Lesson The longer is the chord the larger its central angle is

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<H2>The longer is the chord the larger its central angle is</H2>

In this lesson you will find the proofs to the following statements:

&nbsp;&nbsp;&nbsp;&nbsp;1)&nbsp; the longer is a chord the larger its central angle is;

&nbsp;&nbsp;&nbsp;&nbsp;2)&nbsp; the larger is a central angle the longer the chord is;

&nbsp;&nbsp;&nbsp;&nbsp;3)&nbsp; if two chords in a circle are congruent then their corresponding central angles are congruent;

&nbsp;&nbsp;&nbsp;&nbsp;4)&nbsp; if two central angles in a circle are congruent then their corresponding chords are congruent.

<H3>Theorem 1</H3>The longer is a chord in a circle the larger the corresponding central angle is.

(Surely, &nbsp;when we are talking on the corresponding central angle to a given chord we mean the lesser of the two central angles).

<TABLE> 
 <TR>
 <TD>
Proof

We are given a circle with the center &nbsp;P&nbsp; and two chords &nbsp;AB&nbsp; and &nbsp;CD&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

in the circle &nbsp;(see the &nbsp;Figure 1a&nbsp; or the &nbsp;Figure 1b), &nbsp;where &nbsp;CD&nbsp;

is longer than &nbsp;AB. &nbsp;Let us draw the radii connecting the endpoints of

the chords with the center of the circle. &nbsp;We need to prove that the

central angle &nbsp;LCPD&nbsp; is greater than the central angle &nbsp;LAPB.

For the proof, &nbsp;let me remind you the &nbsp;SSS inequality theorem:

&nbsp;&nbsp;&nbsp;&nbsp;If two sides of one triangle are congruent to the corresponding two  
&nbsp;&nbsp;&nbsp;&nbsp;sides of another triangle, &nbsp;but the third side of the first triangle 
&nbsp;&nbsp;&nbsp;&nbsp;is longer than the third side of the second triangle, &nbsp;then the angle 
&nbsp;&nbsp;&nbsp;&nbsp;included between the two sides of the first triangle is greater than 
&nbsp;&nbsp;&nbsp;&nbsp;the angle included between the two sides of the second triangle.

</TD>
 <TD>
{{{drawing( 200, 200, 0.5, 5.5, -0.5, 4.5, 
 circle(3.0, 2.0, 0.05, 0.05),
 locate ( 3, 2.4, P),

circle(3.0, 2.0, 2.236, 2.236),

line( 2, 0, 4, 0 ), 
        red(line(2,    0, 3, 2)),
        red(line(4,    0, 3, 2)),
            locate ( 1.9, 0, A),
            locate ( 4.0, 0, B),

line( 1, 1, 5, 1 ), 
        red(line(1,    1, 3, 2)),
        red(line(5,    1, 3, 2)),
            locate ( 0.85, 1, C),
            locate ( 5.0,  1, D)
)}}}

Figure 1a. &nbsp;To the Theorem 1&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

</TD>
 <TD>

{{{drawing( 200, 200, 0.5, 5.5, -0.5, 4.5, 
            circle(3.0, 2.0, 0.05, 0.05),
            locate ( 3.15, 2.25, P),

circle(3.0, 2.0, 2.236, 2.236),

line( 2, 0, 4, 0 ), 
        red(line(2,    0, 3, 2)),
        red(line(4,    0, 3, 2)),
            locate ( 1.9, 0, A),
            locate ( 4.0, 0, B),

line( 1, 1, 4, 4 ), 
        red(line(1,    1, 3, 2)),
        red(line(4,    4, 3, 2)),
            locate ( 0.85, 1,   C),
            locate ( 4.0,  4.3, D)
)}}}

Figure 1b. &nbsp;To the Theorem 1
 </TD>
 </TR>
</TABLE>

It was proved in the lesson &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Triangles/Angles-and-sides-inequality-theorems-for-triangles.lesson>Angles and sides inequality theorems for triangles</A>&nbsp; under the topic &nbsp;Triangles&nbsp; of the section &nbsp;Geometry&nbsp; in this site.

Now consider the triangles &nbsp;{{{DELTA}}}CPD&nbsp; and &nbsp;{{{DELTA}}}APB. &nbsp;They both are isosceles triangles and their lateral sides are congruent to the radius of the circle.

Therefore, &nbsp;the &nbsp;Theorem 1&nbsp; directly follows the &nbsp;SSS inequality theorem.

The proof is completed.

<H3>Theorem 2</H3>The larger is a central angle in a circle the longer the corresponding chord is.

(We consider here central angles that are lesser the straight angle (180�)).

<TABLE> 
 <TR>
 <TD>
Proof

We are given a circle with the center &nbsp;P&nbsp; and two central angles &nbsp;LAPB&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

and &nbsp;LCPD&nbsp; in the circle &nbsp;(see the &nbsp;Figure 2a&nbsp; or the &nbsp;Figure 2b), &nbsp;where

LCPD&nbsp; is greater than &nbsp;LAPB. &nbsp;We need to prove that the chord &nbsp;CD&nbsp; is

longer than the chord &nbsp;AB.

For the proof, &nbsp;use the &nbsp;SAS inequality theorem:

&nbsp;&nbsp;&nbsp;&nbsp;If two sides of one triangle are congruent to the corresponding two sides 
&nbsp;&nbsp;&nbsp;&nbsp;of another triangle, &nbsp;and the included angle of the first triangle is greater 
&nbsp;&nbsp;&nbsp;&nbsp;than the included angle of the second triangle, &nbsp;then the third side of the  
&nbsp;&nbsp;&nbsp;&nbsp;first triangle is longer than the third side of the second triangle.

</TD>
 <TD>
{{{drawing( 200, 200, 0.5, 5.5, -0.5, 4.5, 
 circle(3.0, 2.0, 0.05, 0.05),
 locate ( 3, 2.4, P),

circle(3.0, 2.0, 2.236, 2.236),

line( 2, 0, 4, 0 ), 
        red(line(2,    0, 3, 2)),
        red(line(4,    0, 3, 2)),
            locate ( 1.9, 0, A),
            locate ( 4.0, 0, B),

line(1.42, 0.42, 5, 1 ), 
        red(line(1.42, 0.42, 3, 2)),
        red(line(5,    1, 3, 2)),
            locate ( 1.2, 0.5, C),
            locate ( 5.0,  1, D)
)}}}

Figure 2a. &nbsp;To the Theorem 2&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

</TD>
 <TD>
{{{drawing( 200, 200, 0.5, 5.5, -0.5, 4.5, 
 circle(3.0, 2.0, 0.05, 0.05),
 locate ( 3.15, 2.25, P),

circle(3.0, 2.0, 2.236, 2.236),

line( 2, 0, 4, 0 ), 
        red(line(2,    0, 3, 2)),
        red(line(4,    0, 3, 2)),
            locate ( 1.9, 0, A),
            locate ( 4.0, 0, B),

line( 1, 1, 4, 4 ), 
        red(line(1,    1, 3, 2)),
        red(line(4,    4, 3, 2)),
            locate ( 0.85, 1,   C),
            locate ( 4.0,  4.3, D)
)}}}

Figure 2b. &nbsp;To the Theorem 2
 </TD>
 </TR>
</TABLE>

Therefore, &nbsp;the &nbsp;Theorem 2&nbsp; directly follows the &nbsp;SAS inequality theorem.

The proof is completed.

<H3>Theorem 3</H3>If two chords in a circle are congruent then their corresponding central angles are congruent.

Proof

The proof is similar to that of the &nbsp;Theorem 1.

Instead of referring to the &nbsp;SSS inequality theorem, &nbsp;simply use the &nbsp;SSS test for the triangles congruency.

<H3>Theorem 4</H3>If two central angles in a circle are congruent then their corresponding chords are congruent.

Proof

The proof is similar to that of the &nbsp;Theorem 2.

Instead of referring to the &nbsp;SAS inequality theorem, &nbsp;simply use the &nbsp;SAS test for the triangles congruency.

<H3>Summary</H3>Two chords in a circle are congruent if and only if their corresponding central angles are congruent.

Two chords in a circle are congruent if and only if their corresponding arcs are congruent.

In a circle, &nbsp;the longer chord has larger corresponding central angle and longer corresponding arc; &nbsp;the shorter chord has smaller corresponding central angle and shorter corresponding arc.

My other lessons on circles in this site are

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Circles/A-circle-its-chords-tangent-and-secant-lines-the-major-definitions.lesson>A circle, its chords, tangent and secant lines - the major definitions</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Circles/The-chords-in-a-circle-and-the-radii-perpendicular-to-the-chords.lesson>The chords of a circle and the radii perpendicular to the chords</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Circles/A-tangent-line-to-a-circle-is-perpendicular-to-the-radius-drawn-to-the-tangent-point.lesson>A tangent line to a circle is perpendicular to the radius drawn to the tangent point</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Circles/An-inscribed-angle.lesson>An inscribed angle in a circle</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Circles/Two-parallel-secants-to-a-circle-cut-off-congruent-arcs.lesson>Two parallel secants to a circle cut off congruent arcs</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Circles/The-angle-between-two-chords-intersecting-inside-a-circle.lesson>The angle between two chords intersecting inside a circle</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Circles/The-angle-between-two-secants-intersecting-outside-a-circle.lesson>The angle between two secants intersecting outside a circle</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Circles/The-angle-between-a-chord-and-a-tangent-line-to-a-circle.lesson>The angle between a chord and a tangent line to a circle</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Circles/Tangent-segments-to-a-circle-from-a-point-outside-the-circle.lesson>Tangent segments to a circle from a point outside the circle</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Circles/The-converse-theorem-on-inscribed-angles.lesson>The converse theorem on inscribed angles</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Circles/The-parts-of-chords-intersecting-inside-a-circle.lesson>The parts of chords that intersect inside a circle</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Circles/The-metric-relations-for-secants-intersecting-outside-a-circle.lesson>Metric relations for secants intersecting outside a circle</A> &nbsp;and

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Circles/Metric-relations-for-a-tangent-and-a-secant-lines-released-from-a-point-outside-a-circle.lesson>Metric relations for a tangent and a secant lines released from a point outside a circle</A>

under the current topic, &nbsp;and

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/HOW-TO-bisect-an-arc-in-a-circle-using-a-compass-and-a-ruler.lesson>HOW TO bisect an arc in a circle using a compass and a ruler</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/HOW-TO-find-the-center-of-a-circle-given-by-two-chords.lesson>HOW TO find the center of a circle given by two chords</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-a-radius-and-a-tangent-line-to-a-circle.lesson>Solved problems on a radius and a tangent line to a circle</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-inscribed-angles.lesson>Solved problems on inscribed angles</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/The-property-of-the-angles-of-a-quadrilateral-inscribed-in-a-circle.lesson>A property of the angles of a quadrilateral inscribed in a circle</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Isosceles-trapezoid-can-be-inscribed-in-a-circle.lesson>An isosceles trapezoid can be inscribed in a circle</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/HOW-TO-construct-a-tangent-line-to-a-circle.lesson>HOW TO construct a tangent line to a circle at a given point on the circle</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/HOW-TO-construct-a-tangent-line-to-a-circle-through-a-given-point-outside-the-circle.lesson>HOW TO construct a tangent line to a circle through a given point outside the circle</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/HOW-TO-construct-a-common-exterior-tangent-line-to-two-circles.lesson>HOW TO construct a common exterior tangent line to two circles</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/HOW-TO-construct-a-common-interior-tangent-line-to-two-circles.lesson>HOW TO construct a common interior tangent line to two circles</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-chords-that-intersect-within-a-circle.lesson>Solved problems on chords that intersect within a circle</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-secants-that-intersect-outside-a-circle.lesson>Solved problems on secants that intersect outside a circle</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-a-tangent-and-a-secant-lines-released-from-a-point-outside-a-circle.lesson>Solved problems on a tangent and a secant lines released from a point outside a circle</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/geometry/The-radius-of-a-circle-inscribed-into-a-right-angled-triangle.lesson>The radius of a circle inscribed into a right angled triangle</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-tangent-lines-released-from-a-point-outside-a-circle.lesson>Solved problems on tangent lines released from a point outside a circle</A>

under the topic &nbsp;Geometry&nbsp; of the section &nbsp;Word problems.

The overview of lessons on Properties of Circles is in this file &nbsp;<A HREF=https://www.algebra.com/algebra/homework/Circles/PROPERTIES-OF-CIRCLES-THEIR-CHORDS-SECANTS-AND-TANGENTS.lesson>PROPERTIES OF CIRCLES, THEIR CHORDS, SECANTS AND TANGENTS</A>.
You can use the overview file or the list of links above to navigate over these lessons.

To navigate over all topics/lessons of the Online Geometry Textbook use this file/link &nbsp;<A HREF=https://www.algebra.com/algebra/homework/Triangles/GEOMETRY-your-online-textbook.lesson>GEOMETRY - YOUR ONLINE TEXTBOOK</A>.