The points `A,B,C,D,E` are marked on the circumference of a circle in clockwishdirection such that `?_ABC = 130^@ and /_CDE=110^@`. The measure of `/_ACE` degress is

Options

A. `50^(@)`
B. `60^(@)`
C. `70^(@)`
D. `80^(@)`

Answer

To find the measure of angle \( \angle ACE \) given the angles \( \angle ABC = 130^\circ \) and \( \angle CDE = 110^\circ \), we can follow these steps:

### Step 1: Identify the Given Angles
We have:
- \( \angle ABC = 130^\circ \)
- \( \angle CDE = 110^\circ \)

### Step 2: Analyze the Quadrilateral \( ABCE \)
Since points \( A, B, C, E \) are on the circumference of the circle, we can form quadrilateral \( ABCE \). In any cyclic quadrilateral, the sum of the opposite angles is \( 180^\circ \).

### Step 3: Find \( \angle AEC \)
Using the property of cyclic quadrilaterals:
\[ \angle ABC + \angle AEC = 180^\circ \]
Substituting the known value:
\[ 130^\circ + \angle AEC = 180^\circ \]
Solving for \( \angle AEC \):
\[ \angle AEC = 180^\circ - 130^\circ = 50^\circ \]

### Step 4: Analyze the Quadrilateral \( CDEA \)
Now, consider quadrilateral \( CDEA \). Again, using the property of cyclic quadrilaterals:
\[ \angle CDE + \angle CAE = 180^\circ \]
Substituting the known value:
\[ 110^\circ + \angle CAE = 180^\circ \]
Solving for \( \angle CAE \):
\[ \angle CAE = 180^\circ - 110^\circ = 70^\circ \]

### Step 5: Find \( \angle ACE \)
Now we can use the angles in triangle \( ACE \):
\[ \angle CAE + \angle AEC + \angle ACE = 180^\circ \]
Substituting the known values:
\[ 70^\circ + 50^\circ + \angle ACE = 180^\circ \]
Combining the angles:
\[ 120^\circ + \angle ACE = 180^\circ \]
Solving for \( \angle ACE \):
\[ \angle ACE = 180^\circ - 120^\circ = 60^\circ \]

### Final Answer
The measure of \( \angle ACE \) is \( 60^\circ \).

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