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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 angles, we know that the angles between the rays must sum to 90° since there is a right angle between the first ray (vertical) and the third ray (horizontal).

Let’s denote:

• The angle between the first ray and the second ray as \( (x + 4)° \).
• The angle between the second ray and the third ray as \( (3x + 2)° \).

Since these two angles are adjacent and their sum is equal to \( 90° \), we can set up the following equation:

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

Now, simplify and solve for \( x \):

\[ x + 4 + 3x + 2 = 90 \] \[ 4x + 6 = 90 \] \[ 4x = 90 - 6 \] \[ 4x = 84 \] \[ x = 21 \]

Now, substitute \( x = 21 \) back into the expressions to find the actual angle measures:

1. For \( (x + 4)° \): \[ (x + 4) = 21 + 4 = 25° \]

2. For \( (3x + 2)° \): \[ (3x + 2) = 3(21) + 2 = 63 + 2 = 65° \]

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

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