Circumferential angle number theorem, how to understand the circumferential angle theorem?

In a circle, the degree of the central angle of the arc of a circle is 360°, the central angle of the semicircle is 180°, and the central angle of the arc of 1 4 is 90°....That is, the degree of the arc is equal to the degree of the central angle of the circle.

In the same circle, if you have made the diameter of the vertices of the circumferential corner, you can see that since the radius is equal, the central angle of the circle is equal to twice the circumferential angle (the same arc), so the degree of the circumferential angle is equal to half of the degree of the central angle to which it is opposed.

Thus, the degree of the circumferential angle is equal to half the degree of the arc it opposes. I don't know if you understand?

Vernacular translation: Translate the circumferential angle first.

1. The circumferential angle is the angle formed by the connection between the two ends of an arc and any point other than these two endpoints on the circle.

2. The degree of the arc: The degree of the arc is the degree of the central angle of the circle.

Then the theorem of the number of circumferential angles is translated as: the degree of the angle of the same arc is twice the number of the circumferential angle of the arc; It can also be said that the number of circumferential angles of the same arc is half of the degree of the angle (the central angle) of the arc.

Make a circle first, then make a circumferential angle with the diameter of the circle as one side, set the center of the circle as O, the two intersection points of the diameter and the circle are b and c, and the point A is another point on the triangle ABC, and intersect with the circle at the point A.

Proof: Connect AO because the radius of the circles is equal.

So oa=ob=oc

So the triangle OAB and the triangle OAC are isosceles triangles, the angle OBA = angle OAB, and the angle OAC = the angle OCA

Angular BAC = Angular OAB + Angular OAC

180 degrees-angle aob) 2+(180 degrees-angle aob) 2 (180 degrees-angle aob) 2+[180 degrees-(180-angle aob) 2 (180 degrees-angle aob) 2+(180 degrees-180 + angle aob) 2 (180 degrees-angle aob + 180 degrees -180 degrees + angle aob) 2180 degrees 2

90 degrees. If you don't know, ask me again.

I understand the picture as soon as I see it, but unfortunately I can't spend it.

The circumferential angle theorem states that the circumferential angle of an arc is equal to half of the angle to which it is opposed. This theorem is called the circumferential angle theorem. This theorem reflects the relationship between the circumferential angle and the central angle of the circle.

1.In the same circle or equal circle, the circumferential angles of the same or equal arcs are equal, and the arcs of the circumferential angles of the same are also equal.

2.The circumferential angle of the semicircle (diameter) is a right angle; The chord to which the circumferential angle of 90° is aligned is the diameter.

3.The diagonal diagonal of the circumscribed quadrilateral of the circle complements each other, and any one outer angle is equal to its inner diagonal.

Circumferential Angle:

1) Definition of circumferential angle:

The angle where the vertex is on the circle and both sides intersect the circle is called the circumferential angle.

2) Circumferential angle theorem:

The circumferential angle of an arc is equal to half of the central angle of the circle to which it is opposed.

Corollary: The circumferential angles of the same or equal arcs are equal.

The circumferential angle of the semicircle (or diameter) is a right angle, and the chord of the circumferential angle of 90° is the diameter.

In a circle or an equal circle, one of the two circumferential angles, the two central angles, the two arcs, and the two strings are equal in one group, and the other groups of quantities corresponding to them are also equal.

3) Modification of the multi-sided belt within the circle:

If all the vertices of a polygon are on the same circle, the polygon is called the inner polygon of the circle, and the circle is called the outer circle of the polygon.

4) The nature of the circumscribed quadrilateral within the circle:

The circle is complemented diagonally by the quadrilaterals.

The above content refers to: Encyclopedia - Circumferential Angle Theorem.

The law of circumferential angles is as follows:

The circumferential angle theorem states that the circumferential angle of an arc is equal to one and a half of the central angle of the circle. This theorem is called the circumferential angle theorem. This theorem reflects the relationship between the circumferential angle and the central angle of the circle.

Proof: It is known that in O, BOC and the circumferential angle BAC are the same as the arc BC, and the eggplant year is verified: BOC=2 BAC.

oa=oc；bac= aco (equilateral equilateral).

Theorem corollary: 1. The circumferential angle of an arc is equal to half the age of the central angle of the circle it opposes.

2. The degree of circumference is equal to half of the radian of which it is opposed.

3. In the same circle or equal circle, the circumferential angle of the same arc or equal arc is equal; The arcs opposite the circumferential angles that are equal are also equal.

4. The circumferential angle of the semicircle (diameter) is a right angle.

The circumferential angle of the chord is the diameter.

6. Equal arc pairs equal circumferential angles. Note that in a circle, there are an infinite number of circumferential angles of the same string.

Circumferential Angle Definition:

The circumferential angle was originally called the Janet angle because its vertex was on the circumference of the circle and the two sides intersected the circle, so it was renamed the Annette angle. In the same circle or equal circle, if the circumferential angles of the two circles are equal, then the strings (or arcs) to which they are paired are also equal; Conversely, the circumferential angles of the equal arcs are equal. The circumferential angles of the equal chord are equal or complementary, and the degree of the circumferential angle is equal to half of the degree of the arc to which it opposes.

For a circumferential angle, there must be an arc inside the corner, and the circumferential angle is usually said to be the circumferential angle on the arc, or the circumferential angle to which the arc is opposed. In addition, there is an arc on the outside of the angle, and we also say that the circumferential angle is the circumferential angle contained in this arc.

Circumferential angle theorem: The degree of a circumferential angle is equal to half of the number of angles at the center of the arc to which it is opposed.

Proof of theorem. It is known that in O, BOC and the circumferential angle BAC are the same arc BC, and it is verified that BOC=2 Bac

Proof: Case 1:

Figure 1oa, oc are the radius.

Solution: oa=oc

bac= aco (equilateral equilateral).

BOC is the outer corner of AOC.

boc=∠bac+∠aco=2∠bac

Scenario 2: Connect the AO and extend the AO to D

Figure 2oa, ob, oc are the radii.

Solution: oa=ob=oc

bad= abo, cad= aco

BOD and COD are the outer angles of AOB and AOC respectively.

bod= bad+ abo=2 bad (the outer angles of the triangle are equal to the sum of two non-adjacent inner angles).

cod = cad + aco = 2 cad (the outer angles of a triangle are equal to the sum of two non-adjacent inner angles).

boc=∠bod+∠cod=2(∠bad+∠cad)=2∠bac

Case 3: <>

Figure 3 connects AO and extends AO to D to connect OA, OB.

Solution: OA, OB, OC, are the radius.

oa=ob=oc

bad= abo(isosceles triangles have equal bottom angles), cad= aco(oa=oc).

dob and doc are the outer angles of aob and aoc, respectively.

dob= bad+ abo=2 bad(the outer corner of the triangle is equal to the sum of the two adjacent inner angles).

doc = cad + aco = 2 cad (the outer angles of the triangle are equal to the sum of two non-adjacent inner angles).

boc=∠doc-∠dob=2(∠cad-∠bad)=2∠bac

What about when the central angle of the circle is equal to 180 degrees?

Looking at the diagram of case 1, the central angle of the circle is aob=180 degrees, and the circumferential angle is acb, obviously because oca= oac= boc 2

ocb=∠obc=∠aoc/2

So oca+ ocb=( boc+ abc) 2=90 degrees.

So 2 ACB = AOC

What about when the central angle of the circle is greater than 180 degrees?

Looking at the diagram of case 3, the central angle of the circle is (360 degrees - aob), Wang Heng refers to the circumferential angle is ACB, as long as the co-intersection is extended at the point E, it can be seen from the case that the central angle of the circle is equal to 180 degrees cae= cbe=90 degrees.

So ACB+AEB=180 degrees, i.e. ACB=180 degrees- AEB

From case 2, it can be seen that AOB = 2 AEB

So 360 degrees - AOB = 2 (180 degrees - AEB) = 2 ACB

Circumferential angle theorem: The circumferential angle of the same arc is equal to half of the angle of the center of the circle to which it is opposite.

Corollary of the circumferential angle theorem:

The circumferential angles of the same or equal arcs are equal; In the same circle or equal circle, the arc opposite the equal circumference of the sail angle is the equal arc.

The circumferential angle of the semicircle or diameter is a right angle; The circumferential angle is the semicircle of the arc to which the right angle is right, and the chord to which is the diameter.

If the middle line on one side of the triangle is equal to half of this side, then the triangle is a right triangle.

Circumferential angle theorem: The circumferential angle of an arc is equal to half of the angle of the center of the circle to which it opposes.

Proof: It is known that in O, Boc and the circumferential angle Bac are the same arc BC, and it is verified that BoC=2 Bac

Proof: Case 1:

Figure 1<>

oa and oc are radius.

Solution: oa=oc

bac= aco (equilateral equilateral).

BOC is the outer corner of AOC.

boc=∠bac+∠aco=2∠bac

Scenario 2: Connect the AO and extend the AO to D

Figure 2<>

oa, ob, oc are the radius.

Solution: oa=ob=oc

bad= abo, cad= aco

BOD and COD are the outer angles of AOB and AOC respectively.

bod= bad+ abo=2 bad (the outer angles of the triangle are equal to the sum of two non-adjacent inner angles).

cod = cad + aco = 2 cad (the outer angles of a triangle are equal to the sum of two non-adjacent inner angles).

boc=∠bod+∠cod=2(∠bad+∠cad)=2∠bac

Situation 3: Figure no sail 3

Connect AO and extend AO to D Connect OA, OB.

Solution: OA, OB, OC, are the radius.

oa=ob=oc

bad= abo(equilateral equilateral angle), cad= aco(oa=oc).

dob and doc are the outer angles of aob and aoc, respectively.

dob= bad+ abo=2 bad(the outer angles of the triangle are equal to the sum of two non-adjacent inner angles).

doc = cad + aco = 2 cad (the outer angles of the triangle are equal to the sum of two non-adjacent inner angles).

boc=∠doc-∠dob=2(∠cad-∠bad)=2∠bac

This proves that boc=2 bac