given: chord AB is parallel to chord CD
prove if two chords of a circle are parallel, the two arcs between the chords are congruent.
Prove: arc AC is congruent to arc BD
Draw a figure by hand and insert all the points, lines as angles as directed below in order to follow the proof.
Insert the centre of the circle and call it O, the radius of the circle is r.
Draw another line parallel to AB and CD through O and call it EF, where E is between A and C, F is between B and D.
Consider isosceles triangle ABO where AO=BO=r:
α=∠OAB=∠OBA (isosceles triangle)
Now consider parallel lines AB and EOF.
∠AOE=∠BAO (alternate angles)
∠BAO=∠ABO (isosceles triangle)
∠ABO=∠BOF (alternate angles)
Therefore
∠EOA=∠FOB
Similarly,
∠EOC=∠FOD
Thus
∠AOC=∠BOD
Length of arc AEC=r*∠AOC
Length of arc BFD=r*∠BOD
Therefore arc AEC = arc BFD
A circle has a diameter of 20 inches and a central angle AOB that measures 160°. What is the length of the intercepted arc AB? Use 3.14 for pi and round your answer to the nearest tenth
To prove that if two chords of a circle are parallel, the two arcs between the chords are congruent, we can use the following steps:
Step 1: Given that chord AB is parallel to chord CD.
Step 2: Draw a line segment from the center of the circle to point E on chord AB and another line segment from the center of the circle to point F on chord CD. Label the center of the circle as O.
Step 3: Since AB is parallel to CD, angle EOA is congruent to angle FOD. This is because alternate interior angles are congruent when two lines are parallel.
Step 4: Since angle EOA is congruent to angle FOD, angle EOA is a central angle of arc AC and angle FOD is a central angle of arc BD.
Step 5: According to the definition of a central angle, the measure of a central angle is equal to the measure of its intercepted arc.
Step 6: Therefore, arc AC is congruent to arc BD, as the central angles EOA and FOD are congruent.
Thus, we have proven that if two chords of a circle are parallel, the two arcs between the chords are congruent.
To prove that arc AC is congruent to arc BD, we can use the fact that when two chords of a circle are parallel, the arcs between the chords are congruent. Here's how we can prove it step by step:
1. Given that chord AB is parallel to chord CD.
2. Draw a line that intersects both chords, such as line EF, so that it cuts through both arcs AC and BD.
3. Since chord AB is parallel to chord CD, and line EF intersects both chords, we can apply the property of parallel lines that the corresponding angles formed by the line and the chords are congruent.
4. Let's call the angle between chords AB and CD as angle θ.
5. Due to the congruence of corresponding angles, we can establish that angle AEF is equal to angle BFE, and angle CEF is equal to angle DFE.
6. Now, if we consider arcs AC and BD, we can observe that each arc corresponds to an angle on the circle.
7. From step 5, we know that angle AEF is congruent to angle BFE, which implies that the arc AC, corresponding to angle AEF, is congruent to the arc BD, corresponding to angle BFE.
8. Similarly, based on the congruence of angles CEF and DFE, we can conclude that arc AC is also congruent to arc BD.
Therefore, we have successfully proven that if two chords of a circle are parallel, the two arcs between the chords are congruent. In this case, arc AC is congruent to arc BD, as stated.