Interior/Exterior Angles of a Polygon Worksheet - Free Printable

To find the missing angles in these polygons, we need to use two main rules:

1. Sum of Interior Angles: For any polygon with $n$ sides, the sum of all interior angles is $(n - 2) \times 180^\circ$.
2. Straight Line Rule: Angles on a straight line add up to $180^\circ$. This helps us find interior angles if an exterior angle is given, or vice versa.

Let's solve them one by one, going from left to right, top to bottom.

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Figure 1 (Top Left): Quadrilateral

* Shape: 4-sided polygon (Quadrilateral).
* Rule: Sum of angles = $(4 - 2) \times 180^\circ = 360^\circ$.
* Given Angles:
* Top-left: $85^\circ$
* Bottom-right interior: The exterior angle is not explicitly given as a number, but there is a straight line at the bottom right vertex. Wait, looking closely at the first figure:
* Top-left: $85^\circ$
* Top-right: $90^\circ$ (indicated by the square symbol)
* Bottom-left: $100^\circ$
* Bottom-right: There is an exterior angle marked, but no value. However, usually, in these problems, if one angle is missing, we calculate it. Let's re-examine.
* Actually, looking at the diagram again:
* Angle 1: $85^\circ$
* Angle 2: $90^\circ$ (right angle symbol)
* Angle 3: $100^\circ$
* Angle 4: Unknown ($x$)
* Calculation:
$$85 + 90 + 100 + x = 360$$
$$275 + x = 360$$
$$x = 360 - 275$$
$$x = 85^\circ$$

*Note: The line extending from the bottom right suggests an exterior angle might be involved, but since the interior angle is the one inside the shape, and we have 3 interior angles given, we just solve for the 4th interior angle.*

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Figure 2 (Top Right): Pentagon

* Shape: 5-sided polygon (Pentagon).
* Rule: Sum of angles = $(5 - 2) \times 180^\circ = 540^\circ$.
* Given Angles:
* Three interior angles are $90^\circ$ (indicated by square symbols).
* Two base angles are adjacent to a straight line. The exterior angles are not given numbers, but the vertices lie on a straight line. Wait, let's look closer.
* The bottom side lies on a straight line. The two bottom corners have interior angles marked with squares ($90^\circ$). So, we have three $90^\circ$ angles? No, let's trace carefully.
* Left vertical side meets bottom horizontal side: $90^\circ$.
* Right vertical side meets bottom horizontal side: $90^\circ$.
* Top-left angle: $90^\circ$.
* Top-right angle: $90^\circ$.
* Top-middle angle: Unknown ($x$).
* Let's recount the sides. It looks like a house shape.
* Bottom-left corner: $90^\circ$
* Top-left corner: $90^\circ$
* Top peak: $x$
* Top-right corner: $90^\circ$
* Bottom-right corner: $90^\circ$
* This is a pentagon.
* Sum = $540^\circ$.
* Known angles: $90 + 90 + 90 + 90 = 360^\circ$.
* Calculation:
$$360 + x = 540$$
$$x = 540 - 360$$
$$x = 180^\circ$$
* *Correction*: Looking at the image again, the top "peak" is an angle pointing up. If the side angles are 90, and the bottom angles are 90, the shape is essentially a rectangle with a triangle on top? No, it's a single polygon.
* Let's re-read the diagram.
* Bottom-left interior: $90^\circ$
* Top-left interior: $90^\circ$
* Top-right interior: $90^\circ$
* Bottom-right interior: $90^\circ$
* Top vertex: $x$
* If four angles are 90, the sum is 360. The fifth angle must be $540 - 360 = 180$. An interior angle of 180 means the top two sides form a straight line, making it effectively a rectangle. However, visually it has a peak.
* *Alternative Interpretation*: Perhaps the bottom angles are NOT interior angles of the polygon but angles between the polygon side and the external line? No, the square symbol is inside.
* Let's look at the labels again. Ah, the bottom line is a straight line passing through the base. The angles marked $90^\circ$ are the interior angles at the base. The angles at the "shoulders" are also marked $90^\circ$.
* If this is the case, the calculation holds: $x = 180^\circ$. But geometrically, that would mean the top point is flat.
* Let's look really closely at the second image. It's a pentagon.
* Angle 1 (bottom left): $90^\circ$
* Angle 2 (top left): $90^\circ$
* Angle 3 (top middle): $x$
* Angle 4 (top right): $90^\circ$
* Angle 5 (bottom right): $90^\circ$
* Sum = 540. $90+90+90+90 = 360$. $540-360=180$.
* Is it possible the top angle is reflex? No.
* Is it possible the side angles are not 90? The square symbol denotes 90.
* Let's assume the standard interpretation: The sum is 540. The four known angles are 90. The missing angle is 180. (This implies the "roof" is flat, which contradicts the drawing slightly, but mathematically follows the symbols).
* *Wait*, looking at similar problems online, sometimes the "base" angles are exterior? No, the squares are inside.
* Let's reconsider the shape. Maybe it's not a pentagon? 1,2,3,4,5 sides. Yes.
* Let's look at the third possibility: The top angle is the only unknown. The other 4 are 90. Result: 180.

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Figure 3 (Second Row, Left): Hexagon

* Shape: 6-sided polygon (Hexagon).
* Rule: Sum of angles = $(6 - 2) \times 180^\circ = 720^\circ$.
* Given Angles:
* Four angles are marked with squares ($90^\circ$).
* One angle is $120^\circ$.
* One angle is unknown ($x$).
* Calculation:
Sum of known angles = $90 + 90 + 90 + 90 + 120 = 480^\circ$.
$$480 + x = 720$$
$$x = 720 - 480$$
$$x = 240^\circ$$
*Note: An interior angle of $240^\circ$ is a reflex angle, which matches the visual "indentation" or concave nature if the angle was reflex, but here the angle marked $x$ looks obtuse, not reflex. Let's re-read the diagram.*

Looking at Figure 3 again:
It's a hexagon.
Angles:
- Top-left: $90^\circ$
- Bottom-left: $90^\circ$
- Bottom-right: $120^\circ$
- Right-side vertical-ish angle: $90^\circ$ ?? No, the square is on the top-right vertex.
- Top-right vertex: $90^\circ$
- Top-middle vertex: $90^\circ$ ??

Let's trace the perimeter clockwise from top-left:
1. Top-Left: Square ($90^\circ$)
2. Top-Right: Square ($90^\circ$)
3. Right-Middle: Angle $x$
4. Bottom-Right: $120^\circ$
5. Bottom-Left: Square ($90^\circ$)
6. Left-Middle: Square ($90^\circ$) -- Wait, is that a square? Yes.

So we have five known angles: $90, 90, 90, 90, 120$.
Sum = $480^\circ$.
Total sum for hexagon = $720^\circ$.
Missing angle $x = 720 - 480 = 240^\circ$.

Visually, angle $x$ is the large interior angle on the right. If it were convex, it would be less than 180. But $240 > 180$. This implies the polygon is concave at that vertex. Looking at the drawing, the vertex for $x$ points *inwards*? No, it points outwards.

Let's re-evaluate the number of sides.
1 (top), 2 (top right slant), 3 (bottom right slant), 4 (bottom), 5 (bottom left vertical), 6 (top left vertical).
Actually, let's look at the shape again.
It looks like a rectangle with a trapezoid attached?
Vertices:
1. Top Left ($90^\circ$)
2. Bottom Left ($90^\circ$)
3. Bottom Right ($120^\circ$)
4. Right "indent"? No.
5. Top Right ($90^\circ$)
6. Top Middle ($90^\circ$)?

Let's count the squares. There are THREE squares clearly visible.
1. Top Left
2. Bottom Left
3. Top Right? Or is that Top Middle?

Let's assume the standard regular-ish layout:
- Left side is vertical. Top and Bottom are horizontal. This creates two $90^\circ$ angles on the left.
- Top side goes right, then turns down ($90^\circ$).
- Then a side goes down-right.
- Then a side goes down-left to meet the bottom.
- Bottom side goes left.

If this is the case:
Sides:
1. Left Vertical
2. Top Horizontal
3. Short Vertical Down
4. Slanted Side
5. Bottom Horizontal

That's a Pentagons (5 sides).
Sum = $540^\circ$.
Angles:
1. Top-Left: $90^\circ$
2. Bottom-Left: $90^\circ$
3. Top-Right (corner of the step): $90^\circ$
4. Bottom-Right: $120^\circ$
5. The "inner" corner where the slant meets the short vertical? Let's call it $x$.

If it is a 5-sided polygon:
Sum = 540.
Known: $90 + 90 + 90 + 120 = 390$.
$x = 540 - 390 = 150^\circ$.

Does the image show 5 or 6 sides?
Looking at crop 3:
- Left vertical line.
- Top horizontal line.
- Right vertical line (short).
- Slanted line going down-left.
- Bottom horizontal line.
- Left vertical line closes it.

Yes, this is a Pentagon (5 sides). The "step" creates the 5th vertex.
So, $n=5$. Sum = 540.
Angles given:
- Top-Left: $90^\circ$
- Bottom-Left: $90^\circ$
- Top-Right (the outer corner of the step): $90^\circ$
- Bottom-Right (acute/obtuse?): Marked $120^\circ$.
- The remaining angle is the "inner" corner of the step (reflex? or interior?). The angle marked $x$ is the interior angle at the vertex connecting the short vertical and the slant.

Calculation:
$$90 + 90 + 90 + 120 + x = 540$$
$$390 + x = 540$$
$$x = 150^\circ$$

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Figure 4 (Second Row, Right): Pentagon

* Shape: 5-sided polygon.
* Rule: Sum of angles = $540^\circ$.
* Given Angles:
* Bottom-Left: $90^\circ$ (square symbol).
* Top-Left: $90^\circ$ (square symbol).
* Bottom-Right: $130^\circ$.
* Top-Right: $x$ (unknown).
* Top-Middle: This vertex connects to the top-left and top-right. Wait, let's trace the sides.
1. Bottom horizontal.
2. Left vertical.
3. Top horizontal? No, it slants.

Let's look at the vertices again.
1. Bottom-Left: $90^\circ$.
2. Top-Left: $90^\circ$.
3. Top-Middle/Right: There is a vertex with an exterior angle? No, $x$ is inside.
4. Bottom-Right: $130^\circ$.
5. There is a 5th vertex?

Let's count sides:
1. Bottom
2. Right slant
3. Top slant
4. Left vertical
... Wait, that's 4 sides. A quadrilateral?
If it's a quadrilateral:
Sum = 360.
Angles: $90, 90, 130, x$.
$90+90+130+x = 360$
$310 + x = 360$
$x = 50^\circ$.

Let's check the image "Figure 4" (second row, right).
It has a vertical left side. A horizontal bottom side. A slanted right side. A slanted top side.
It looks like a right trapezoid or general quadrilateral.
There are 4 vertices.
- Bottom Left: Square ($90^\circ$)
- Top Left: Square ($90^\circ$)
- Bottom Right: $130^\circ$
- Top Right: $x$

Yes, it is a Quadrilateral.
Sum = $360^\circ$.
$$90 + 90 + 130 + x = 360$$
$$310 + x = 360$$
$$x = 50^\circ$$

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Figure 5 (Third Row, Left): Parallelogram-like shape with transversals?

Actually, this looks like two parallel lines intersected by transversals, forming a parallelogram in the middle? Or just a parallelogram?
Let's look at the markings.
- It's a quadrilateral.
- Opposite sides appear parallel.
- Angle at bottom-left: $65^\circ$.
- Angle at top-right: $120^\circ$? No, that's an exterior angle.
- Angle at top-left: $x$? No, $x$ is an interior angle.
- Angle at bottom-right: $y$? No, $y$ is an interior angle.

Let's look closer at Crop 5.
It shows a parallelogram.
- Bottom-left interior angle: $65^\circ$.
- Top-right exterior angle: $120^\circ$.
- We need to find $x$ (top-left interior) and $y$ (bottom-right interior).

Properties of a Parallelogram:
1. Opposite angles are equal.
2. Consecutive angles add up to $180^\circ$.
3. Exterior angle + Interior angle = $180^\circ$.

Step 1: Find the interior angle at the top-right.
Exterior angle = $120^\circ$.
Interior angle = $180^\circ - 120^\circ = 60^\circ$.

Wait, if it's a parallelogram, opposite angles are equal.
Bottom-left is $65^\circ$. So Top-right should be $65^\circ$.
But we calculated Top-right interior as $60^\circ$.
$65 \neq 60$.
So, it is NOT a parallelogram. It is just a general quadrilateral or a trapezoid.

Let's look at the arrowheads on the lines.
- The top and bottom lines have arrowheads indicating they are parallel.
- The left and right lines do NOT have arrowheads.
- So, this is a Trapezoid (or trapezium) with parallel top and bottom bases.

Properties of Trapezoid with parallel horizontal sides:
- Consecutive interior angles between the parallel sides (same-side interior angles) are supplementary (add to $180^\circ$).
- Left side acts as a transversal: Angle(bottom-left) + Angle(top-left) = $180^\circ$.
- Right side acts as a transversal: Angle(bottom-right) + Angle(top-right) = $180^\circ$.

Given:
- Bottom-left interior = $65^\circ$.
- Top-right exterior = $120^\circ$.

Find $x$ (Top-left interior):
Since top and bottom lines are parallel:
$$x + 65^\circ = 180^\circ$$
$$x = 180 - 65$$
$$x = 115^\circ$$

Find $y$ (Bottom-right interior):
First, find the Top-right interior angle.
Top-right interior + Top-right exterior = $180^\circ$ (straight line).
Top-right interior = $180 - 120 = 60^\circ$.

Now, use the parallel line property for the right side:
Top-right interior + Bottom-right interior ($y$) = $180^\circ$.
$$60 + y = 180$$
$$y = 120^\circ$$

So, $x = 115^\circ$, $y = 120^\circ$.

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Figure 6 (Third Row, Right): Triangle/Polygon mix?

This figure shows two intersecting lines forming triangles?
No, it looks like a single polygon with some extended lines.
Let's trace the polygon.
It has 4 sides?
1. Bottom horizontal.
2. Right vertical.
3. Top slanted.
4. Left slanted.

Let's look at the angles provided.
- Bottom-right corner: Square symbol ($90^\circ$). So the bottom and right sides are perpendicular.
- Top vertex: Exterior angle $70^\circ$.
- Left vertex: Exterior angle $x$? No, $x$ is the interior angle? Or exterior? The arc is outside. So $x$ is an exterior angle.
- Bottom-left vertex: Exterior angle $y$? No, $y$ is inside? The arc is inside. So $y$ is an interior angle.
- Wait, let's look at the labels again.
- Top vertex: Interior angle is adjacent to exterior $70^\circ$. So Interior = $180 - 70 = 110^\circ$.
- Right vertex: Interior is $90^\circ$.
- Bottom-left vertex: Interior is $y$.
- Left vertex: Exterior is $x$. So Interior is $180 - x$.

Is it a quadrilateral?
Sides:
1. Bottom (horizontal)
2. Right (vertical)
3. Top (slanted)
4. Left (slanted)
Yes, 4 sides. Sum = $360^\circ$.

We have 4 interior angles:
1. Top: $110^\circ$
2. Right: $90^\circ$
3. Bottom-Left: $y$
4. Top-Left: $180 - x$

We have one equation: $110 + 90 + y + (180 - x) = 360$.
$380 + y - x = 360$.
$y - x = -20$ or $x - y = 20$.

We need another relationship. Are any lines parallel?
The bottom line is horizontal. The right line is vertical.
There are no explicit parallel markers on the left/top sides.
However, often in these diagrams, if it looks like a specific shape, it might be.
But wait, look at the bottom-left vertex. There is a line extending to the left. The angle $y$ is interior.
Look at the top-left vertex. There is a line extending to the left. The angle $x$ is exterior.

Is there a missing piece of info?
Let's re-examine the image.
Maybe the left side and right side are parallel? No.
Maybe the top and bottom are parallel?
If Top || Bottom:
Then consecutive interior angles sum to 180.
Left side transversal: Interior(Top-Left) + Interior(Bottom-Left) = 180.
$(180 - x) + y = 180 \Rightarrow y = x$.
Right side transversal: Interior(Top-Right) + Interior(Bottom-Right) = 180.
Top-Right is part of the top vertex? No, the right vertex is the 90 degree one.
If Top || Bottom, then the right side (vertical) is perpendicular to Bottom, so it must be perpendicular to Top.
This would mean the Top-Right interior angle is $90^\circ$.
But we calculated the Top interior angle (at the peak) as $110^\circ$.
A quadrilateral has 4 vertices.
Vertex 1: Top Peak.
Vertex 2: Right Corner ($90^\circ$).
Vertex 3: Bottom Left Corner ($y$).
Vertex 4: Left Corner.

If Top and Bottom are parallel:
The angle at Vertex 1 (Top Peak) and Vertex 2 (Right Corner) are on the same side? No.
Let's assume the standard trapezoid orientation: Parallel sides are Top and Bottom.
Then the angles on the leg (left side) sum to 180.
Angle(Top-Left Interior) + Angle(Bottom-Left Interior) = 180.
$(180-x) + y = 180 \rightarrow y = x$.

The angles on the other leg (right side) sum to 180.
Angle(Top-Right Interior) + Angle(Bottom-Right Interior) = 180.
Here, the "Top-Right" vertex is the Peak? No.
The polygon has 4 vertices.
1. Left
2. Top
3. Right
4. Bottom

If Top and Bottom sides are parallel:
This geometry doesn't fit a simple "Top and Bottom are parallel" unless the "Top" vertex is actually a side.
Looking at the shape, it has 4 distinct corners.
Corner 1 (Left): Ext $x$, Int $180-x$.
Corner 2 (Top): Ext $70$, Int $110$.
Corner 3 (Right): Int $90$.
Corner 4 (Bottom): Int $y$.

If we assume the Left and Right sides are parallel?
Then Top and Bottom are transversals.
Top transversal: Int(Left) + Int(Right... wait, Right is a vertex, not a side end for the top).
This is getting complicated. Let's look for parallel markers.
In Figure 5, there were arrows. In Figure 6, there are NO arrows.
However, notice the line at the bottom extends to the right. The line at the top extends to the left.
Usually, if no parallel markers are present, we can't assume parallel lines.
BUT, look at the right angle.
And look at the exterior angle $70$.
Is it possible this is a Triangle?
No, 4 sides.

Let's look at the sum of exterior angles.
Sum of exterior angles of any convex polygon is $360^\circ$.
Exterior angles:
1. Top: $70^\circ$
2. Right: $90^\circ$ (since interior is 90, exterior is 90)
3. Bottom-Left: Exterior is $180 - y$.
4. Top-Left: Exterior is $x$.

Sum: $70 + 90 + (180 - y) + x = 360$.
$160 + 180 - y + x = 360$.
$340 - y + x = 360$.
$x - y = 20$.
Or $y = x - 20$.

We still have two variables. Is there a constraint I'm missing?
Let's look at the shape again.
Does the left side look parallel to the right side? No.
Does the top side look parallel to the bottom side?
If Top || Bottom, then Interior(Top-Left) + Interior(Bottom-Left) = 180.
$(180-x) + y = 180 \Rightarrow y=x$.
If $y=x$ and $y=x-20$, then $x=x-20 \Rightarrow 0=-20$, which is impossible.
So Top and Bottom are NOT parallel.

What if Left || Right?
Then Interior(Top-Left) + Interior(Top-Right?? No).
Interior(Top-Left) + Interior(Bottom-Left)... no.
If Left || Right, then Interior(Top-Left) + Interior(Top-Right vertex?) No.
Transversal is Top Side: Int(Left) + Int(Right... no, the right side meets the top side at the Top Vertex? No, there is a distinct Right Vertex).

Let's reconsider the polygon type.
Maybe it's a Right Trapezoid?
If it is a right trapezoid, usually two adjacent angles are 90.
Here only one is 90.

Let's look at the provided solution in similar online homework.
Often, these diagrams imply parallel lines if they look like it, even without arrows, OR there is a typo in my reading.
Let's re-read Figure 6.
- Top vertex: Exterior 70.
- Right vertex: Interior 90.
- Bottom vertex: Interior y.
- Left vertex: Exterior x.

Is it possible the bottom line and the top line are parallel?
If so, we proved it leads to a contradiction ($0=-20$).
Is it possible the Left and Right sides are parallel?
If Left || Right:
Then the Top side is a transversal.
Int(Left) + Int(Top... wait, the angle at the top is between Top Side and Right Side? No, Top Side and Left Side).
Let's name vertices CCW starting from Bottom-Left.
V1: Bottom-Left. Int $y$.
V2: Right. Int $90$.
V3: Top. Int $110$.
V4: Left. Int $180-x$.

Sides: V1-V2 (Bottom), V2-V3 (Right), V3-V4 (Top), V4-V1 (Left).

If Left (V4-V1) is parallel to Right (V2-V3):
Then the transversal V1-V2 (Bottom) creates consecutive interior angles? No, V1 and V2 are on the same line segment.
The transversal V3-V4 (Top) connects the parallels.
Interior angles on the same side of the transversal?
No, the property is: Consecutive interior angles between parallel lines sum to 180.
The angles "between" the parallel lines Left and Right are:
At Top: Angle between Top and Left? No. Angle between Top and Right?
Let's use vectors or slopes.
Let Right side be vertical (slope undefined). Since Int(V2)=90 and Bottom is horizontal, Right is vertical.
If Left is parallel to Right, Left is vertical.
If Left is vertical, then Int(V4) and Int(V1) depend on the Top and Bottom slopes.
If Left is vertical, and Bottom is horizontal, then Int(V1) = $90^\circ$. So $y=90$.
If Left is vertical, and Top connects to it...
We know Int(V3) = 110. This is the angle between Top and Right.
Since Right is vertical, the Top side makes an angle of $110-90=20$ degrees with the horizontal? Or $90-110$?
Let's place V2 at origin $(