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A triangle a b c is shown with angle a measuring 4 x plus 14, angle b measuring 5 x plus 4, and angle c measuring 6 x minus 3.

Use the Triangle Angle Sum Theorem to find the measure of the angles for the triangle in the diagram.

(1 point)
Responses

59°
, 58°
, and 63°
59 degrees , 58 degrees , and 63 degrees

57°
, 54°
, and 69°
57 degrees , 54 degrees , and 69 degrees

54°
, 44°
, and 82°
54 degrees , 44 degrees , and 82 degrees

59°
, 63°
, and 48°
59 degrees , 63 degrees , and 48 degrees
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1 answer

To find the measures of the angles in triangle ABC using the Triangle Angle Sum Theorem, we know that the sum of the angles in a triangle is always 180 degrees.

The angles given are:

  • Angle A = \(4x + 14\)
  • Angle B = \(5x + 4\)
  • Angle C = \(6x - 3\)

Setting up the equation using the Triangle Angle Sum Theorem: \[ (4x + 14) + (5x + 4) + (6x - 3) = 180 \]

Combining like terms: \[ 4x + 5x + 6x + 14 + 4 - 3 = 180 \] \[ 15x + 15 = 180 \]

Subtracting 15 from both sides: \[ 15x = 165 \]

Dividing by 15: \[ x = 11 \]

Now, substituting back to find the measures of the angles:

  • Angle A: \[ 4x + 14 = 4(11) + 14 = 44 + 14 = 58^\circ \]

  • Angle B: \[ 5x + 4 = 5(11) + 4 = 55 + 4 = 59^\circ \]

  • Angle C: \[ 6x - 3 = 6(11) - 3 = 66 - 3 = 63^\circ \]

Thus, the measures of the angles are:

  • Angle A = 58°
  • Angle B = 59°
  • Angle C = 63°

The correct response is: 59°, 58°, and 63°