Question

1. consider the polygon shown.

what is the value of ( x^circ + y^circ )?
(options: ( 154^circ ), ( 142^circ ), ( 116^circ ), ( 64^circ ))

Explanation:

Step1: Recall the sum of exterior angles of a polygon.

The sum of the exterior angles of any polygon is \(360^\circ\). But here, we can also use the fact that for a pentagon (since it's a 5 - sided polygon), the sum of interior angles is \((5 - 2)\times180^\circ= 540^\circ\). However, we can also work with the exterior and interior angle relationships. Let's first find the interior angles corresponding to the given exterior angles and then find \(x\) and \(y\).

First, let's identify the angles. We know that an interior angle and its corresponding exterior angle are supplementary (sum to \(180^\circ\)).

We have some angles: \(116^\circ\) (interior), \(90^\circ\) (right angle, interior), \(67^\circ\) (exterior, so interior is \(180 - 67=113^\circ\)), \(75^\circ\) (exterior, so interior is \(180 - 75 = 105^\circ\)), and then we have the interior angles related to \(x\) and \(y\). Wait, maybe a better approach: the sum of interior angles of a pentagon is \((5-2)\times180=540^\circ\). Let's list the interior angles:

- One angle is \(116^\circ\)
- One is \(90^\circ\) (right angle)
- One interior angle corresponding to \(67^\circ\) exterior: \(180 - 67=113^\circ\)
- One interior angle corresponding to \(75^\circ\) exterior: \(180 - 75 = 105^\circ\)
- Let the fifth interior angle be \(z\), then \(116+90 + 113+105+z=540\)
- Calculate \(116 + 90=206\), \(206+113 = 319\), \(319+105=424\), so \(z=540 - 424 = 116^\circ\)

Now, \(x\) is an exterior angle corresponding to the \(90^\circ\) interior angle? Wait, no. Wait, \(x\) is an exterior angle, so \(x = 180 - 90=90^\circ\)? No, wait the diagram: there is a right angle (interior), so its exterior angle \(x\) would be \(90^\circ\)? Wait, no, maybe I messed up. Wait, let's re - examine.

Wait, the polygon is a pentagon. Let's consider the exterior angles. Wait, the sum of exterior angles of any polygon is \(360^\circ\). But in the diagram, we have some exterior angles and some interior angles. Wait, maybe the angles given: \(116^\circ\) (interior), \(90^\circ\) (interior, right angle), \(67^\circ\) (exterior), \(75^\circ\) (exterior), and then the angles related to \(x\) (exterior) and \(y\) (exterior). Wait, no, let's count the number of sides. The figure has 5 sides (pentagon). The sum of interior angles of a pentagon is \((5 - 2)\times180=540^\circ\).

Let's list the interior angles:

1. \(116^\circ\) (given interior)
2. \(90^\circ\) (right angle, interior)
3. Interior angle adjacent to \(67^\circ\) exterior: \(180 - 67 = 113^\circ\)
4. Interior angle adjacent to \(75^\circ\) exterior: \(180 - 75=105^\circ\)
5. Interior angle adjacent to \(y\) exterior: let's call it \(180 - y\)
6. Wait, no, pentagon has 5 interior angles. Wait, I think I made a mistake. The figure: let's see the angles. There is a \(116^\circ\) interior, a right angle ( \(90^\circ\) interior), a \(67^\circ\) exterior, a \(75^\circ\) exterior, and then two angles: one with exterior \(x\) and one with exterior \(y\). Wait, no, pentagon has 5 vertices. So 5 interior angles and 5 exterior angles.

Wait, maybe the correct approach: the sum of interior angles of a pentagon is \(540^\circ\). Let's denote the interior angles as \(A = 116^\circ\), \(B = 90^\circ\), \(C\) (adjacent to \(67^\circ\) exterior, so \(C=180 - 67 = 113^\circ\)), \(D\) (adjacent to \(75^\circ\) exterior, so \(D = 180 - 75=105^\circ\)), and \(E\) (adjacent to \(y\) exterior, so \(E = 180 - y\)), and the exterior angle \(x\) is adjacent to \(B = 90^\circ\) interior, so \(x=180 - 90 = 90^\circ\)? No, that can't be. Wait, maybe the \(116^\circ\) is an interior angle, the right angle is an interior angle, the \(67^\circ\) and \(75^\circ\) are exterior angles, and then we have two angles: one with exterior \(x\) and one with interior - exterior pair \(y\).

Wait, let's use the sum of interior angles:

\(116+90+(180 - 67)+(180 - 75)+(180 - y)=540\)

Wait, because the interior angle adjacent to \(67^\circ\) exterior is \(180 - 67\), adjacent to \(75^\circ\) is \(180 - 75\), adjacent to \(y\) exterior is \(180 - y\), and the interior angle adjacent to \(x\) exterior is \(180 - x\), but we also have the \(116^\circ\) and \(90^\circ\) interior angles. Wait, no, the \(90^\circ\) is an interior angle, so its exterior angle is \(x = 180 - 90=90^\circ\). Then the interior angle adjacent to \(x\) is \(90^\circ\).

So let's list all interior angles:

- \(116^\circ\)
- \(90^\circ\) (interior, so exterior \(x = 90^\circ\))
- \(180 - 67=113^\circ\) (interior, exterior \(67^\circ\))
- \(180 - 75 = 105^\circ\) (interior, exterior \(75^\circ\))
- \(180 - y\) (interior, exterior \(y\))

Sum of interior angles: \(116+90 + 113+105+(180 - y)=540\)

Calculate left - hand side: \(116+90 = 206\), \(206+113 = 319\), \(319+105 = 424\), \(424+180 - y=604 - y\)

Set equal to 540: \(604 - y=540\) → \(y = 604 - 540=64^\circ\)

Now, we know \(x = 90^\circ\) (since the interior angle is \(90^\circ\), exterior angle \(x = 180 - 90 = 90^\circ\))

Then \(x + y=90 + 64 = 154^\circ\)

Step2: Verify the calculation.

Sum of interior angles: \(116+90+113+105+(180 - 64)=116 + 90+113+105 + 116\)

\(116+90 = 206\), \(206+113 = 319\), \(319+105 = 424\), \(424+116 = 540\), which matches the sum of interior angles of a pentagon \((5 - 2)\times180 = 540^\circ\). And the sum of exterior angles: \(x + 67+75 + y+ \) (exterior angle of \(116^\circ\) interior) \(=90 + 67+75 + 64+(180 - 116)=90+67 = 157\), \(157+75 = 232\), \(232+64 = 296\), \(296+(64)=360^\circ\) (since \(180 - 116 = 64^\circ\) exterior angle), which matches the sum of exterior angles of any polygon (\(360^\circ\)).

Answer:

\(154^\circ\)