Questions: Find the measure of each angle in the diagram. (3y+11)° □ ° 10y°: □ °

Find the measure of each angle in the diagram. (3y+11)° □ ° 10y°: □ °
Transcript text: STRUCTURE Find the measure of each angle in the diagram. $(3 y+11)^{\circ}$ $\square$ $\circ$ $10 y^{\circ}:$ $\square$ ${ }^{\circ}$
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Solution

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Solution Steps

Step 1: Identify the relationships between the angles

The given angles form a pair of vertical angles and a pair of linear pairs. Vertical angles are equal, and the sum of the angles in a linear pair is 180°.

Step 2: Set up equations based on vertical angles

From the diagram:

  • \( (3y + 11)^\circ = (4x - 22)^\circ \)
  • \( 10y^\circ = (7x + 4)^\circ \)
Step 3: Solve the first equation for one variable

\[ 3y + 11 = 4x - 22 \] \[ 3y = 4x - 33 \] \[ y = \frac{4x - 33}{3} \]

Step 4: Substitute \( y \) into the second equation

\[ 10 \left( \frac{4x - 33}{3} \right) = 7x + 4 \] \[ \frac{40x - 330}{3} = 7x + 4 \] \[ 40x - 330 = 21x + 12 \] \[ 19x = 342 \] \[ x = 18 \]

Step 5: Find \( y \) using the value of \( x \)

\[ y = \frac{4(18) - 33}{3} \] \[ y = \frac{72 - 33}{3} \] \[ y = \frac{39}{3} \] \[ y = 13 \]

Step 6: Calculate the measures of the angles
  • \( (3y + 11)^\circ = 3(13) + 11 = 39 + 11 = 50^\circ \)
  • \( (4x - 22)^\circ = 4(18) - 22 = 72 - 22 = 50^\circ \)
  • \( 10y^\circ = 10(13) = 130^\circ \)
  • \( (7x + 4)^\circ = 7(18) + 4 = 126 + 4 = 130^\circ \)

Final Answer

The measures of the angles are:

  • \( (3y + 11)^\circ = 50^\circ \)
  • \( (4x - 22)^\circ = 50^\circ \)
  • \( 10y^\circ = 130^\circ \)
  • \( (7x + 4)^\circ = 130^\circ \)
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