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

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

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

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

57°
, 54°
, and 69°

Answer 1

To find the measures of the angles in triangle ABC, we can use 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

We can set up the equation based on the Triangle Angle Sum Theorem:

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

Now, let's combine like terms:

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

This simplifies to:

\[ 15x + 15 = 180 \]

Next, we subtract 15 from both sides:

\[ 15x = 165 \]

Now, divide by 15:

\[ x = 11 \]

Now we can find the measures of each angle by substituting \(x\) back into the equations for angle a, angle b, and angle c.

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

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

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

Thus, the measures of the angles in the triangle are 58°, 59°, and 63°.

Therefore, the correct response is:

59°, 58°, and 63° (this option is repeated in the format given, so either is correct).