Question

polygon.
3
147° 70°
102° 128°
x°
138°
x° =
6
x°
140° 125°
210°
32°
x° =

Answer:

Problem 3 (Hexagon? Wait, no, let's check the number of sides. Wait, the first figure: let's count the angles. Wait, the first polygon has angles: 102°, 147°, 70°, 128°, 138°, and x°. Wait, that's 6 angles? Wait, no, maybe a hexagon? Wait, the formula for the sum of interior angles of a polygon is $(n - 2) \times 180^\circ$, where $n$ is the number of sides.

Wait, first, let's identify the number of sides. Let's count the vertices. The first figure: let's see, the angles are 102, 147, 70, 128, 138, and x. So that's 6 angles? Wait, no, maybe a hexagon? Wait, no, maybe a pentagon? Wait, no, let's count again. Wait, the first figure: let's see, the angles are 102°, 147°, 70°, 128°, 138°, and x°. Wait, that's 6 angles, so n=6. Then sum of interior angles is (6-2)180 = 4180 = 720°.

Now sum the given angles: 102 + 147 + 70 + 128 + 138. Let's calculate that:

102 + 147 = 249

249 + 70 = 319

319 + 128 = 447

447 + 138 = 585

Then x = 720 - 585 = 135? Wait, no, wait, maybe I miscounted the number of sides. Wait, maybe it's a hexagon? Wait, no, maybe a pentagon? Wait, no, let's check again. Wait, the first figure: let's see, the angles are 102, 147, 70, 128, 138, and x. So 6 angles, so n=6. Sum is (6-2)*180 = 720. Then sum of given angles: 102 + 147 + 70 + 128 + 138 = let's recalculate:

102 + 147 = 249

249 + 70 = 319

319 + 128 = 447

447 + 138 = 585

Then x = 720 - 585 = 135? Wait, but maybe I made a mistake. Wait, maybe it's a pentagon? Wait, no, 5 angles? Wait, no, the angles are 102, 147, 70, 128, 138, x. That's 6 angles. So n=6. So sum is 720. Then 102 + 147 = 249; 249 + 70 = 319; 319 + 128 = 447; 447 + 138 = 585; 585 + x = 720; so x = 720 - 585 = 135. Wait, but let's check again.

Wait, maybe the first polygon is a hexagon (6 sides). So sum of interior angles: (6-2)*180 = 720. Then sum of given angles: 102 + 147 + 70 + 128 + 138 = 102+147=249; 249+70=319; 319+128=447; 447+138=585. Then x = 720 - 585 = 135. So x = 135°.

Problem 6: Let's identify the polygon. The angles are 140°, 90° (right angle), 210°? Wait, no, 210° is a reflex angle? Wait, no, maybe it's a heptagon? Wait, no, let's count the angles. The angles are: 140°, 90° (right angle), 210°, 32°, 125°, and x. Wait, no, maybe a hexagon? Wait, no, let's use the formula for the sum of interior angles, but also, for a polygon with n sides, the sum is (n-2)180. But if there are reflex angles (greater than 180°), we still use the same formula. Wait, but let's count the number of sides. Let's see the figure: it has a right angle (90°), 140°, 210°, 32°, 125°, and x. Wait, that's 6 angles? Wait, no, maybe 7? Wait, no, let's check again. Wait, the figure has a right angle, 140°, 210°, 32°, 125°, and x. Wait, maybe 6 sides? Wait, no, let's calculate the sum. Wait, maybe it's a hexagon, but with a reflex angle (210°). Wait, the sum of interior angles for a hexagon is (6-2)180 = 720. But 210° is a reflex angle, but in the formula, we still use the interior angle (which is 210° here, even though it's reflex). Wait, no, actually, the interior angle is the angle inside the polygon, so if it's a reflex angle, the interior angle is 210°, but that's possible in a concave polygon.

Wait, let's list all angles: 140°, 90° (right angle), 210°, 32°, 125°, and x. Wait, that's 6 angles? Wait, no, maybe 7? Wait, no, let's count the vertices. Let's see: the figure has a right angle, 140°, x, 125°, 32°, 210°, and then back? Wait, maybe 7 sides? Wait, (7-2)*180 = 900°. Let's check:

Angles: 140°, 90°, 210°, 32°, 125°, x, and... Wait, no, maybe I miscounted. Wait, the figure: let's see, the angles are 140°, 90° (right angle), 210°, 32°, 125°, and x. Wait, that's 6 angles? No, maybe 7. Wait, let's calculate:

Wait, maybe the correct approach is to use the formula for the sum of interior angles, considering the number of sides. Let's count the number of sides by the number of angles. Let's see:

• Right angle (90°)
• 140°
• x°
• 125°
• 32°
• 210°

Wait, that's 6 angles? No, maybe 7. Wait, maybe the figure is a heptagon (7 sides). Then sum is (7-2)*180 = 900°.

Now sum the given angles: 140 + 90 + 210 + 32 + 125. Let's calculate:

140 + 90 = 230

230 + 210 = 440

440 + 32 = 472

472 + 125 = 597

Then x = 900 - 597 = 303? No, that can't be. Wait, maybe I made a mistake. Wait, maybe the 210° is an exterior angle? No, no, the interior angle is what's inside. Wait, maybe the figure is a hexagon, but with a reflex angle. Wait, hexagon sum is 720. Let's check: 140 + 90 + 210 + 32 + 125 + x = 720.

Sum of given angles: 140 + 90 = 230; 230 + 210 = 440; 440 + 32 = 472; 472 + 125 = 597. Then x = 720 - 597 = 123. But that seems low. Wait, maybe the 210° is not an interior angle. Wait, no, the figure shows a concave polygon with a reflex angle (210°) inside. Wait, maybe the correct number of sides is 6. Let's recalculate:

140 + 90 + 210 + 32 + 125 + x = 720

140 + 90 = 230

230 + 210 = 440

440 + 32 = 472

472 + 125 = 597

x = 720 - 597 = 123. But that seems off. Wait, maybe I made a mistake in the number of sides.

Wait, alternatively, maybe the figure is a hexagon, but the 210° is a reflex angle, so the interior angle is 210°, and the sum is 720. Then:

140 + 90 + 210 + 32 + 125 + x = 720

140 + 90 = 230

230 + 210 = 440

440 + 32 = 472

472 + 125 = 597

x = 720 - 597 = 123. So x = 123°?

Wait, but let's check again. Maybe the first problem is a hexagon (6 sides) with angles 102, 147, 70, 128, 138, x. Sum is (6-2)*180 = 720. Sum of given angles: 102 + 147 + 70 + 128 + 138 = 102+147=249; 249+70=319; 319+128=447; 447+138=585. Then x = 720 - 585 = 135. So x = 135°.

For problem 6: Let's count the number of sides. Let's see, the figure has a right angle (90°), 140°, x, 125°, 32°, 210°. Wait, that's 6 angles? No, maybe 7. Wait, (7-2)180 = 900. Then sum of angles: 140 + 90 + 210 + 32 + 125 + x + ... Wait, no, maybe I missed an angle. Wait, the figure: let's see, the angles are 140°, 90°, 210°, 32°, 125°, and x. Wait, that's 6 angles. Then sum is (6-2)180 = 720. Then 140 + 90 + 210 + 32 + 125 + x = 720.

140 + 90 = 230

230 + 210 = 440

440 + 32 = 472

472 + 125 = 597

x = 720 - 597 = 123. So x = 123°.

Wait, but let's confirm with another approach. Maybe the first problem is a hexagon (6 sides), sum is 720. Angles: 102, 147, 70, 128, 138, x. Sum: 102+147=249; 249+70=319; 319+128=447; 447+138=585; 585+x=720; x=135. Correct.

Problem 6: Let's check again. The angles are 140°, 90° (right angle), 210°, 32°, 125°, and x. Wait, that's 6 angles, so n=6. Sum is 720. Then 140 + 90 + 210 + 32 + 125 + x = 720.

140 + 90 = 230

230 + 210 = 440

440 + 32 = 472

472 + 125 = 597

x = 720 - 597 = 123. So x = 123°.

So the answers are 135° for problem 3 and 123° for problem 6.

But let's check problem 3 again. Wait, maybe it's a pentagon (5 sides). Then sum is (5-2)*180 = 540. Let's see: angles 102, 147, 70, 128, 138, x. Wait, that's 6 angles, so n=6. So sum is 720. So my initial calculation is correct.

Final Answers:

Problem 3: $\boldsymbol{135^\circ}$

Problem 6: $\boldsymbol{123^\circ}$