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Geometry · G.C.A.2

Arcs and central angles

Two points on a circle cut it into two curved pieces. How do you say which piece you mean, and how much of the circle it is?

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Why this exists

A polygon has corners, so every piece of it has a name: side \overline{AB}, angle \angle B. A circle has no corners. Put two points A and B on it and the circle falls into two curved pieces, and both of them run from A to B. “The curve from A to B” no longer says which one you mean.

There is a second problem. A curved piece cannot be laid against a ruler, and how long it is depends on how big the circle is. What stays the same when a circle grows is how much of the full turn the piece takes up: a piece that is a quarter of one circle is a quarter of any circle.

This lesson settles both. It names the pieces, called arcs, so that each name picks out exactly one, and it measures them in Degrees, by the angle they make at the center. That single choice ties angles and circles together. Almost every later fact about circles, from arc length to the inscribed angle theorem, is stated in arc measures.

Definition

Take a circle with center O, written \odot O, and two different points A and B on it.

An arc is an unbroken piece of the circle. The points A and B split the circle into two arcs, and each of them has endpoints A and B. An arc is a curve, made of points on the circle and including its endpoints. It is not the region inside.

A central angle is an angle whose vertex is the center O and whose sides pass through two points of the circle. For A and B it is \angle AOB, and its sides run along the radii \overline{OA} and \overline{OB}.

The central angle decides which arc is which.

  • When A and B are not the two ends of a diameter, \angle AOB is less than 180^\circ. The arc lying inside \angle AOB is the minor arc, the shorter of the two. It is named by its two endpoints, \overset{\frown}{AB}, read “arc AB”.
  • The other arc, lying outside \angle AOB, is the major arc, the longer one. It is named with three letters: the two endpoints with any point C of the arc written between them, \overset{\frown}{ACB}. The middle letter says which way round to go.
  • When \overline{AB} is a diameter, the two arcs are the same size and each is a semicircle. A semicircle is also named with three letters, because two letters would not say which half.

The segment \overline{AB} joining the endpoints is a Chords, secants and tangents. A chord and its two arcs come together: the chord cuts the circle into the minor arc on one side and the major arc on the other.

The measure of an arc, written m\overset{\frown}{AB}, is a number of degrees.

  • A minor arc measures the same as its central angle: m\overset{\frown}{AB} = m\angle AOB.
  • A semicircle measures 180^\circ.
  • A major arc measures 360^\circ minus the minor arc with the same endpoints: m\overset{\frown}{ACB} = 360^\circ - m\overset{\frown}{AB}.

Two arcs are congruent, written \overset{\frown}{AB} \cong \overset{\frown}{CD}, when they have the same measure and lie on the same circle, or on two circles with equal radii.

65^\circ
 A B C O
\overset{\frown}{AB}
\overset{\frown}{ACB}
Figure 1The central angle \angle AOB picks out the minor arc \overset{\frown}{AB}. The major arc is named through a point on it, \overset{\frown}{ACB}.

In Figure 1, m\angle AOB = 65^\circ, so m\overset{\frown}{AB} = 65^\circ and m\overset{\frown}{ACB} = 360^\circ - 65^\circ = 295^\circ.

Intuition

Stand at the center facing A, and turn until you face B the short way. The minor arc is the stretch of rim you swept past, and its measure is simply how far you turned. Turning all the way round is 360^\circ, so an arc’s measure is a fraction of the full turn written in degrees: a 90^\circ arc is \tfrac{90}{360} = \tfrac{1}{4} of its circle, and a 120^\circ arc is a third.

That is why the measure does not care how big the circle is. In Figure 2 one central angle of 80^\circ cuts an arc from each of two circles with the same center. Both arcs measure 80^\circ and each is \tfrac{80}{360} = \tfrac{2}{9} of its own circle, but the outer one is plainly longer. The measure says how much of the turn an arc takes. Its length depends on the radius as well, through the Circumference, and working it out is arc length.

80^\circ
 O
Figure 2Both heavy arcs measure 80^\circ, and each is \tfrac{2}{9} of its own circle. Their lengths differ.

The naming rule comes from the same picture. Two letters are enough when there is an obvious choice, and the short way round is the obvious one. To mean the long way round, you name a point you pass on the way.

Results

Range of measures. A minor arc measures more than 0^\circ and less than 180^\circ. A semicircle measures exactly 180^\circ. A major arc measures more than 180^\circ and less than 360^\circ.

An arc and its partner fill the circle. The two arcs with the same endpoints, going opposite ways round, have measures that add to a full turn:

m\overset{\frown}{AB} + m\overset{\frown}{ACB} = 360^\circ

Arc Addition Postulate. If B is a point on an arc from A to C, the two pieces it makes add to the whole arc:

m\overset{\frown}{ABC} = m\overset{\frown}{AB} + m\overset{\frown}{BC}

Here the pieces are minor arcs; when a piece is itself major or a semicircle, it is named with three letters and the rule is the same. This is segment addition bent round a circle.

Congruent arcs and central angles. In one circle, or in two circles with equal radii, two minor arcs are congruent exactly when their central angles are congruent.

An arc as a fraction. An arc measuring a^\circ is \tfrac{a}{360} of its circle.

Why it is true

The arc measures are definitions, so what needs an argument is that they behave the way the Results say: that pieces add up, and that equal central angles really give arcs that match.

Arc addition. Take B on an arc from A to C, and suppose first that both pieces, \overset{\frown}{AB} and \overset{\frown}{BC}, are minor arcs. Their central angles \angle AOB and \angle BOC sit side by side at O, sharing the ray OB. Call their total s.

  • If s < 180^\circ, ray OB lies inside \angle AOC, so the angle addition postulate gives m\angle AOC = s. The whole arc is the minor arc from A to C, so it measures s.
  • If s = 180^\circ, the two angles form a linear pair, so A, O and C lie on one line. Then \overline{AC} is a diameter, the whole arc is a semicircle, and it measures 180^\circ = s.
  • If s > 180^\circ, the whole arc is a major arc. The rest of the circle is the minor arc from C back to A, whose central angle is \angle COA. The three angles \angle AOB, \angle BOC and \angle COA go once round O without overlapping, so m\angle COA = 360^\circ - s, and the major arc measures 360^\circ - (360^\circ - s) = s.

In every case the whole arc measures s. If a piece is itself a major arc, cut it at its midpoint into two halves, each less than 180^\circ, and use the argument once for each half.

Congruent arcs. A rotation about O keeps every point at the same distance from O, so it carries the circle onto itself. Suppose \angle AOB \cong \angle COD. The rotation that turns ray OA onto ray OC turns ray OB onto ray OD, because the angle between the rays is kept (if the two angles open in opposite directions, a reflection in a line through O does the same job). It carries A to C, B to D, and every point of \overset{\frown}{AB} onto \overset{\frown}{CD}, so the arcs match exactly. Running the argument backwards, arcs that match have matching central angles. For two circles with equal radii, first slide one center onto the other.

Worked examples

Example 1

A minor arc and its major arc

In Figure 3, m\angle POQ = 64^\circ and R lies on the long way round from P to Q. Name the minor arc and the major arc with endpoints P and Q, and find their measures.

64^\circ
 P Q R O
Figure 3Example 1. The central angle is given; R is on the long way round.
Solution

The minor arc is the one inside the central angle \angle POQ, and two letters name it: \overset{\frown}{PQ}. A minor arc measures the same as its central angle, so

m\overset{\frown}{PQ} = 64^\circ

The major arc is the other way round, through R, so it needs three letters: \overset{\frown}{PRQ}. A major arc measures 360^\circ minus its minor arc:

m\overset{\frown}{PRQ} = 360^\circ - 64^\circ = 296^\circ

Check. 296^\circ is between 180^\circ and 360^\circ, as a major arc must be, and 64 + 296 = 360.

Example 2

A diameter and arc addition

In Figure 4, \overline{AD} is a diameter of \odot O, m\angle AOB = 48^\circ and m\angle BOC = 72^\circ. Find m\overset{\frown}{BC}, m\overset{\frown}{AC}, m\overset{\frown}{CD}, m\overset{\frown}{BCD} and m\overset{\frown}{CAB}.

48^\circ
72^\circ
 A B C D O
Figure 4Example 2. A diameter and two central angles above it.
Solution

\overset{\frown}{BC} is a minor arc with central angle \angle BOC, so m\overset{\frown}{BC} = 72^\circ. In the same way m\overset{\frown}{AB} = 48^\circ.

\overset{\frown}{AC} is the short way from A to C, which passes through B. By arc addition the two pieces add to the whole:

\begin{aligned} m\overset{\frown}{AC} &= m\overset{\frown}{AB} + m\overset{\frown}{BC} \\ &= 48^\circ + 72^\circ = 120^\circ \end{aligned}

A and D are the ends of a diameter, so the upper half \overset{\frown}{ACD} is a semicircle and measures 180^\circ. Arc addition splits it at C:

m\overset{\frown}{CD} = 180^\circ - 120^\circ = 60^\circ

\overset{\frown}{BCD} runs from B through C to D, so it is the two pieces \overset{\frown}{BC} and \overset{\frown}{CD}:

m\overset{\frown}{BCD} = 72^\circ + 60^\circ = 132^\circ

\overset{\frown}{CAB} starts at C and goes through A to B: the long way, past D and round the lower half. It is the major arc whose minor arc is \overset{\frown}{BC}:

m\overset{\frown}{CAB} = 360^\circ - 72^\circ = 288^\circ

Check. Add the pieces along the way instead: \overset{\frown}{CD}, then the lower semicircle from D to A, then \overset{\frown}{AB}, which is 60 + 180 + 48 = 288.

Example 3

Three arcs that fill the circle

In Figure 5 the radii \overline{OA}, \overline{OB} and \overline{OC} cut the circle into three arcs. Going round, the arc from A to B measures (4x + 12)^\circ, the arc from B to C measures (5x - 3)^\circ, and the arc from C back to A measures (6x + 6)^\circ. Find x, the three arcs, and m\overset{\frown}{ACB}.

A B C O
(4x+12)^\circ
(5x-3)^\circ
(6x+6)^\circ
Figure 5Example 3. Three arcs that together go once round the circle.
Solution

The three arcs go once round the circle and meet only at their ends, so by arc addition they make the whole circle, 360^\circ. Adding the three expressions gives (4x + 12) + (5x - 3) + (6x + 6) = 15x + 15, so

\begin{aligned} 15x + 15 &= 360 \\ 15x &= 345 \\ x &= 23 \end{aligned}

The three arcs measure 4(23) + 12 = 104^\circ, 5(23) - 3 = 112^\circ and 6(23) + 6 = 144^\circ. Each is under 180^\circ, so they are the minor arcs \overset{\frown}{AB}, \overset{\frown}{BC} and \overset{\frown}{CA}, and the two-letter names are right.

\overset{\frown}{ACB} runs from A through C to B, which is the arc from A to C followed by the arc from C to B:

m\overset{\frown}{ACB} = 144^\circ + 112^\circ = 256^\circ

Check. 104 + 112 + 144 = 360, and 256 = 360 - 104, as the major arc partner of \overset{\frown}{AB} must be.

Example 4

The hands of a clock

A clock face is a circle whose center is the pivot of the hands. Each hand, carried on in a straight line, points at one spot on the rim. Find the measures of the minor arc and the major arc between those two spots at 4:00 and at 7:30.

Solution

The twelve numbers are equally spaced round the rim, so they cut it into twelve congruent arcs. Congruent arcs have equal measures that together make 360^\circ, so each step from one number to the next is \tfrac{360^\circ}{12} = 30^\circ.

At 4:00 the minute hand points at 12 and the hour hand at 4. The short way between them is four steps:

4 \times 30^\circ = 120^\circ \qquad\qquad 360^\circ - 120^\circ = 240^\circ

so the minor arc is 120^\circ and the major arc 240^\circ.

At 7:30 the minute hand points at 6. The hour hand is not at 7: half of the hour has passed, so it is halfway from 7 to 8. Measuring clockwise from 12, the minute hand is at 6 \times 30^\circ = 180^\circ and the hour hand at 7.5 \times 30^\circ = 225^\circ, so

225^\circ - 180^\circ = 45^\circ \qquad\qquad 360^\circ - 45^\circ = 315^\circ

The minor arc is 45^\circ and the major arc 315^\circ.

Example 5

Two diameters

In Figure 6, \overline{PR} and \overline{QS} are diameters of \odot O, m\overset{\frown}{PQ} = (3x + 14)^\circ and m\overset{\frown}{RS} = (5x - 22)^\circ. Find x and the measures of all four arcs between neighboring points.

P Q R S O
(3x+14)^\circ
(5x-22)^\circ
Figure 6Example 5. Two diameters cut the circle into four arcs.
Solution

The two diameters lie on two lines crossing at O, so the central angles \angle POQ and \angle ROS sit opposite each other. They are vertical angles, and so they are congruent. Minor arcs of one circle with congruent central angles are congruent, so their measures are equal:

\begin{aligned} 3x + 14 &= 5x - 22 \\ 36 &= 2x \\ x &= 18 \end{aligned}

Then m\overset{\frown}{PQ} = 3(18) + 14 = 68^\circ, and m\overset{\frown}{RS} = 5(18) - 22 = 68^\circ agrees.

P and R are the ends of a diameter, so \overset{\frown}{PQR} is a semicircle. Splitting it at Q,

m\overset{\frown}{QR} = 180^\circ - 68^\circ = 112^\circ

and the semicircle \overset{\frown}{RSP} gives m\overset{\frown}{SP} = 112^\circ the same way.

Check. 68 + 112 + 68 + 112 = 360.

Example 6

A piece that turns out to be a major arc

Points A, B and C on a circle cut it into three arcs: from A to B, from B to C, and from C back to A. Their measures are in the ratio 2 : 3 : 7. Find the three arcs, and find m\overset{\frown}{CA}.

Solution

Write the three measures as 2k, 3k and 7k. They go once round the circle, so they add to 360^\circ:

12k = 360 \qquad\Longrightarrow\qquad k = 30

The arcs measure 60^\circ, 90^\circ and 210^\circ.

The third one, from C back to A, measures more than 180^\circ, so it is a major arc. That changes what \overset{\frown}{CA} means. Two letters name the minor arc with endpoints C and A, and that is the other way round, through B:

\begin{aligned} m\overset{\frown}{CA} &= m\overset{\frown}{CB} + m\overset{\frown}{BA} \\ &= 90^\circ + 60^\circ = 150^\circ \end{aligned}

which is 360^\circ - 210^\circ, as it should be. To name the 210^\circ arc you would need a point D on it, and then it is \overset{\frown}{CDA}.

Example 7

Equal arcs stay equal when the same piece is added

Going round a circle you meet A, B, C and D in that order, and the arcs between neighbors \overset{\frown}{AB} and \overset{\frown}{CD} are congruent. Show that \overset{\frown}{ABC} \cong \overset{\frown}{BCD}.

Solution

First, the two-letter names are right. The arc from A to B and the arc from C to D do not overlap, have equal measures, and leave the arc from B to C between them. If each were 180^\circ or more, those three would add to more than 360^\circ, more than the whole circle. So both are minor arcs.

\overset{\frown}{ABC} is the arc from A to B followed by the arc from B to C, and \overset{\frown}{BCD} is the arc from B to C followed by the arc from C to D. By arc addition,

\begin{aligned} m\overset{\frown}{ABC} &= m\overset{\frown}{AB} + m\overset{\frown}{BC} \\ m\overset{\frown}{BCD} &= m\overset{\frown}{BC} + m\overset{\frown}{CD} \end{aligned}

Congruent arcs have equal measures, so m\overset{\frown}{AB} = m\overset{\frown}{CD}, and the two right-hand sides are equal. The two arcs have the same measure and lie on the same circle, so \overset{\frown}{ABC} \cong \overset{\frown}{BCD}.

This is the circle form of a fact about segments: add the same piece to two equal pieces and the results are equal.

Boundaries

The measure is the angle at the center, and only there. An angle whose vertex is somewhere else but opens onto the same arc does not equal the arc’s measure. With its vertex on the circle it is half of it, which is the inscribed angle theorem, and from outside the circle it is something else again (angles outside a circle).

Degrees are not length. Two arcs with the same measure on circles of different sizes take the same fraction of their circles but are not the same length, and they are not congruent. Congruent arcs must lie on one circle, or on circles with equal radii.

Two letters mean the minor arc. They cannot name a major arc, and they cannot name a semicircle, since the two halves have the same endpoints. Before reading m\overset{\frown}{AB} off a figure, check that the arc you mean really is the short way round.

Arc addition needs pieces that follow on. The two pieces must meet only at their shared endpoint and continue the same way round. Two arcs that overlap, or that both leave B in the same direction, do not add. An arc with two different endpoints measures less than 360^\circ and more than 0^\circ; if solving for x gives arcs like that, the figure described cannot exist.

The endpoints must be two different points. One point does not cut a circle into two pieces. The whole circle measures 360^\circ, but it has no endpoints and is not a minor arc, a major arc or a semicircle.

An arc is a curve, not a region. The region bounded by two radii and the arc between them is a sector, and its area is a separate question: sectors and their areas.

Where this goes next

An arc’s measure is a fraction of the full turn, and that fraction is the key to two measurements. Applied to the Circumference, it gives the length of the curve, which is arc length. Applied to the area of the whole circle, it gives the area of the slice, which is sector area.

Equally spaced points on a circle cut it into congruent arcs, as on the clock face. Join neighboring points and you have a regular polygon whose central angle is 360^\circ divided by the number of sides.

Arcs are also what angles in a circle are measured against. An angle with its vertex on the circle is half the arc it opens onto, the inscribed angle theorem, and angles formed by chords, secants and tangents are worked out from the arcs they cut off. Congruent arcs go with congruent chords, which the chord theorems make precise.

Exercises

Naming arcs

Use Figure 7 for Exercises 1 to 7. Points A, B, C, D and E lie on \odot O, and \overline{AD} is a diameter.

A B C D E O
70^\circ
75^\circ
35^\circ
105^\circ
75^\circ
Figure 7For Exercises 1 to 7. Each number is the measure of the arc between two neighboring points.
  1. Name the two semicircles with endpoints A and D, and check from the figure that each measures 180^\circ.
  2. Is \overset{\frown}{AC} a minor arc, a major arc or a semicircle? Explain.
  3. Name the major arc with endpoints B and C in two different ways, and find its measure.

Measures from a figure

Use Figure 7. Find each measure.

  1. m\overset{\frown}{AC}
  2. m\overset{\frown}{CE}
  3. m\overset{\frown}{ACE}
  4. m\overset{\frown}{EBD}

From central angles

  1. In \odot O, m\angle POQ = 37^\circ. Find the measures of the minor arc and the major arc with endpoints P and Q.
  2. A major arc measures 241^\circ. Find the measure of the central angle of its minor arc.
  3. \overline{AB} is a diameter, and C is a point on the semicircle \overset{\frown}{ACB} with m\overset{\frown}{AC} = 64^\circ. Find m\overset{\frown}{CB}.

Using algebra

  1. Three radii cut a circle into three arcs measuring (2x)^\circ, (3x + 15)^\circ and (4x - 6)^\circ. Find x and the three arcs.
  2. A minor arc measures (3x + 12)^\circ and the major arc with the same endpoints measures (7x - 2)^\circ. Find x and both arcs.
  3. \overline{AC} and \overline{BD} are diameters of one circle, with A, B, C, D in order round it. m\overset{\frown}{AB} = (2x + 30)^\circ and m\overset{\frown}{CD} = (5x - 33)^\circ. Find x and all four arcs between neighboring points.
  4. Three points cut a circle into three arcs whose measures are in the ratio 2 : 5 : 8. Find the three arcs, and say which of them is a major arc.
  5. Points A and B cut a circle into two arcs measuring (x^2)^\circ and (2x)^\circ. Find every value of x that gives a real figure, and both arcs.

Clocks, wheels and the equator

  1. At 5:00, find the minor arc and the major arc between the spots on the rim that the two hands of a clock point at.
  2. Do the same at 9:30.
  3. A big wheel has 16 cars equally spaced round its rim, numbered 1 to 16 in order. Find the arc measure between neighboring cars, and the minor arc between car 1 and car 7.
  4. Treat the equator as a circle whose center is the center of the Earth. A place’s longitude is the central angle between it and the spot where the equator meets the prime meridian, counted east or west up to 180^\circ. Two places on the equator have longitudes 20^\circ east and 75^\circ west. Find the minor arc of the equator between them.
  5. Two places on the equator have longitudes 150^\circ east and 120^\circ west. Find the minor arc of the equator between them, and say which way round it runs.

Reasoning

  1. Explain why a major arc has to be named with three letters, while two are enough for a minor arc.
  2. Two circles have radii 2 and 5, and an arc of each measures 50^\circ. Are the arcs congruent? Explain.
  3. \overline{AC} and \overline{BD} are diameters of \odot O, with A, B, C, D in order round the circle. Show that \overset{\frown}{AB} \cong \overset{\frown}{CD}.
  4. Can a minor arc be congruent to the major arc with the same endpoints? Explain.
  5. Three points cut a circle into three arcs. Show that at most one of the three can be a major arc.
Answers
  1. \overset{\frown}{ABD} and \overset{\frown}{AED} (the first can also be written \overset{\frown}{ACD}). Going through B and C: 70 + 75 + 35 = 180. Going through E: 105 + 75 = 180.
  2. A minor arc. The short way from A to C passes through B and measures 70^\circ + 75^\circ = 145^\circ, which is less than 180^\circ.
  3. \overset{\frown}{BAC}, \overset{\frown}{BDC} or \overset{\frown}{BEC}, any two of these; it measures 360^\circ - 75^\circ = 285^\circ. Any point on the long way round can stand in the middle.
  4. 145^\circ. Through B: 70^\circ + 75^\circ.
  5. 140^\circ. The two ways from C to E measure 35 + 105 = 140 (through D) and 75 + 70 + 75 = 220 (through B and A); two letters mean the shorter.
  6. 285^\circ. From A through B, C and D to E: 70 + 75 + 35 + 105 = 285, which is 360 - 75.
  7. 255^\circ. From E through A, B and C to D: 75 + 70 + 75 + 35 = 255, which is 360 - 105.
  8. 37^\circ and 323^\circ. The minor arc equals its central angle, and the major arc is 360^\circ - 37^\circ.
  9. 119^\circ. The minor arc is 360^\circ - 241^\circ = 119^\circ, and its central angle measures the same.
  10. 116^\circ. The semicircle measures 180^\circ, and arc addition gives m\overset{\frown}{CB} = 180^\circ - 64^\circ.
  11. x = 39; the arcs are 78^\circ, 132^\circ and 150^\circ. They fill the circle: 2x + (3x + 15) + (4x - 6) = 360, so 9x + 9 = 360 and 9x = 351.
  12. x = 35; the arcs are 117^\circ and 243^\circ. A minor arc and its major arc add to 360^\circ: 10x + 10 = 360. The smaller is under 180^\circ, as a minor arc must be.
  13. x = 21; m\overset{\frown}{AB} = m\overset{\frown}{CD} = 72^\circ and m\overset{\frown}{BC} = m\overset{\frown}{DA} = 108^\circ. \angle AOB and \angle COD are vertical angles, so the arcs are equal: 2x + 30 = 5x - 33, giving 3x = 63. Each neighbor completes a semicircle: 180^\circ - 72^\circ = 108^\circ.
  14. 48^\circ, 120^\circ and 192^\circ; the 192^\circ arc is the major arc. From 2k + 5k + 8k = 360, 15k = 360 and k = 24.
  15. x = 18, giving arcs of 324^\circ and 36^\circ. The two arcs make the whole circle: x^2 + 2x = 360, so (x - 18)(x + 20) = 0. The other root, x = -20, gives (2x)^\circ = -40^\circ, which no arc can measure.
  16. 150^\circ and 210^\circ. The hands point at 12 and 5, five steps of 30^\circ apart.
  17. 105^\circ and 255^\circ. Clockwise from 12, the minute hand is at 180^\circ and the hour hand halfway from 9 to 10, at 9.5 \times 30^\circ = 285^\circ; the difference is 105^\circ.
  18. 22.5^\circ between neighbors, since 360^\circ \div 16 = 22.5^\circ, and 135^\circ from car 1 to car 7. That is six steps, 6 \times 22.5^\circ = 135^\circ; the other way is ten steps, 225^\circ.
  19. 95^\circ. The two places are on opposite sides of the prime meridian, so the central angle between them is 20^\circ + 75^\circ. The major arc is 265^\circ.
  20. 90^\circ, running across the 180^\circ line, not across the prime meridian. From 150^\circ east to 180^\circ is 30^\circ, and from 180^\circ on to 120^\circ west is 60^\circ. The way across the prime meridian is 150^\circ + 120^\circ = 270^\circ.
  21. Two endpoints always give two arcs, so a name with only the endpoints is ambiguous unless a rule picks one. The rule is that two letters mean the shorter, minor arc. To mean the other one, you must say which way to go, and naming a point on it does that.
  22. No. Both are \tfrac{50}{360} of their circles, but the arc on the radius 5 circle is \tfrac{5}{2} times as long, so no rotation, reflection or slide can lay one on the other. Congruent arcs need equal measures on one circle or on circles with equal radii.
  23. The diameters lie on two lines crossing at O, so \angle AOB and \angle COD are vertical angles and are congruent. Neither is a straight angle, so \overset{\frown}{AB} and \overset{\frown}{CD} are minor arcs of one circle with congruent central angles, and so they are congruent.
  24. No. A minor arc measures less than 180^\circ and a major arc more than 180^\circ, so they never have equal measures. The only way to split a circle into two equal arcs is with a diameter, and then both halves are semicircles, which are neither minor nor major.
  25. The three arcs go once round the circle, so by arc addition their measures add to 360^\circ. A major arc measures more than 180^\circ, so two of them would already add to more than 360^\circ, leaving the third arc nothing, or less than nothing. So at most one is major.
kg1128 · updated 2026-10-04 · CC-BY-SA-4.0