Question

triangle
at the top
(8x + 5) degrees
at the left corner
(5x - 1) degrees
at the right corner
(4x + 6) degrees.
Use the Triangle Angle Sum Theorem to find the measures of each angle for the triangle shown in the diagram. Responses:
46° 50° and 85°
42°, 44°, and 79°
50°, 54°, and 93°
46°, 49°, and 85°

Answer 1

The Triangle Angle Sum Theorem states that the sum of the angles in a triangle is always \(180^\circ\). Let's denote the three angles of the triangle as follows:

- Angle at the top: \( (8x + 5)^\circ \)
- Angle at the left corner: \( (5x - 1)^\circ \)
- Angle at the right corner: \( (4x + 6)^\circ \)

According to the Triangle Angle Sum Theorem, we can set up the following equation:

\[
(8x + 5) + (5x - 1) + (4x + 6) = 180
\]

Now, let's simplify the equation step by step:

1. Combine like terms:
- \( 8x + 5x + 4x = 17x \)
- \( 5 - 1 + 6 = 10 \)

So the equation becomes:

\[
17x + 10 = 180
\]

2. Subtract 10 from both sides:

\[
17x = 170
\]

3. Divide by 17:

\[
x = 10
\]

Now that we know \(x\), we can find the measures of each angle by substituting \(x = 10\) back into the angle expressions:

1. Angle at the top:
\[
8x + 5 = 8(10) + 5 = 80 + 5 = 85^\circ
\]

2. Angle at the left corner:
\[
5x - 1 = 5(10) - 1 = 50 - 1 = 49^\circ
\]

3. Angle at the right corner:
\[
4x + 6 = 4(10) + 6 = 40 + 6 = 46^\circ
\]

Now we have the measures of the angles: \(85^\circ\), \(49^\circ\), and \(46^\circ\).

Thus, the angles of the triangle are:
- \(46^\circ\)
- \(49^\circ\)
- \(85^\circ\)

The correct response from the options provided is:
**46°, 49°, and 85°.**