Question

1. 80° y° z° 53° 62° 35° 95° x° x = __ y = z = __

Explanation:

Step1: Recall the formula for exterior angles of a polygon

The sum of the exterior angles of any polygon is \( 360^\circ \). But here, we can also use the linear pair (supplementary angles) for each angle. For a linear pair, the sum of an interior angle and its corresponding exterior angle is \( 180^\circ \).

Step2: Find \( x \)

We know that \( 95^\circ \) and \( x^\circ \) are supplementary (linear pair), so \( 95 + x = 180 \).
Solving for \( x \): \( x = 180 - 95 = 85 \).

Step3: Find \( y \)

\( 80^\circ \) and \( y^\circ \) are supplementary, so \( 80 + y = 180 \).
Solving for \( y \): \( y = 180 - 80 = 100 \).

Step4: Find \( z \)

First, let's list all the exterior angles or use the sum of exterior angles. Wait, actually, we can use the sum of all the angles around the polygon. Wait, no, let's check the given angles. Wait, the angles given are \( 80^\circ \), \( 62^\circ \), \( 35^\circ \), \( 53^\circ \), and the angles related to \( x \), \( y \), \( z \). Wait, actually, the sum of all the exterior angles (the ones given and the ones we need to find? No, wait, the figure is a hexagon? Wait, no, the exterior angles sum to \( 360^\circ \). Wait, let's list the exterior angles: \( 80^\circ \), \( 62^\circ \), \( 35^\circ \), \( 53^\circ \), and the exterior angles corresponding to \( y \) and \( x \)? Wait, no, actually, the angles \( 80^\circ \), \( 62^\circ \), \( 35^\circ \), \( 53^\circ \) are exterior angles, and the angles adjacent to \( y \), \( x \), \( z \) are interior? Wait, no, let's re-examine.

Wait, the angle \( 80^\circ \) and \( y^\circ \) form a linear pair, so \( y = 180 - 80 = 100 \) (as before). The angle \( 95^\circ \) and \( x^\circ \) form a linear pair, so \( x = 180 - 95 = 85 \) (as before). Now, for \( z \), let's consider the sum of all the exterior angles. Wait, the exterior angles of any polygon sum to \( 360^\circ \). The given exterior angles are \( 80^\circ \), \( 62^\circ \), \( 35^\circ \), \( 53^\circ \), and we need to find the remaining exterior angle? Wait, no, the angle \( z^\circ \) is adjacent to a straight line, so the sum of \( z \), \( 53^\circ \), and the other angles? Wait, maybe I made a mistake. Wait, let's count the number of sides. The figure has 6 sides? Wait, no, let's see the angles: there are 6 angles? Wait, the angles given are \( 80^\circ \), \( 62^\circ \), \( 35^\circ \), \( 95^\circ \), \( 53^\circ \), and the angles \( y \), \( x \), \( z \). Wait, maybe it's a hexagon, so the sum of interior angles is \( (6 - 2) \times 180 = 720^\circ \). But maybe easier to use exterior angles.

Wait, the exterior angles sum to \( 360^\circ \). The exterior angles are \( 80^\circ \), \( 62^\circ \), \( 35^\circ \), \( 53^\circ \), and the exterior angles corresponding to \( y \) and \( x \)? No, wait, \( y \) is an interior angle, so its exterior angle is \( 80^\circ \), \( x \) is an interior angle, so its exterior angle is \( 95^\circ \)? Wait, no, I think I mixed up. Let's start over.

For any angle, the interior angle and its adjacent exterior angle are supplementary (sum to \( 180^\circ \)).

So:

- For the angle with \( 80^\circ \) exterior: interior angle \( y = 180 - 80 = 100^\circ \)
- For the angle with \( 62^\circ \) exterior: interior angle \( = 180 - 62 = 118^\circ \)
- For the angle with \( 35^\circ \) exterior: interior angle \( = 180 - 35 = 145^\circ \)
- For the angle with \( 53^\circ \) exterior: interior angle \( = 180 - 53 = 127^\circ \)
- For the angle with \( x^\circ \) exterior: interior angle \( = 95^\circ \), so \( x = 180 - 95 = 85^\circ \) (since they are supplementary)
- Now, the sum of interior angles of a hexagon (6 sides) is \( (6 - 2) \times 180 = 720^\circ \)
- So sum of interior angles: \( y + 118 + 145 + 127 + 95 + z = 720 \)
- We know \( y = 100 \), so:
\( 100 + 118 + 145 + 127 + 95 + z = 720 \)
- Calculate the sum of known angles: \( 100 + 118 = 218 \); \( 218 + 145 = 363 \); \( 363 + 127 = 490 \); \( 490 + 95 = 585 \)
- So \( 585 + z = 720 \)
- Then \( z = 720 - 585 = 135 \)

Wait, but let's check with exterior angles. The sum of exterior angles is \( 360^\circ \). The exterior angles are \( 80^\circ \), \( 62^\circ \), \( 35^\circ \), \( 53^\circ \), \( x^\circ \), and the exterior angle corresponding to \( z \). Wait, the exterior angle corresponding to \( z \) is \( 180 - z \). So:

\( 80 + 62 + 35 + 53 + x + (180 - z) = 360 \)

We know \( x = 85 \), so:

\( 80 + 62 + 35 + 53 + 85 + 180 - z = 360 \)

Calculate left side: \( 80 + 62 = 142 \); \( 142 + 35 = 177 \); \( 177 + 53 = 230 \); \( 230 + 85 = 315 \); \( 315 + 180 = 495 \)

So \( 495 - z = 360 \)

Then \( z = 495 - 360 = 135 \), which matches the previous result.

So:

- \( x = 85 \)
- \( y = 100 \)
- \( z = 135 \)

Answer:

\( x = \boxed{85} \), \( y = \boxed{100} \), \( z = \boxed{135} \)