Angle measures add, and finding a missing angle
An angle is cut into smaller angles. How is the size of the whole tied to the sizes of the parts, and how do you find a part that nobody measured?
Where you are 5 in Measuring angles
Why this exists
A protractor measures one angle at a time. But angles are often built out of smaller angles. The spray of a lawn sprinkler sweeps across a sidewalk and then across the lawn. The slats of a folding hand fan cut its opening into small gaps. Two pattern blocks fit together at one corner.
Each time, two questions come up. If you know the parts, how big is the whole? And if you know the whole and some of the parts, how big is the part nobody measured? Both have short answers, because angle measures add the way lengths do.
This lesson takes two days. Part 1 is joining angles: the measure of the whole is the sum of the parts. Part 2 runs that backward to find a missing angle, with an equation and a letter for the unknown, and uses three wholes you already know: a right angle, a straight angle and a full turn.
Definition
An angle is split into parts when one or more rays from its vertex run between its two arms. The parts then sit side by side. Each part shares the vertex, neighboring parts share an arm, no part reaches into another (no overlap), and together they fill the whole angle (no gap).
Angle measures add. When an angle is split into parts that do not overlap, the measure of the whole angle is the sum of the measures of the parts.
In symbols: if ray BC lies inside \angle ABD, then
Recall that m\angle ABD means the measure of \angle ABD in degrees (naming angles).
Intuition
Cut two wedges out of paper, one of 30^\circ and one of 45^\circ. Lay them on the table so their points meet and one edge of each lies along the same line, with no overlap. The outside edges now make one angle, and it measures 75^\circ.
Why 75? A 30^\circ angle is thirty one-degree angles side by side, and a 45^\circ angle is forty-five more (Degrees). Placed side by side, the two wedges hold all of them, none twice and none missing:
It is the same as laying two paper strips end to end: the total length is the two lengths added.
Results
Part 1: joining angles
When angles sit side by side at one vertex, with no overlap and no gap, the angle they make is the sum of their measures:
Any number of parts may be joined this way. Pattern blocks make this easy to try, because their corners have set sizes.
| Block | Its corners |
|---|---|
| square | all four 90^\circ |
| green triangle | all three 60^\circ |
| blue rhombus | two of 60^\circ, two of 120^\circ |
| tan rhombus | two of 30^\circ, two of 150^\circ |
| yellow hexagon | all six 120^\circ |
You can see where these sizes come from by fitting. Six green triangle corners fit exactly around a point, and a full turn is 360^\circ, so each is 360 \div 6 = 60 degrees. Three tan rhombus narrow corners fill a square corner, so each is 90 \div 3 = 30 degrees.
Part 2: finding a missing angle
If the whole and all the parts but one are known, the missing part is what is left of the whole:
A letter can stand for the unknown measure (a letter for the unknown). If a 130^\circ angle is split into 45^\circ and an unknown part x, then
Three wholes come up again and again:
- Parts that fill a right angle add to 90^\circ.
- Parts along a straight line, all on one side of it, fill a straight angle and add to 180^\circ (a straight angle is a half turn).
- Parts that go once around a point fill a full turn and add to 360^\circ.
Why it is true
An angle measures n degrees when it is made of n one-degree angles side by side, because a degree is the turn that, taken 360 times, makes a full turn. Suppose ray BC splits \angle ABD, with \angle ABC made of 40 one-degree angles and \angle CBD made of 60. They share arm BC, so there is no gap between them, and neither reaches into the other, so no one-degree angle is counted twice. All of \angle ABD is then made of
one-degree angles, so it measures 100^\circ.
The same thing seen as a turn: swing a ray from BA to BC, a turn of 40^\circ, then keep going from BC to BD, 60^\circ more. The whole swing from BA to BD is the two turns one after the other, 100^\circ.
Finding a missing part rests on how addition and subtraction undo each other. The facts 45 + 85 = 130 and 130 - 45 = 85 are the same fact read two ways (Fact families). So if 45 + x = 130, then x = 130 - 45.
Worked examples
Examples 1 to 3 go with Part 1, and Examples 4 to 7 with Part 2.
Measuring the parts and the whole
Laoise draws \angle ABD with a ray BC inside it (the figure). She measures \angle ABC, then \angle CBD, then the whole \angle ABD. Show that the measures add.
Laoise puts the center mark on B and the base line along BA. Arm BA runs to the right end, where the inner row starts at 0, so she reads the inner row. Ray BC crosses one long mark past 30, so m\angle ABC = 40^\circ.
To measure \angle CBD on its own, she turns the protractor until the base line runs along BC, with the center mark still on B. Ray BD now crosses at 60, so m\angle CBD = 60^\circ.
For the whole, she puts the base line back along BA. Ray BD crosses one long mark past 90, so m\angle ABD = 100^\circ. The parts add to the whole:
Check. With the base line along BA, the two readings were 40 and 100. The turn from BC to BD is the gap between them, 100 - 40 = 60, just as Exercise 25 of the protractor lesson found.
Two pattern blocks at one corner
Gearoid sets a square and a green triangle so that a corner of each meets at point P and they share one side, with no overlap. What angle do the two outside sides make at P?
The square’s corner is 90^\circ and the triangle’s corner is 60^\circ. They sit side by side at P, share a side, and do not overlap, so their measures add:
The outside sides make a 150^\circ angle.
Check. The angle is more than a right angle and less than a straight angle, as the picture shows. A straight line would need 180 - 150 = 30 degrees more, which is exactly a tan rhombus narrow corner: one would fit in the space left.
The folding hand fan
Padraig’s folding hand fan has 8 slats on one pin. When it is open, each slat is 20^\circ from the next. What is the angle between the two outside slats?
The angle between the outside slats is split into the gaps between neighboring slats. Count the gaps, not the slats: 8 slats have 8 - 1 = 7 gaps between them, just as 8 fence posts in a row have 7 spaces. Each gap is 20^\circ, and the gaps sit side by side with no overlap, so the whole is 7 twenties added together. Adding the same number again and again is multiplication:
The outside slats make a 140^\circ angle.
Check. 140 is less than 180, and the open fan in the picture is short of a straight line.
The lawn sprinkler’s sweep
Laoise’s lawn sprinkler at P sweeps back and forth across an angle of 130^\circ. Starting from ray PA, the first 45^\circ of the sweep sprays the sidewalk, and the rest sprays the lawn. Write an equation with a letter for the unknown, and find how many degrees of the sweep spray the lawn.
Ray PB splits the whole sweep into the sidewalk part and the lawn part, with no overlap. Call the lawn part x degrees. The parts add to the whole:
Subtraction undoes adding 45, so x is what is left of 130 after taking away 45:
The lawn gets 85^\circ of the sweep.
Check. 45 + 85 = 130.
Three angles along a straight line
Two rays rise from one point on a straight line and split the half turn on that side into three angles: 48^\circ, x and 67^\circ. Find x.
The three angles lie along a straight line, all on one side, so together they fill a straight angle, 180^\circ:
Gather the known parts first. Changing the order of the numbers (the commutative property) lets 48 and 67 sit together, and they add to 115. So 115 + x = 180, and
The missing angle is 65^\circ.
Check. 48 + 65 + 67 = 180.
All the way around a point
Four angles go once around a point: a right angle, 110^\circ, 85^\circ and x. Find x.
The four angles go once around the point with no overlap and no gap, so they fill a full turn:
The known parts add to 90 + 110 = 200, then 200 + 85 = 285. The missing angle is what is left of the full turn:
Check. 285 + 75 = 360. And x looks a little less than a square corner, as 75^\circ is.
One known part and two equal parts
Rays OB and OC split the straight angle \angle AOD into three angles. m\angle AOB = 40^\circ, and \angle BOC and \angle COD are equal. Find their measure.
The two equal angles have the same measure, so the same letter x stands for both. The three angles fill a straight angle:
Take the known part away first: the two equal parts together are 180 - 40 = 140 degrees. Two equal parts that make 140 are each half of it:
So m\angle BOC = m\angle COD = 70^\circ.
Check. 40 + 70 + 70 = 180.
Boundaries
The parts must not overlap. If two angles share some of their opening, adding their measures counts the shared part twice, and the sum is more than the angle they cover (Exercise 23).
The parts must leave no gap and share the vertex. Two angles at different points, or with a space between them, do not join into one angle, and their sum is not the measure of anything drawn.
A sum past 180^\circ is a turn, not an inside opening. Joining a 110^\circ wedge and a 100^\circ wedge makes a turn of 210^\circ from one outside edge to the other. A protractor reads the inside opening between those edges, which is the rest of the full turn (Exercise 25).
The three wholes need exact shapes. Parts add to 180^\circ only when the outside arms make a truly straight line, and to 360^\circ only when the parts go once around with no overlap and no gap. A line that only looks straight in a sketch does not give 180.
A missing angle is as exact as what it is found from. Stated measures give an exact answer. Measures read off a drawing are close, so a part found from them is close too.
Where this goes next
Angles that sit side by side get a name of their own, and two side by side along a line are studied in adjacent angles and linear pairs. Two angles whose measures add to 90^\circ or to 180^\circ are named in complementary and supplementary angles, and the sorting of angles and their pairs is gathered in angle types and pairs. When two lines cross, the angles that face each other turn out equal, in Vertical angles. Longer chains of these steps, through several angles in one figure, are finding unknown angle measures. And the three angles of any triangle add to a straight angle, in the triangle angle sum.
Exercises
Part 1: joining angles
Use the figure of five rays for Exercises 1 to 5.
- Find m\angle AOC.
- Find m\angle BOD.
- Find m\angle AOD.
- Find m\angle COE.
- Find m\angle AOE. What kind of angle is it, and what does that say about A, O and E?
- Gearoid puts two tan rhombus narrow corners and one square corner side by side at one point. What angle do they make?
- Laoise puts a blue rhombus wide corner and a green triangle corner side by side. What angle do they make, and what do the two outside sides look like?
- How many yellow hexagon corners fit around one point with no gap?
- A folding hand fan has 9 slats, each 15^\circ from the next. What is the angle between the outside slats?
- Padraig lays paper wedges of 25^\circ, 25^\circ and 40^\circ side by side. What angle do they make, and what is its name?
- Laoise joins a 57^\circ wedge and a 38^\circ wedge side by side. Is the angle they make acute, right or obtuse?
Part 2: finding a missing angle
In Exercises 12 to 16, write an equation with a letter for the missing angle, then find it.
- A right angle is split into two parts. One part is 52^\circ.
- A straight angle is split into two parts. One part is 107^\circ.
- A straight angle is split into three parts: 35^\circ, a right angle, and the missing part.
- Three angles go once around a point: 120^\circ, 95^\circ and the missing angle.
- Three angles go once around a point. Two of them are 100^\circ each.
- Gearoid’s lawn sprinkler sweeps 160^\circ. The first 65^\circ sprays the driveway and the rest sprays the lawn. How many degrees of the sweep spray the lawn?
- Padraig’s hand fan is open to 150^\circ. He folds it partly closed, and now it is open 95^\circ. Through how many degrees did he close it?
- Laoise sets her sprinkler to sweep 75^\circ, then widens the sweep by 40^\circ. What is the new sweep? How many more degrees would make it sweep a straight angle?
- A ray splits a right angle into two equal parts. Find each part.
- Two rays split a straight angle into three equal parts. Find each part. Which pattern block corner would fit each part exactly?
- A sprinkler head turns 85^\circ from facing the house to facing the garage, then 140^\circ more to face the gate, then on round the same way back to the house. How far is that last turn?
Reasoning
- Gearoid adds m\angle AOC and m\angle BOD from the figure of five rays and expects to get m\angle AOD. Find his sum, and explain why it is not m\angle AOD.
- Explain why angles along a straight line, all on one side of it, always add to 180^\circ, however many rays split them.
- Padraig joins a 110^\circ wedge and a 100^\circ wedge side by side. His protractor reads 150^\circ for the angle between the outside edges. Explain why, and show the numbers agree.
Answers
- 75^\circ. 30 + 45 = 75.
- 90^\circ. 45 + 45 = 90.
- 120^\circ. 30 + 45 + 45 = 120.
- 105^\circ. 45 + 60 = 105.
- 180^\circ, a straight angle. 30 + 45 + 45 + 60 = 180, so OA and OE point in opposite directions and A, O and E lie on one straight line.
- 150^\circ. 30 + 30 + 90 = 150.
- 180^\circ. 120 + 60 = 180, a straight angle, so the two outside sides make one straight line.
- 3. Each corner is 120^\circ, and 120 + 120 + 120 = 360, a full turn.
- 120^\circ. 9 slats have 9 - 1 = 8 gaps, and 8 \times 15 = 120.
- 90^\circ, a right angle. 25 + 25 + 40 = 90.
- Obtuse. 57 + 38 = 95, which is more than 90 and less than 180.
- 38^\circ. 52 + x = 90, so x = 90 - 52 = 38.
- 73^\circ. 107 + x = 180, so x = 180 - 107 = 73.
- 55^\circ. 35 + 90 + x = 180; the known parts make 125, so x = 180 - 125 = 55.
- 145^\circ. 120 + 95 + x = 360; the known parts make 215, so x = 360 - 215 = 145.
- 160^\circ. 100 + 100 + x = 360, so x = 360 - 200 = 160.
- 95^\circ. 65 + x = 160, so x = 160 - 65 = 95.
- 55^\circ. The part still open and the part closed make the old opening: 95 + x = 150, so x = 150 - 95 = 55.
- 115^\circ, and 65^\circ more. 75 + 40 = 115, then 115 + y = 180 gives y = 180 - 115 = 65.
- 45^\circ. x + x = 90, so x = 90 \div 2 = 45.
- 60^\circ each. x + x + x = 180, so x = 180 \div 3 = 60. A green triangle corner fits each part, and so does a narrow corner of the blue rhombus.
- 135^\circ. The three turns make one full turn: 85 + 140 + x = 360; 85 + 140 = 225, so x = 360 - 225 = 135.
- His sum is 75 + 90 = 165, but m\angle AOD = 120^\circ. The two angles overlap: \angle BOC, 45^\circ, is inside both, so his sum counts it twice. Taking one copy away gives 165 - 45 = 120.
- The angles share the vertex, sit side by side and fill the opening on that side of the line, so their measures add to the measure of the whole opening. That whole opening is the turn from one direction of the line to the opposite direction, a half turn, which is 180^\circ. Adding more rays only cuts the same opening into more parts; the total stays 180^\circ.
- Side by side, the wedges make a turn of 110 + 100 = 210 degrees from one outside edge to the other, more than a half turn. A protractor reads only the inside opening between two arms. The two openings, the 210^\circ one and the inside one, go once around the point together, so the inside one is 360 - 210 = 150 degrees, as the protractor shows.