Question

1. find the measure of y. 116° 129° 120° 130° 135° 125° 145° 129° 130° 134°

Explanation:

Step1: Recall the formula for the sum of interior angles of a polygon.

The formula for the sum of interior angles of an \( n \)-sided polygon is \( S=(n - 2)\times180^{\circ} \). Here, the polygon is a heptagon (7 - sided), so \( n = 7 \).
Calculate the sum: \( S=(7 - 2)\times180^{\circ}=5\times180^{\circ} = 900^{\circ} \).

Step2: Sum the given angles.

Add the known angles: \( 129^{\circ}+116^{\circ}+120^{\circ}+135^{\circ}+125^{\circ}+130^{\circ}+y^{\circ} \)? Wait, no, wait. Wait, let's count the angles. Wait, the figure is a heptagon? Wait, no, wait the number of angles: 129, 116, 120, y, 125, 135, 130. Wait, that's 7 angles? Wait, no, wait the figure: let's count the vertices. Wait, the angles given are 129, 130, 135, 125, y, 120, 116. So 7 angles, so \( n = 7 \). So sum of interior angles is \( (7 - 2)\times180=900^{\circ} \).

Now sum the known angles: \( 129+130+135+125+120+116 \). Let's calculate:

\( 129+130 = 259 \); \( 259+135 = 394 \); \( 394+125 = 519 \); \( 519+120 = 639 \); \( 639+116 = 755 \).

Step3: Solve for \( y \).

We know that the sum of all interior angles is \( 900^{\circ} \), so \( 755 + y=900 \).
Subtract 755 from both sides: \( y = 900 - 755=145 \)? Wait, no, wait that can't be. Wait, wait I must have miscounted the number of sides. Wait, let's check the figure again. Wait, the figure: let's count the sides. Wait, the angles are 129, 130, 135, 125, y, 120, 116. Wait, that's 7 angles? Wait, no, maybe it's a hexagon? Wait, no, the figure looks like a heptagon? Wait, no, maybe I made a mistake. Wait, let's count the number of angles: 129, 130, 135, 125, y, 120, 116. That's 7 angles. Wait, but maybe it's a hexagon? Wait, no, the formula for sum of interior angles: for \( n \) sides, \( (n - 2)\times180 \). Let's check the sum again. Wait, maybe I added wrong. Let's add again:

129 + 130 = 259; 259 + 135 = 394; 394 + 125 = 519; 519 + 120 = 639; 639 + 116 = 755. Then 755 + y = (n - 2)180. Wait, if n=7, (7-2)180=900. So y=900-755=145. But 145 is one of the options (first option). Wait, but let's confirm the number of sides. Wait, the figure: let's see, the angles are 129, 130, 135, 125, y, 120, 116. That's 7 angles, so 7 sides. So sum is 900. So 755 + y = 900 => y=145.

Answer:

145° (corresponding to the first option: 145°)