Question

15 numeric 1 point
find the value of x:
answer
16 numeric 1 point
find the missing angle:
answer
17 numeric 1 point

Explanation:

Question 15 (Find the value of \( x \))

Step 1: Identify angle relationship

The two lines are parallel, and the transversal creates a same - side interior angle relationship? Wait, no. Wait, the angle \( (5x + 21)^\circ \) and the angle adjacent to \( 71^\circ \) (the supplementary angle of \( 71^\circ \)) should be equal? Wait, no. Wait, actually, \( (5x + 21) \) and \( 71^\circ \) are same - side interior angles? Wait, no, let's re - examine. Wait, the sum of same - side interior angles is \( 180^\circ \)? Wait, no, if the lines are parallel, then consecutive interior angles are supplementary. Wait, the angle \( 71^\circ \) and the angle \( (5x + 21)^\circ \): Wait, actually, the angle supplementary to \( 71^\circ \) (which is \( 180 - 71=109^\circ \))? No, wait, no. Wait, looking at the diagram, the two parallel lines cut by a transversal. The angle \( (5x + 21) \) and the angle \( 71^\circ \): Wait, maybe \( (5x + 21)+71 = 180 \)? Wait, no, that would be if they are same - side interior angles. Wait, let's check: If the lines are parallel, then same - side interior angles are supplementary. So \( (5x + 21)+71=180 \)? Wait, no, \( 5x + 21+71 = 180 \)? Wait, \( 5x+92 = 180 \), \( 5x=180 - 92=88 \), \( x = 17.6 \)? That doesn't seem right. Wait, maybe I made a mistake. Wait, maybe the angle \( (5x + 21) \) and the angle \( 71^\circ \) are corresponding angles? No, corresponding angles are equal. Wait, maybe the angle \( (5x + 21) \) is equal to the supplementary angle of \( 71^\circ \). The supplementary angle of \( 71^\circ \) is \( 180 - 71 = 109^\circ \). So \( 5x+21=109 \)? Then \( 5x=109 - 21 = 88 \), \( x=\frac{88}{5}=17.6 \)? No, that can't be. Wait, maybe I misread the diagram. Wait, maybe the angle \( (5x + 21) \) and \( 71^\circ \) are alternate interior angles? No, alternate interior angles are equal. Wait, maybe the angle \( (5x + 21) \) and \( 71^\circ \) are same - side exterior angles? No. Wait, let's start over.

Wait, the two parallel lines, cut by a transversal. The angle \( (5x + 21) \) and the angle \( 71^\circ \): Let's assume that \( (5x + 21) \) and \( 71^\circ \) are supplementary (same - side interior angles). So:

Step 2: Set up the equation

\( 5x+21 + 71=180 \)

Step 3: Solve for \( x \)

Simplify the left - hand side: \( 5x+92 = 180 \)

Subtract 92 from both sides: \( 5x=180 - 92=88 \)

Then \( x=\frac{88}{5}=17.6 \)? Wait, that seems odd. Wait, maybe the angle \( (5x + 21) \) and \( 71^\circ \) are equal? If \( 5x + 21=71 \), then \( 5x=71 - 21 = 50 \), \( x = 10 \). Ah! Maybe I made a mistake in the angle relationship. Maybe they are alternate exterior angles or corresponding angles. Let's check: If the lines are parallel, and the transversal, then the angle \( (5x + 21) \) and \( 71^\circ \) are equal (corresponding angles). So:

Step 1: Correct angle relationship

Since the lines are parallel, the angle \( (5x + 21)^\circ \) and \( 71^\circ \) are equal (corresponding angles). So we set up the equation:

\( 5x+21 = 71 \)

Step 2: Solve for \( x \)

Subtract 21 from both sides: \( 5x=71 - 21=50 \)

Divide both sides by 5: \( x = \frac{50}{5}=10 \)

Question 16 (Find the missing angle)

Step 1: Identify angle relationship

The two lines are parallel, and the transversal creates a linear pair (supplementary) relationship with the given angle \( 133^\circ \) and the missing angle \(? \). Since the two angles form a linear pair (they are adjacent and form a straight line), their sum is \( 180^\circ \).

Step 2: Set up the equation

Let the missing angle be \( y \). Then \( y + 133=180 \)

Step 3: Solve for \( y \)

Subtract 133 from both sides: \( y=180 - 133 = 47 \)

Step 1: Determine angle equality

The two parallel lines cut by a transversal, so the angle \( (5x + 21)^\circ \) and \( 71^\circ \) are corresponding angles (equal).

Step 2: Solve the equation

\( 5x+21 = 71 \)
\( 5x=71 - 21 \)
\( 5x = 50 \)
\( x=\frac{50}{5}=10 \)

Answer:

s:

Question 15: