Use the image to answer the question.

Question

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

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

59°
, 63°
, and 48°
59 degrees , 63 degrees , and 48 degrees

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

59°
, 58°
, and 63°

Answer 1

To find the measures of the angles in triangle ABC, we can apply the Triangle Angle Sum Theorem, which states that the sum of the angles in a triangle is always 180 degrees.

Given:

• Angle a = \(4x + 14\)
• Angle b = \(5x + 4\)
• Angle c = \(6x - 3\)

According to the Triangle Angle Sum Theorem:

\[ (4x + 14) + (5x + 4) + (6x - 3) = 180 \]

Now, combine like terms:

\[ 4x + 5x + 6x + 14 + 4 - 3 = 180 \]

This simplifies to:

\[ 15x + 15 = 180 \]

Next, subtract 15 from both sides:

\[ 15x = 165 \]

Now, divide by 15:

\[ x = 11 \]

Now we can substitute \(x = 11\) back into the expressions for the angles:

• Angle a: \[ 4x + 14 = 4(11) + 14 = 44 + 14 = 58° \]

• Angle b: \[ 5x + 4 = 5(11) + 4 = 55 + 4 = 59° \]

• Angle c: \[ 6x - 3 = 6(11) - 3 = 66 - 3 = 63° \]

Thus, the measures of the angles in triangle ABC are \(58°\), \(59°\), and \(63°\).

So, the correct response is: 59°, 58°, and 63°.