Question

Given: circle k(O), m
AM
=125°,
m
EF
=31°, m∠MAF=75°
Find: m∠AME

Answer 1

Answer:

$m<A.M.E=58\°$

Step-by-step explanation:

step 1

Find the measure of angle M.E.F

we know that

In an inscribed quadrilateral opposite angles are in fact supplements for each other

so

$m<M.A.F+m<M.E.F=180\°$

$m<M.E.F=180\°-75\°=105\°$

step 2

Find the measure of arc M.A.F

we know that

The inscribed angle measures half that of the arc comprising

so

$m<M.E.F=\frac{1}{2}(arc\ M.A.F)$

we have

$m<M.E.F=105\°$

substitute

$105\°=\frac{1}{2}(arc\ M.A.F)$

$arc\ M.A.F=210\°$

step 3

Find the measure of arc A.F

$arc\ M.A.F=arc\ A.M+arc\ A.F$

we have

$arc\ M.A.F=210\°$

$arc\ A.M=125\°$

substitute

$210\°=125\°+arc\ A.F$

$arc\ A.F=210\°-125\°=85\°$

step 4

Find the measure of angle A.M.E

we know that

The inscribed angle measures half that of the arc comprising

so

$m<A.M.E=\frac{1}{2}(arc\ A.F.E)$

we have

$arc\ A.F.E=arc\ A.F+arc\ E.F=85\°+31\°=116\°$

substitute

$m<A.M.E=\frac{1}{2}(116\°)=58\°$

Answer 2

The angle ∠AME is the angle subtended at the circumference by the arc $m \widehat{EFA}$.

Correct response:

• m∠AME is 58°

Methods used for the calculation:

The given parameters are;

$m \widehat{AM}$ = 125°

$m\widehat{EF}$ = 31°

m∠MAF = 75°

Required:

m∠AME

Solution:

$m \widehat{MEF}$ = 2 × m∠MAF

Therefore;

$m \widehat{MEF}$ = 2 × 75° = 150°

$m \widehat{AM}$ + $m \widehat{MEF}$ + $m \widehat{FA}$ = 360° sum of arcs of a circle postulate

Therefore;

$m \widehat{FA}$ = 360° - ($\mathbf{m \widehat{AM}}$ + $\mathbf{m\widehat{MEF}}$)

Which gives;

$m\widehat{FA}$ = 360° - (125° + 150°) = 85°

• $m \widehat{EFA}$ = $\mathbf{m \widehat{FA}}$ + $\mathbf{m \widehat{EF}}$

Therefore;

$m\widehat{EFA}$ = 85° + 31° = 116°

$m \widehat{EFA}$ = 2 × m∠AME (angle at center is twice angle at the circumference)

Therefore;

$m\angle AME = \mathbf{\dfrac{m \widehat{EFA}}{2} }$

$m \angle AME = \dfrac{116^{\circ}}{2} = 58^{\circ}$

• m∠AME = 58°

Learn more about circle theorems here:

https://brainly.com/question/16879446