Question

Question

Use the image to answer the question.

An illustration shows three rays extending from a common origin. The angle measures between them are labeled.An illustration shows three rays extending from a common origin. The angles between them are labeled. The first ray extends vertically upward from the vertex, the second ray is inclined upward from the vertex to the right, and the third ray extends horizontally to the right. The angle between the first and the third rays is marked with a right angle symbol. The angle between the first and the second rays is labeled as left parenthesis x plus 4 right parenthesis degrees. The angle between the second and the third rays is labeled as left parenthesis 3x plus 2 right parenthesis degrees.

What are the two unknown angle measures that form the 90° angle?

(1 point)
(x+4)°=
°, (3x+2)°=
°

Answer 1

To solve for the unknown angle measures, we know that the angles between the three rays should sum up to 90 degrees since the first ray (vertical) and the third ray (horizontal) form a right angle.

The angle between the first and second rays is labeled as \( (x+4)^\circ \) and the angle between the second and third rays is labeled as \( (3x+2)^\circ \).

We can express the relationship as:

\[ (x + 4) + (3x + 2) = 90 \]

Now, let's simplify the equation:

\[ x + 4 + 3x + 2 = 90 \]

Combining like terms gives us:

\[ 4x + 6 = 90 \]

Next, we isolate \(4x\) by subtracting 6 from both sides:

\[ 4x = 90 - 6 \] \[ 4x = 84 \]

Now, divide by 4:

\[ x = 21 \]

Now we can calculate the two unknown angle measures:

1. For the angle \( (x + 4)^\circ \): \[ (x + 4) = 21 + 4 = 25^\circ \]

2. For the angle \( (3x + 2)^\circ \): \[ (3x + 2) = 3(21) + 2 = 63 + 2 = 65^\circ \]

Thus, the two unknown angle measures that form the 90° angle are:

\[ (x+4)^\circ = 25^\circ, \quad (3x+2)^\circ = 65^\circ \]