# CONGRUENT CHORDS AND CENTRAL ANGLES

If two chords of a circle are congruent, then they determine central angles which are equal in measure.

Problem 1 :

Find the value of x. Solution :

PQ = SR

mSCR = 60˚

So, mSCR = mPCQ = 60˚

x = 60˚

Problem 2 : Solution :

The arc lengths are the same and the corresponding angles are the same.

UV = YZ

x + 10˚ = 40˚

x = 40˚ - 10˚

x = 30˚

Problem 3 : Solution :

The arc lengths are the same and the corresponding angles are the same.

MN = KJ

x + 5˚ = 50˚

x = 50˚ - 5˚

x = 45˚

Problem 4 : Solution :

The arc lengths are the same and the corresponding angles are the same.

AB = DE

x + 6˚ = 35˚

x = 35˚ - 6˚

x = 29˚

Problem 5 : Solution :

The arc lengths are the same and the corresponding angles are the same.

AB = DE

x + 45˚ = 4x˚

4x - x = 45˚

3x = 45˚

x = 15˚

Problem 6 :

In the diagram below, AD and BE are diameters of ʘF. Find the measures. 1) m arc DE            2) m arc BC

3) m arc AE            4) m arc BCD

5)m arc ABC           6) m arc ADE

Solution :

40 + ∠BFC + 65 = 180

105 + ∠BFC = 180

∠BFC = 180 - 105

∠BFC = 75

∠BFC + ∠CFD + ∠DFE = 180

75 + 65 + ∠DFE = 180

140 + ∠DFE = 180

∠DFE = 180 - 140

∠DFE = 40

1) m arc DE  = 40

2) m arc BC = 75

3) m arc AE :

∠AFB + ∠BFC + ∠CFD + ∠DFE = Measure of arc AE

40 + 75 + 65 + 40 = Measure of arc AE

Measure of arc AE = 220

4) m arc BCD :

∠BFC + ∠BCD = Measure of arc BCD

75 + 65 = Measure of arc BCD

Measure of BCD = 140

5) m arc ABC :

∠AFB + ∠BFC = Measure of arc ABC

40 + 75 = Measure of arc ABC

Measure of arc ABC = 115

∠DFE = 40 = m arc ADE

## Recent Articles 1. ### Graphing Exponential Growth and Decay Worksheet

Dec 08, 23 08:03 AM

Graphing Exponential Growth and Decay Worksheet

2. ### Finding the Specific Term from the Given Explicit Formula Worksheet

Dec 08, 23 07:32 AM

Finding the Specific Term from the Given Explicit Formula Worksheet