Find the Value of x, and the Missing Angle Measures - YouTube - Free Printable

Problem Analysis:

We are tasked with finding the value of \( x \) and the missing angles in two geometric diagrams. Let's solve each part step by step.

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Part (a):

The diagram shows a point where several angles meet. The angles around a point sum to \( 360^\circ \). The given angles are:
- \( x^\circ \)
- \( 2x^\circ \)
- \( 89^\circ \)
- \( 76^\circ \)

#### Step 1: Set up the equation
The sum of all angles around a point is \( 360^\circ \):
\[
x + 2x + 89 + 76 = 360
\]

#### Step 2: Simplify the equation
Combine like terms:
\[
3x + 165 = 360
\]

#### Step 3: Solve for \( x \)
Subtract 165 from both sides:
\[
3x = 195
\]
Divide by 3:
\[
x = 65
\]

#### Step 4: Find the missing angles
- The angle labeled \( x^\circ \) is \( 65^\circ \).
- The angle labeled \( 2x^\circ \) is \( 2 \times 65 = 130^\circ \).

Thus, the missing angles are:
- \( x = 65^\circ \)
- \( 2x = 130^\circ \)

---

Part (b):

The diagram shows a point where several angles meet. The angles around a point sum to \( 360^\circ \). The given angles are:
- \( x + 8^\circ \)
- \( 2x - 8^\circ \)
- \( y^\circ \)
- \( 4x - 24^\circ \)

#### Step 1: Set up the equation
The sum of all angles around a point is \( 360^\circ \):
\[
(x + 8) + (2x - 8) + y + (4x - 24) = 360
\]

#### Step 2: Simplify the equation
Combine like terms:
\[
x + 2x + 4x + 8 - 8 - 24 + y = 360
\]
\[
7x - 24 + y = 360
\]

#### Step 3: Express \( y \) in terms of \( x \)
Rearrange the equation to solve for \( y \):
\[
y = 360 - 7x + 24
\]
\[
y = 384 - 7x
\]

#### Step 4: Use the property of vertical angles
In the diagram, the angles \( x + 8^\circ \) and \( 2x - 8^\circ \) are vertical angles, which means they are equal:
\[
x + 8 = 2x - 8
\]

#### Step 5: Solve for \( x \)
Subtract \( x \) from both sides:
\[
8 = x - 8
\]
Add 8 to both sides:
\[
x = 16
\]

#### Step 6: Find the missing angles
- The angle labeled \( x + 8^\circ \) is \( 16 + 8 = 24^\circ \).
- The angle labeled \( 2x - 8^\circ \) is \( 2 \times 16 - 8 = 24^\circ \).
- The angle labeled \( 4x - 24^\circ \) is \( 4 \times 16 - 24 = 64 - 24 = 40^\circ \).
- The angle labeled \( y^\circ \) is \( 384 - 7x = 384 - 7 \times 16 = 384 - 112 = 272^\circ \).

Thus, the missing angles are:
- \( x = 16^\circ \)
- \( x + 8 = 24^\circ \)
- \( 2x - 8 = 24^\circ \)
- \( 4x - 24 = 40^\circ \)
- \( y = 272^\circ \)

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Final Answers:

\[
\boxed{65, 24}
\]