Geometry worksheet with angle calculation problems at different difficulty levels.

Here are the step-by-step solutions for each problem in the grid.

Secure Row

1. Calculate value of $x$ (Left)
* Step 1: The angles lie on a straight line, so they add up to $180^\circ$.
* Step 2: Set up the equation: $x + (2x + 15) = 180$.
* Step 3: Combine like terms: $3x + 15 = 180$.
* Step 4: Subtract 15 from both sides: $3x = 165$.
* Step 5: Divide by 3: $x = 55$.

2. Calculate value of $x$ (Middle)
* Step 1: The angles around a point add up to $360^\circ$.
* Step 2: Set up the equation: $(x - 10) + x + (x + 40) = 360$.
* Step 3: Combine like terms ($x+x+x$ and $-10+40$): $3x + 30 = 360$.
* Step 4: Subtract 30 from both sides: $3x = 330$.
* Step 5: Divide by 3: $x = 110$.

3. Calculate value of $x$ (Right)
* Step 1: The angles lie on a straight line, so they add up to $180^\circ$.
* Step 2: Set up the equation: $(x + 10) + 10x + x = 180$.
* Step 3: Combine like terms ($x+10x+x$): $12x + 10 = 180$.
* Step 4: Subtract 10 from both sides: $12x = 170$.
* Step 5: Divide by 12: $x = \frac{170}{12} = 14.166...$ which rounds to 14.17.

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Developing Row

4. Calculate the value of $V$ (Left)
* Step 1: Look at the triangle on the left. It has two equal sides (marked with lines), making it an isosceles triangle. This means the base angles are equal. Let's call them $t$.
* Step 2: Angles in a triangle sum to $180^\circ$: $38 + t + t = 180 \rightarrow 2t = 142 \rightarrow t = 71^\circ$.
* Step 3: Angle $u$ and angle $t$ form a straight line. So, $u = 180 - 71 = 109^\circ$.
* Step 4: Angles $u$ and $v$ are vertically opposite (they share the same vertex formed by crossing lines). Therefore, $v = u$.
* Step 5: $v = 109$.

5. Do the following three angles make a triangle? (Middle)
* Step 1: To form a triangle, the sum of the interior angles must be exactly $180^\circ$.
* Step 2: Add the given angles: $55 + 60 + 55$.
* Step 3: Calculation: $55 + 55 = 110$, and $110 + 60 = 170$.
* Step 4: Since $170^\circ$ is not equal to $180^\circ$, these angles cannot form a triangle.
* Answer: No.

6. Calculate the value of [missing angle] (Right)
*(Note: The text "Calculate the value of" is cut off, but we can solve for the missing third angle).*
* Step 1: Angles in a triangle add up to $180^\circ$.
* Step 2: Add the known angles: $20 + 110 = 130$.
* Step 3: Subtract from 180: $180 - 130 = 50$.
* Answer: The missing angle is $50^\circ$.

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Foundation Row

7. Calculate the value of $d$ (Left)
* Step 1: Angles in a triangle add up to $180^\circ$.
* Step 2: Add the known angles: $51 + 69 = 120$.
* Step 3: Subtract from 180: $180 - 120 = 60$.
* Answer: $d = 60$.

8. Calculate the value of $x$ (Middle)
* Step 1: The angles around a central point add up to $360^\circ$.
* Step 2: Identify the angles: There is a right angle (square symbol) which is $90^\circ$, an angle of $131^\circ$, and angle $x$.
* Step 3: Add the known angles: $90 + 131 = 221$.
* Step 4: Subtract from 360: $360 - 221 = 139$.
* Answer: $x = 139$.

9. Calculate the value of $x$ (Right)
* Step 1: The angles lie on a straight line, so they add up to $180^\circ$.
* Step 2: Set up the equation: $132 + x = 180$.
* Step 3: Subtract 132 from 180: $180 - 132 = 48$.
* Answer: $x = 48$.

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Final Answer:
Secure Row:
1. $x = 55$
2. $x = 110$
3. $x \approx 14.17$ (or $\frac{85}{6}$)

Developing Row:
4. $v = 109$
5. No (Sum is $170^\circ$, not $180^\circ$)
6. Missing angle = $50^\circ$

Foundation Row:
7. $d = 60$
8. $x = 139$
9. $x = 48$