle B + m\angle C. Subtracting m\angle B from both sides gives m\angle A = m\angle C, so \angle A \cong \angle C.
Congruent angles have equal measures, so call each x^\circ. Being supplementary, x + x = 180, so x = 90, and both are right angles.
The angles of a linear pair add to 180^\circ. Complementary angles add to 90^\circ, and one sum cannot be both 180^\circ and 90^\circ.
For an acute angle of x^\circ the supplement is (180 - x)^\circ and the complement is (90 - x)^\circ. Their difference is (180 - x) - (90 - x) = 90, whatever x is. The complement exists only when x < 90, which is why the angle must be acute.