Triangle Angle Sum Theorem Quick Check

Question

5 of 55 of 5 Items

Question
Use the image to answer the 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°
, 58°
, and 63°
59 degrees , 58 degrees , and 63 degrees

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

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

Answer 1

To solve for 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 180°. Given:

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

We can set up the equation:

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

Now, we'll combine like terms:

\[ 4x + 5x + 6x + 14 + 4 - 3 = 180 \] \[ 15x + 15 = 180 \]

Next, we subtract 15 from both sides:

\[ 15x = 165 \]

Now, we divide by 15:

\[ x = 11 \]

Now, we can find the measures of the angles:

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

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

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

So the angles are \( 58° \), \( 59° \), and \( 63° \).

The corresponding response from the options provided is:

59°, 58°, and 63°