CALCU-consider ⊙U at the right. line IQ is tangent

consider ⊙U at the right. line IQ is tangent to ⊙U at S. m∠SGD=40; m⌒GS=140

Solution by Steps

step 1

Identify the relationship between the angle formed by a tangent and a chord and the arcs it intercepts

step 2

Use the fact that the measure of the angle formed by a tangent and a chord (m∠SGD) is half the measure of the intercepted arc (m⌒GS)

step 3

Calculate the measure of angle SGD:

m∠SGD = \frac{1}{2} m⌒GS

step 4

Substitute the given values:

40 = \frac{1}{2} \times 140

step 5

Solve for m⌒GS:

m⌒GS = 2 \times 40

step 6

Calculate m⌒GS:

m⌒GS = 80

Answer

The measure of arc GS (m⌒GS) is 80 degrees.

Key Concept

The measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc.

Explanation

Since m∠SGD is given as 40 degrees and it is formed by a tangent and a chord, the measure of the intercepted arc GS is twice that of the angle, which is 80 degrees.

consider ⊙U at the right. line IQ is tangent to ⊙U at S. m∠SGD=40; m⌒GS=140 1. What is m∠GSQ?

Solution by Steps

step 1

The measure of the angle formed by a tangent and a chord through the point of contact is equal to half the measure of the intercepted arc

step 2

Apply this property to find

m\angle GSQ

using

m\overset{\frown}{GS}

:

m\angle GSQ = \frac{1}{2} m\overset{\frown}{GS}

step 3

Substitute the given measure of the intercepted arc

m\overset{\frown}{GS} = 140^\circ

:

m\angle GSQ = \frac{1}{2} \times 140^\circ

step 4

Calculate the measure of

m\angle GSQ

:

m\angle GSQ = 70^\circ

Answer

The measure of angle

m\angle GSQ

is

70^\circ

.

Key Concept

The measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc.

Explanation

The angle

\angle GSQ

is formed by the tangent line IQ and the chord GS, so its measure is half of the measure of the intercepted arc

\overset{\frown}{GS}

, which is

140^\circ

. Therefore,

m\angle GSQ

is

70^\circ

.

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