Question

vertical/adjacent/complementary angles l2
score: 0/1 penalty: none
question
find the measure of the missing angles.
answer attempt 1 out of 2
x =

y =

Explanation:

Step1: Find x (complementary angles)

Complementary angles sum to \( 90^\circ \). So \( x + 68^\circ = 90^\circ \). Solving for \( x \): \( x = 90^\circ - 68^\circ = 22^\circ \).

Step2: Find y (vertical angles or complementary)

Notice the right angle ( \( 90^\circ \)) and vertical angles. Since \( x = 22^\circ \), and the angle adjacent to \( y \) and \( x \) forms a right angle? Wait, actually, the angle with the right angle symbol and \( 68^\circ \) and \( x \): Wait, no, looking at the diagram, the angle with \( x \), \( 68^\circ \), and the right angle? Wait, no, the right angle ( \( 90^\circ \)) and the angle \( y \) and \( x \): Wait, actually, \( y \) is equal to \( 68^\circ \)? No, wait, vertical angles? Wait, no, let's re-examine. The angle between the two lines with the right angle: Wait, the angle with \( x \) and \( 68^\circ \) is complementary (since there's a right angle symbol), so \( x = 90 - 68 = 22 \). Then, the angle \( y \) and \( x \) and the right angle? Wait, no, the angle opposite to \( 68^\circ \) would be \( y \)? Wait, no, the vertical angle to \( 68^\circ \) is \( y \)? Wait, no, the diagram: two intersecting lines, with a right angle ( \( 90^\circ \)) and angles \( x \), \( 68^\circ \), and \( y \). Wait, actually, the angle containing \( x \) and \( 68^\circ \) is a right angle? Wait, the right angle symbol is between two lines, so that angle is \( 90^\circ \). So \( x + 68^\circ = 90^\circ \), so \( x = 22^\circ \). Then, \( y \) is equal to \( 68^\circ \) because they are vertical angles? Wait, no, maybe \( y \) is equal to \( 68^\circ \)? Wait, no, let's see: the angle opposite to \( 68^\circ \) is \( y \), so vertical angles are equal. Wait, but also, the angle with \( x \) and \( y \) and the right angle? Wait, maybe I made a mistake. Wait, the right angle is \( 90^\circ \), so the angle between the two lines with the right angle is \( 90^\circ \). So \( x + 68^\circ = 90^\circ \), so \( x = 22^\circ \). Then, \( y \) is equal to \( 68^\circ \) because they are vertical angles? Wait, no, the angle adjacent to \( y \) and \( x \) is \( 90^\circ \)? Wait, no, let's look again. The diagram: two intersecting lines, creating angles. One angle is \( 68^\circ \), another is \( x \), and there's a right angle ( \( 90^\circ \)) between two lines. So the angle that is \( 90^\circ \) has \( x \) and \( 68^\circ \), so \( x = 90 - 68 = 22 \). Then, the angle \( y \) is equal to \( 68^\circ \) because they are vertical angles (opposite angles when two lines intersect are equal). Wait, but also, the angle opposite to \( x \) would be... Wait, no, maybe \( y = 68^\circ \). Wait, let's confirm:

Step 1: Find \( x \)
The angle with \( x \) and \( 68^\circ \) is a right angle ( \( 90^\circ \)) because of the right angle symbol. So:
\( x + 68^\circ = 90^\circ \)
\( x = 90^\circ - 68^\circ = 22^\circ \)

Step 2: Find \( y \)
The angle \( y \) and the angle of \( 68^\circ \) are vertical angles (opposite angles formed by intersecting lines), so vertical angles are equal. Therefore, \( y = 68^\circ \).

Wait, but also, if we check with \( x \): \( x + y + 90^\circ = 180^\circ \)? No, that's a straight line. Wait, a straight line is \( 180^\circ \). So \( x + 90^\circ + y = 180^\circ \)? Wait, no, the diagram: two intersecting lines, so the sum of angles on a straight line is \( 180^\circ \). So if one line is straight, then \( x + 90^\circ + y = 180^\circ \)? Wait, no, the right angle is \( 90^\circ \), so \( x + 68^\circ = 90^\circ \) (as the right angle), then the other line: \( y + x + 90^\circ = 180^\circ \)? Wait, no, that would be \( y + 22 + 90 = 180 \), so \( y = 68 \). Yes, that matches. So \( y = 68^\circ \).

So:

Step1: Calculate \( x \)

Complementary angles sum to \( 90^\circ \).
\( x + 68^\circ = 90^\circ \)
\( x = 90^\circ - 68^\circ = 22^\circ \)

Step2: Calculate \( y \)

Angles on a straight line sum to \( 180^\circ \), and there's a \( 90^\circ \) angle.
\( x + 90^\circ + y = 180^\circ \)
Substitute \( x = 22^\circ \):
\( 22^\circ + 90^\circ + y = 180^\circ \)
\( 112^\circ + y = 180^\circ \)
\( y = 180^\circ - 112^\circ = 68^\circ \)
(Alternatively, vertical angles: \( y = 68^\circ \) because they are vertical to the \( 68^\circ \) angle.)

Answer:

\( x = 22 \), \( y = 68 \)