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Supplementary, complementary and vertical angles, and using them to find what is missing.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
In this lesson you find unknown angles using relationships rather than a protractor: angles on a straight line add to 180°, angles in a right angle add to 90°, and angles opposite each other where two lines cross are equal. Each fact becomes an equation, and you will solve equations with $x$ in them to find the angles and check them.
You can measure angles in degrees, and you know a right angle is 90° and a straight angle is 180°. You can also solve equations such as $3x + 20 = 110$. In this lesson a fact about a picture becomes an equation, and you solve the equation to find an angle you cannot measure. No protractor is needed.
| Term | What it means |
|---|---|
| Adjacent angles | Two angles side by side that share a vertex and a side, with no overlap. |
| Supplementary angles | Two angles that add to 180°, a straight angle. |
| Complementary angles | Two angles that add to 90°, a right angle. |
| Vertical angles | The angles directly opposite each other where two lines cross. They are always equal. |
| Right angle | An angle of 90°, a square corner, often marked with a small square. |
| Straight angle | An angle of 180°, half a turn, the angle along a straight line. |
Most missing angles can be found from four facts.
Each fact turns straight into an equation. If one angle on a line is 115°, the other is $x$ with $115 + x = 180$, so $x = 65$. If two vertical angles are $4x$ and $80$, then $4x = 80$ and $x = 20$. So the work always has two parts: name the relationship, then solve the equation it gives.
Another way: picture
Open a book flat on the table. The two covers make a straight line, 180°. Lift one cover until it stands straight up: it makes a right angle with the table, and the angle on the other side is also 90°, because $90 + 90 = 180$. Lean it a little further and one angle grows exactly as much as the other shrinks.
Another way: steps
When two angles sit side by side and their outer sides make a straight line, together they are half a turn, 180°. They are supplementary. If one is 70°, the other is $180 - 70 = 110$ degrees. If one is 90°, so is the other.
When two adjacent angles fill a square corner, together they are 90°. They are complementary. If one is 25°, the other is $90 - 25 = 65$ degrees. Only acute angles, those smaller than 90°, can have a complement.
The two angles do not need to be next to each other to be supplementary or complementary; the words only describe their sum. Angles of 30° and 60° are complementary even if they are in different pictures. But in a diagram, the usual clue is a pair of adjacent angles on a line or in a marked right angle.
One way to remember which is which: c for corner (90°) comes before s for straight (180°), just as 90 comes before 180.
Two straight lines that cross make four angles. Call them $a$, $b$, $c$ and $d$ going around. The pairs across from each other, $a$ and $c$, and $b$ and $d$, are vertical angles. Here is why they are always equal.
Angles $a$ and $b$ lie on one straight line, so $a + b = 180$. Angles $b$ and $c$ lie on the other line, so $b + c = 180$. Both $a$ and $c$ are what is left when $b$ is taken from 180, so $a = c$. The same argument shows $b = d$.
| Angle $a$ | Opposite, $c$ | Neighbors, $b$ and $d$ | Total |
|---|---|---|---|
| 40° | 40° | 140° each | 360° |
| 72° | 72° | 108° each | 360° |
| 90° | 90° | 90° each | 360° |
So at any crossing you only need one angle to know all four. The angles come in two sizes, and the two sizes add to 180°. The four together make a full turn, 360°.
The full-turn fact works for more than two lines. If three lines meet at one point, they make six angles around it, and all six add to 360°. Suppose four of them are 50°, 70°, 50° and 70°, and the other two are equal. Together the known four make 240°, so the last two share $360 - 240 = 120$ degrees, 60° each. Vertical angles still pair up across the point, which is a quick way to check.
Sometimes an angle is given as an expression such as $2x + 15$. Nothing new is needed: the relationship gives the equation, and algebra solves it.
Remember that $x$ is not always the angle. After finding $x$, substitute it back into each expression to find the angles themselves.
Every problem in this lesson uses the same five moves.
Each move has a reason. Naming the relationship first matters because it decides the number on the right of the equation, 90 or 180, or whether there is a total at all. Solving one step at a time keeps each equation balanced.
How to check. Put your angles back into the relationship: supplementary angles should add to exactly 180, complementary ones to 90, vertical ones should match. Then look at the picture. An angle that looks sharp should be less than 90°; one that looks wide should be more. An answer of 130° for an angle drawn small and pointed means the wrong relationship was used. Finally, no angle in these problems can be negative or more than 180°.
A picture frame is made of four strips of wood that meet at square corners. A carpenter does not cut each strip straight across. She cuts each end at an angle so the two strips meet in a neat diagonal seam called a miter. The two cut ends together must fill the 90° corner, so the angles are complementary. With equal cuts, each is $90 \div 2 = 45$ degrees, which is why a miter saw has a mark at 45°.
A frame with six sides has corners of 120°. Each cut is then $120 \div 2 = 60$ degrees from the edge of the strip. If the saw is set 1° off, at 44° instead of 45°, the two cuts make $44 + 44 = 88$ degrees, not 90, and a gap opens at the corner. Four corners with that error leave the frame 8° short of a full turn, and it will not close.
Safety rules for ladders in the United States use the four-to-one rule: for every 4 feet of height the ladder reaches, its foot should be 1 foot out from the wall. A ladder set this way makes an angle of about 75° with the ground.
The wall stands at a right angle to the ground, so the angle between the ladder and the wall and the angle between the ladder and the ground are the two acute angles of a right triangle. Together they make 90°, so they are complementary. The ladder leans $90 - 75 = 15$ degrees away from the wall. If a worker sets the ladder at 60° to the ground, it leans $90 - 60 = 30$ degrees from the wall, twice as much, and its foot is more likely to slide out.
Two straight roads that cross make an X, so the four corners of the intersection are two pairs of vertical angles. Suppose a map shows Main Street crossing Oak Avenue at 70°. The opposite corner is also 70°, and the two other corners are each $180 - 70 = 110$ degrees. A driver turning from Main onto Oak turns through one of these corners. Turning into the 70° corner is a sharp turn, and turning into the 110° corner is a gentle one.
Road designers try to make streets cross at close to 90°, where all four corners are equal. At a sharp crossing a driver waiting at the corner must look far back over one shoulder to see traffic coming. When an old road meets a new one at a sharp angle, engineers often bend the last stretch of the old road so that it arrives nearly square.
The most common mistake is using the wrong relationship. At a crossing, the angle opposite a 115° angle is also 115°, but the angle next to it is 65°. Say which pair you are looking at before you calculate.
The second is mixing up the totals: supplementary is 180°, complementary is 90°. A right-angle mark means 90; a straight line means 180.
The third is stopping at $x$. If the angles are $3x + 15$ and $2x - 5$, then $x = 34$ is not an angle in the picture; the angles are 117° and 63°. Last, some learners trust how the diagram looks and measure it. Diagrams are often not drawn to scale, so use the relationships, not a ruler or a protractor.
A ray splits a right angle into two parts. One part is 34°. Name the relationship.
$\text{two parts of a right angle} \Rightarrow \text{complementary}$
The small square in the corner marks a right angle, 90°.
Write the equation, calling the missing part $x$.
$x + 34 = 90$
Complementary angles add to 90°.
Subtract 34 from both sides.
$x + 34 - 34 = 90 - 34$
Subtracting undoes the addition and keeps the equation balanced.
Simplify the right side.
$x = 56$
$90 - 34 = 56$.
Check the pair, and the picture.
$34 + 56 = 90$
The two parts fill the right angle, and both are acute, as parts of a right angle must be.
Name the four angles going around the crossing: $a = 40^\circ$, then $b$, $c$ and $d$.
$a = 40^\circ, \quad b, \quad c, \quad d$
Labels make it clear which angles are neighbors and which are opposite.
Find $c$, the angle opposite $a$.
$c = a = 40^\circ$
Vertical angles are equal.
Write an equation for $b$, the neighbor of $a$ on a straight line.
$40 + b = 180$
Adjacent angles along one line are supplementary.
Subtract 40 from both sides.
$b = 180 - 40 = 140^\circ$
Subtracting undoes the addition.
Find $d$, the angle opposite $b$.
$d = b = 140^\circ$
Vertical angles are equal.
Check the full turn.
$40 + 140 + 40 + 140 = 360$
The four angles around a point make a full turn, so all four are right.
Two angles on a straight line measure $(3x + 15)^\circ$ and $(2x - 5)^\circ$. Write the equation.
$(3x + 15) + (2x - 5) = 180$
Angles side by side on a straight line add to 180°.
Combine like terms.
$5x + 10 = 180$
$3x + 2x = 5x$ and $15 - 5 = 10$.
Subtract 10 from both sides.
$5x + 10 - 10 = 180 - 10 \;\Rightarrow\; 5x = 170$
Undo the addition first, keeping the equation balanced.
Divide both sides by 5.
$\dfrac{5x}{5} = \dfrac{170}{5} \;\Rightarrow\; x = 34$
Dividing undoes the multiplication by 5.
Substitute to find the first angle.
$3 \times 34 + 15 = 102 + 15 = 117^\circ$
The question is about the angles, and $x$ is only a step toward them.
Substitute to find the second angle.
$2 \times 34 - 5 = 68 - 5 = 63^\circ$
Each expression gives its own angle.
Check that they are supplementary.
$117 + 63 = 180$
The two angles fill the straight line, so the answer is right.
Write the equation.
$4x + 8 = 72$
Vertical angles are equal.
Subtract 8 from both sides.
Divide both sides by 4.
One angle in a right angle measures $15^\circ$. Which equation finds the other angle $x$?
Two lines cross. One angle measures $(2x + 14)^\circ$, and the angle next to it measures $(x + 55)^\circ$. Complete the worked solution.
Write the equation: the two angles are adjacent on a straight line.
$(2x + 14) + (x + 55) = 180$
Neighboring angles at a crossing are supplementary.
Combine the $x$ terms and the numbers.
$3x + 69 = 180$
$2x + x = 3x$, and the two numbers add.
Subtract $69$ from both sides.
$3x =$ s
Undo the addition first to leave the $x$ term alone.
Divide both sides by 3.
$x =$ x
Dividing undoes the multiplication by 3.
Substitute to find the first angle.
$2x + 14 =$ a
The question is about the angles, so turn $x$ back into degrees.
What do you know about two angles side by side that together make a straight line?
Two angles sit side by side on a straight line. One of them is $80^\circ$. What is the other, in degrees?
Answer:
Two angles together make a right angle. One of them is $11^\circ$. What is the other, in degrees?
Answer:
Two angles on a straight line are $x$ and $3x$. Find $x$ and the larger angle, in degrees.
$x =$ x°, and the larger angle is y°.
A ray of light hits a flat mirror, making an angle of $34^\circ$ with the mirror's surface. It bounces off at the same angle, $34^\circ$, on the other side. How many degrees do the two equal angles make together, and what is the angle between the incoming ray and the outgoing ray?
The two equal angles: t°. Between the rays: y°.
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
The two blades of an open pair of scissors cross like the letter X. The angle between the blades on the handle side is $(2x + 13)^\circ$, and the opposite angle, between the cutting edges, is $(5x - 62)^\circ$. Find $x$, and the size of each of these two angles.
$x =$ x, and each angle is g°.
You can find a missing angle from a relationship. Without looking: two angles are supplementary and one is 115 degrees. What is the other, and what equation did you write? Why are vertical angles always equal?
16. Your turn: vertical angles of (4x + 8)° and 72°, step 2
$4x = 64$
Undo the addition first.
16. Your turn: vertical angles of (4x + 8)° and 72°, step 3
$x = 16$
Dividing undoes the multiplication; check: $4 \times 16 + 8 = 72$.