Find the missing angle x in each of the seven geometric figures.
A worksheet with seven geometry problems, each requiring the calculation of an unknown angle labeled 'x' using given angles and geometric principles.
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Step-by-step solution for: Exterior Angles of Convex Polygons - Opinions Nobody Asked For
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Show Answer Key & Explanations
Step-by-step solution for: Exterior Angles of Convex Polygons - Opinions Nobody Asked For
Here are the step-by-step solutions to find the angle $x$ in each question.
1)
* Step 1: Find the interior angles of the quadrilateral (4-sided shape).
* Top-left: The angle on a straight line is $180^\circ$. So, $180^\circ - 130^\circ = 50^\circ$.
* Bottom-right: Similarly, $180^\circ - 70^\circ = 110^\circ$.
* Bottom-left: Given as $80^\circ$.
* Top-right: This is angle $x$.
* Step 2: Use the sum of angles in a quadrilateral ($360^\circ$).
* $50^\circ + 80^\circ + 110^\circ + x = 360^\circ$
* $240^\circ + x = 360^\circ$
* $x = 360^\circ - 240^\circ$
* $x = 120^\circ$
2)
* Step 1: Find the interior angles of the quadrilateral.
* Bottom-left: There is a right-angle symbol, so it is $90^\circ$.
* Top-left: The angle on a straight line is $180^\circ$. So, $180^\circ - 75^\circ = 105^\circ$.
* Top-right: Similarly, $180^\circ - 60^\circ = 120^\circ$.
* Bottom-right: The angle inside is supplementary to $x$, so it is $(180^\circ - x)$.
* Step 2: Sum of angles is $360^\circ$.
* $90^\circ + 105^\circ + 120^\circ + (180^\circ - x) = 360^\circ$
* $315^\circ + 180^\circ - x = 360^\circ$
* $495^\circ - x = 360^\circ$
* $x = 495^\circ - 360^\circ$
* $x = 135^\circ$
3)
* Step 1: Find the interior angles of the quadrilateral.
* Top-left: $180^\circ - 60^\circ = 120^\circ$.
* Top-right: $180^\circ - 70^\circ = 110^\circ$.
* Bottom-left: $180^\circ - 75^\circ = 105^\circ$.
* Bottom-right: The angle inside is supplementary to $x$, so it is $(180^\circ - x)$. Note: The $35^\circ$ is an exterior angle for the adjacent triangle or just extra info, but looking at the vertex, the interior angle of the quad and $x$ form a linear pair. Wait, looking closely at diagram 3, $x$ is the exterior angle. The interior angle is $180-x$.
* Step 2: Sum of angles is $360^\circ$.
* $120^\circ + 110^\circ + 105^\circ + (180^\circ - x) = 360^\circ$
* $335^\circ + 180^\circ - x = 360^\circ$
* $515^\circ - x = 360^\circ$
* $x = 515^\circ - 360^\circ$
* $x = 155^\circ$
4)
* Step 1: Find the interior angles of the quadrilateral.
* Left: $180^\circ - 140^\circ = 40^\circ$.
* Right: $180^\circ - 120^\circ = 60^\circ$.
* Bottom: Given as $60^\circ$.
* Top: The angle inside is supplementary to $x$, so it is $(180^\circ - x)$.
* Step 2: Sum of angles is $360^\circ$.
* $40^\circ + 60^\circ + 60^\circ + (180^\circ - x) = 360^\circ$
* $160^\circ + 180^\circ - x = 360^\circ$
* $340^\circ - x = 360^\circ$ ... Wait, $340 - x = 360$ gives negative. Let me re-check the image.
* Ah, the angle marked $x$ is inside the shape? No, it looks like an interior angle based on the arc. Let's look closer. The arc for $x$ is inside the corner. The other arcs are outside (exterior).
* Let's re-evaluate based on Exterior Angles Sum = $360^\circ$.
* Exterior angles are: $140^\circ$ (left), $120^\circ$ (right), $60^\circ$ (bottom). The top exterior angle would be $180-x$.
* Sum: $140 + 120 + 60 + (180-x) = 360$?
* $320 + 180 - x = 360 \rightarrow 500 - x = 360 \rightarrow x = 140$.
* Let's try Interior Angles again assuming $x$ is interior.
* Interior Left: $180-140=40$. Interior Right: $180-120=60$. Interior Bottom: $180-60=120$? No, the arc for 60 is inside. So Interior Bottom is 60.
* If Interior Bottom is 60, Interior Left is 40, Interior Right is 60.
* Sum so far: $40+60+60 = 160$.
* Remaining angle $x = 360 - 160 = 200^\circ$? That's a reflex angle. The drawing shows an obtuse angle, not reflex.
* Let's look at the bottom angle again. The arc is between the side and the extension. It is an EXTERIOR angle of $60^\circ$. So Interior is $120^\circ$.
* Let's re-calculate with Interior Bottom = $120^\circ$.
* Interior Left = $40^\circ$. Interior Right = $60^\circ$. Interior Bottom = $120^\circ$.
* Sum = $40 + 60 + 120 = 220^\circ$.
* $x = 360 - 220 = 140^\circ$. This fits the visual.
5)
* Step 1: Identify the shape. It is a pentagon (5 sides) because of the 5 vertices, but one corner has a right angle symbol on the *outside*? No, it's a quadrilateral with a triangle attached? No, it's a single polygon with 5 sides.
* Let's count vertices: Top-left, Top-right (with x), Right-middle (with right angle), Bottom-right, Bottom-left. Yes, it's a pentagon.
* Sum of interior angles of a pentagon = $(5-2) \times 180^\circ = 540^\circ$.
* Step 2: Find interior angles.
* Top-left: Exterior is $65^\circ$, so Interior is $180^\circ - 65^\circ = 115^\circ$.
* Bottom-left: Exterior is $100^\circ$, so Interior is $180^\circ - 100^\circ = 80^\circ$.
* Bottom-right: Exterior is $50^\circ$, so Interior is $180^\circ - 50^\circ = 130^\circ$.
* Right-middle: There is a square symbol. It indicates the interior angle is $90^\circ$.
* Top-right: Angle is $x$.
* Step 3: Calculate $x$.
* $115^\circ + 80^\circ + 130^\circ + 90^\circ + x = 540^\circ$
* $415^\circ + x = 540^\circ$
* $x = 540^\circ - 415^\circ$
* $x = 125^\circ$
6)
* Step 1: This is a pentagon (5 sides). Sum of interior angles = $540^\circ$.
* Step 2: Find interior angles.
* Top-left: Exterior $125^\circ \rightarrow$ Interior $180^\circ - 125^\circ = 55^\circ$.
* Bottom-left: Exterior $60^\circ \rightarrow$ Interior $180^\circ - 60^\circ = 120^\circ$.
* Top-right: Interior is given as $50^\circ$ (arc is inside).
* Bottom-right: Angle is $x$ (arc is inside).
* Middle-right vertex: The line goes straight up. The angle shown is $50^\circ$ inside. Wait, let's look at the shape again.
* Vertices:
1. Top Left (Int: 55)
2. Bottom Left (Int: 120)
3. Bottom Right (Int: x)
4. Top Right (Int: 50)
5. There is a 5th vertex between Top Right and Bottom Right? No, it looks like a quadrilateral with a "bite" taken out, or a concave pentagon.
Let's trace the perimeter:
Start Top-Left -> Bottom-Left -> Bottom-Right -> Top-Right -> Back to Top-Left.
This is a quadrilateral (4 sides).
Let's re-read the diagram.
Side 1: Left vertical-ish.
Side 2: Bottom horizontal-ish.
Side 3: Right vertical-ish.
Side 4: Top diagonal.
It is a quadrilateral.
Interior Angles:
- Top-Left: $180 - 125 = 55^\circ$.
- Bottom-Left: $180 - 60 = 120^\circ$.
- Top-Right: The angle marked $50^\circ$ is interior.
- Bottom-Right: The angle marked $x$ is interior.
Sum = $360^\circ$.
$55 + 120 + 50 + x = 360$
$225 + x = 360$
$x = 135^\circ$.
7)
* Step 1: This is a pentagon (5 sides). Sum of interior angles = $540^\circ$.
* Step 2: Find interior angles.
* Top-left vertex: Exterior angles are $55^\circ$ and part of the straight line. The interior angle is vertically opposite to nothing, it's on a straight line. The interior angle is $180^\circ - 55^\circ = 125^\circ$.
* Top-right vertex: Exterior is $110^\circ$. Interior is $180^\circ - 110^\circ = 70^\circ$.
* Bottom-right vertex: Exterior is $65^\circ$. Interior is $180^\circ - 65^\circ = 115^\circ$.
* Bottom-left vertex: Exterior is $70^\circ$. Interior is $180^\circ - 70^\circ = 110^\circ$.
* Bottom-middle vertex: Angle is $x$. This is an interior angle.
* Step 3: Calculate $x$.
* $125^\circ + 70^\circ + 115^\circ + 110^\circ + x = 540^\circ$
* Sum of knowns: $125 + 70 = 195$. $195 + 115 = 310$. $310 + 110 = 420$.
* $420^\circ + x = 540^\circ$
* $x = 540^\circ - 420^\circ$
* $x = 120^\circ$
Final Answer:
1) 120°
2) 135°
3) 155°
4) 140°
5) 125°
6) 135°
7) 120°
1)
* Step 1: Find the interior angles of the quadrilateral (4-sided shape).
* Top-left: The angle on a straight line is $180^\circ$. So, $180^\circ - 130^\circ = 50^\circ$.
* Bottom-right: Similarly, $180^\circ - 70^\circ = 110^\circ$.
* Bottom-left: Given as $80^\circ$.
* Top-right: This is angle $x$.
* Step 2: Use the sum of angles in a quadrilateral ($360^\circ$).
* $50^\circ + 80^\circ + 110^\circ + x = 360^\circ$
* $240^\circ + x = 360^\circ$
* $x = 360^\circ - 240^\circ$
* $x = 120^\circ$
2)
* Step 1: Find the interior angles of the quadrilateral.
* Bottom-left: There is a right-angle symbol, so it is $90^\circ$.
* Top-left: The angle on a straight line is $180^\circ$. So, $180^\circ - 75^\circ = 105^\circ$.
* Top-right: Similarly, $180^\circ - 60^\circ = 120^\circ$.
* Bottom-right: The angle inside is supplementary to $x$, so it is $(180^\circ - x)$.
* Step 2: Sum of angles is $360^\circ$.
* $90^\circ + 105^\circ + 120^\circ + (180^\circ - x) = 360^\circ$
* $315^\circ + 180^\circ - x = 360^\circ$
* $495^\circ - x = 360^\circ$
* $x = 495^\circ - 360^\circ$
* $x = 135^\circ$
3)
* Step 1: Find the interior angles of the quadrilateral.
* Top-left: $180^\circ - 60^\circ = 120^\circ$.
* Top-right: $180^\circ - 70^\circ = 110^\circ$.
* Bottom-left: $180^\circ - 75^\circ = 105^\circ$.
* Bottom-right: The angle inside is supplementary to $x$, so it is $(180^\circ - x)$. Note: The $35^\circ$ is an exterior angle for the adjacent triangle or just extra info, but looking at the vertex, the interior angle of the quad and $x$ form a linear pair. Wait, looking closely at diagram 3, $x$ is the exterior angle. The interior angle is $180-x$.
* Step 2: Sum of angles is $360^\circ$.
* $120^\circ + 110^\circ + 105^\circ + (180^\circ - x) = 360^\circ$
* $335^\circ + 180^\circ - x = 360^\circ$
* $515^\circ - x = 360^\circ$
* $x = 515^\circ - 360^\circ$
* $x = 155^\circ$
4)
* Step 1: Find the interior angles of the quadrilateral.
* Left: $180^\circ - 140^\circ = 40^\circ$.
* Right: $180^\circ - 120^\circ = 60^\circ$.
* Bottom: Given as $60^\circ$.
* Top: The angle inside is supplementary to $x$, so it is $(180^\circ - x)$.
* Step 2: Sum of angles is $360^\circ$.
* $40^\circ + 60^\circ + 60^\circ + (180^\circ - x) = 360^\circ$
* $160^\circ + 180^\circ - x = 360^\circ$
* $340^\circ - x = 360^\circ$ ... Wait, $340 - x = 360$ gives negative. Let me re-check the image.
* Ah, the angle marked $x$ is inside the shape? No, it looks like an interior angle based on the arc. Let's look closer. The arc for $x$ is inside the corner. The other arcs are outside (exterior).
* Let's re-evaluate based on Exterior Angles Sum = $360^\circ$.
* Exterior angles are: $140^\circ$ (left), $120^\circ$ (right), $60^\circ$ (bottom). The top exterior angle would be $180-x$.
* Sum: $140 + 120 + 60 + (180-x) = 360$?
* $320 + 180 - x = 360 \rightarrow 500 - x = 360 \rightarrow x = 140$.
* Let's try Interior Angles again assuming $x$ is interior.
* Interior Left: $180-140=40$. Interior Right: $180-120=60$. Interior Bottom: $180-60=120$? No, the arc for 60 is inside. So Interior Bottom is 60.
* If Interior Bottom is 60, Interior Left is 40, Interior Right is 60.
* Sum so far: $40+60+60 = 160$.
* Remaining angle $x = 360 - 160 = 200^\circ$? That's a reflex angle. The drawing shows an obtuse angle, not reflex.
* Let's look at the bottom angle again. The arc is between the side and the extension. It is an EXTERIOR angle of $60^\circ$. So Interior is $120^\circ$.
* Let's re-calculate with Interior Bottom = $120^\circ$.
* Interior Left = $40^\circ$. Interior Right = $60^\circ$. Interior Bottom = $120^\circ$.
* Sum = $40 + 60 + 120 = 220^\circ$.
* $x = 360 - 220 = 140^\circ$. This fits the visual.
5)
* Step 1: Identify the shape. It is a pentagon (5 sides) because of the 5 vertices, but one corner has a right angle symbol on the *outside*? No, it's a quadrilateral with a triangle attached? No, it's a single polygon with 5 sides.
* Let's count vertices: Top-left, Top-right (with x), Right-middle (with right angle), Bottom-right, Bottom-left. Yes, it's a pentagon.
* Sum of interior angles of a pentagon = $(5-2) \times 180^\circ = 540^\circ$.
* Step 2: Find interior angles.
* Top-left: Exterior is $65^\circ$, so Interior is $180^\circ - 65^\circ = 115^\circ$.
* Bottom-left: Exterior is $100^\circ$, so Interior is $180^\circ - 100^\circ = 80^\circ$.
* Bottom-right: Exterior is $50^\circ$, so Interior is $180^\circ - 50^\circ = 130^\circ$.
* Right-middle: There is a square symbol. It indicates the interior angle is $90^\circ$.
* Top-right: Angle is $x$.
* Step 3: Calculate $x$.
* $115^\circ + 80^\circ + 130^\circ + 90^\circ + x = 540^\circ$
* $415^\circ + x = 540^\circ$
* $x = 540^\circ - 415^\circ$
* $x = 125^\circ$
6)
* Step 1: This is a pentagon (5 sides). Sum of interior angles = $540^\circ$.
* Step 2: Find interior angles.
* Top-left: Exterior $125^\circ \rightarrow$ Interior $180^\circ - 125^\circ = 55^\circ$.
* Bottom-left: Exterior $60^\circ \rightarrow$ Interior $180^\circ - 60^\circ = 120^\circ$.
* Top-right: Interior is given as $50^\circ$ (arc is inside).
* Bottom-right: Angle is $x$ (arc is inside).
* Middle-right vertex: The line goes straight up. The angle shown is $50^\circ$ inside. Wait, let's look at the shape again.
* Vertices:
1. Top Left (Int: 55)
2. Bottom Left (Int: 120)
3. Bottom Right (Int: x)
4. Top Right (Int: 50)
5. There is a 5th vertex between Top Right and Bottom Right? No, it looks like a quadrilateral with a "bite" taken out, or a concave pentagon.
Let's trace the perimeter:
Start Top-Left -> Bottom-Left -> Bottom-Right -> Top-Right -> Back to Top-Left.
This is a quadrilateral (4 sides).
Let's re-read the diagram.
Side 1: Left vertical-ish.
Side 2: Bottom horizontal-ish.
Side 3: Right vertical-ish.
Side 4: Top diagonal.
It is a quadrilateral.
Interior Angles:
- Top-Left: $180 - 125 = 55^\circ$.
- Bottom-Left: $180 - 60 = 120^\circ$.
- Top-Right: The angle marked $50^\circ$ is interior.
- Bottom-Right: The angle marked $x$ is interior.
Sum = $360^\circ$.
$55 + 120 + 50 + x = 360$
$225 + x = 360$
$x = 135^\circ$.
7)
* Step 1: This is a pentagon (5 sides). Sum of interior angles = $540^\circ$.
* Step 2: Find interior angles.
* Top-left vertex: Exterior angles are $55^\circ$ and part of the straight line. The interior angle is vertically opposite to nothing, it's on a straight line. The interior angle is $180^\circ - 55^\circ = 125^\circ$.
* Top-right vertex: Exterior is $110^\circ$. Interior is $180^\circ - 110^\circ = 70^\circ$.
* Bottom-right vertex: Exterior is $65^\circ$. Interior is $180^\circ - 65^\circ = 115^\circ$.
* Bottom-left vertex: Exterior is $70^\circ$. Interior is $180^\circ - 70^\circ = 110^\circ$.
* Bottom-middle vertex: Angle is $x$. This is an interior angle.
* Step 3: Calculate $x$.
* $125^\circ + 70^\circ + 115^\circ + 110^\circ + x = 540^\circ$
* Sum of knowns: $125 + 70 = 195$. $195 + 115 = 310$. $310 + 110 = 420$.
* $420^\circ + x = 540^\circ$
* $x = 540^\circ - 420^\circ$
* $x = 120^\circ$
Final Answer:
1) 120°
2) 135°
3) 155°
4) 140°
5) 125°
6) 135°
7) 120°
Parent Tip: Review the logic above to help your child master the concept of finding interior angles of a polygon worksheet.
