Question

1. (10x + 26)° y 21° (18x - 2)°

Explanation:

Step1: Identify vertical angles or linear pairs

We can see that the sum of \( (10x + 26)^\circ \), \( 21^\circ \), and \( (18x - 2)^\circ \) should be \( 180^\circ \) because they form a linear pair (a straight line). So the equation is:
\( (10x + 26) + 21 + (18x - 2) = 180 \)

Step2: Simplify the left - hand side

Combine like terms:
\( 10x+18x + 26 + 21-2=180 \)
\( 28x+45 = 180 \)

Step3: Solve for x

Subtract 45 from both sides:
\( 28x=180 - 45 \)
\( 28x = 135 \)
Wait, this seems wrong. Maybe we made a mistake in identifying the angles. Let's re - examine the diagram. Maybe \( (10x + 26)^\circ \) and \( (18x - 2)^\circ \) are related to the \( 21^\circ \) angle through vertical angles or supplementary angles.

Wait, another approach: The angle \( (10x + 26)^\circ \) and the angle formed by \( 21^\circ \) and \( (18x - 2)^\circ \) are vertical angles? No, maybe the sum of \( (10x + 26)^\circ \) and \( 21^\circ \) is equal to \( (18x - 2)^\circ \) because of vertical angles or alternate - interior angles? Wait, let's assume that \( 10x + 26+21=18x - 2 \) (because they are vertical angles or corresponding angles).

So, \( 10x+47 = 18x - 2 \)

Step4: Solve the new equation

Subtract \( 10x \) from both sides:
\( 47=8x - 2 \)

Add 2 to both sides:
\( 49 = 8x \)
No, that's not right. Wait, maybe the sum of \( (10x + 26)^\circ \) and \( (18x - 2)^\circ \) is equal to \( 180^\circ- 21^\circ \)? Wait, the straight line is \( 180^\circ \), so \( (10x + 26)+(18x - 2)=180 - 21 \)

Step5: Correct equation setup

\( 10x+26 + 18x-2=159 \)
Combine like terms:
\( 28x + 24=159 \)

Subtract 24 from both sides:
\( 28x=159 - 24 \)
\( 28x = 135 \)
\( x=\frac{135}{28}\approx4.82 \) (This doesn't seem right. Maybe the diagram has vertical angles. Let's assume that \( 10x + 26=18x - 2+21 \) (if the angle \( (18x - 2)^\circ \) and \( 21^\circ \) are adjacent to the vertical angle of \( (10x + 26)^\circ \)).

So \( 10x+26=18x + 19 \)

Subtract \( 10x \) from both sides:
\( 26 = 8x+19 \)

Subtract 19 from both sides:
\( 7 = 8x \)
\( x=\frac{7}{8}=0.875 \) (Still not right). Wait, maybe the angle \( (10x + 26)^\circ \) and \( (18x - 2)^\circ \) are supplementary to the angle that is vertical to \( 21^\circ \). Wait, perhaps the correct equation is \( (10x + 26)+(18x - 2)=180 - 21\times2 \)? No, that's not.

Wait, let's start over. In a straight line, the sum of angles is \( 180^\circ \). Let's look at the angles around the intersection. The angle of \( 21^\circ \), \( (10x + 26)^\circ \), and the angle supplementary to \( (18x - 2)^\circ \) (or maybe \( (18x - 2)^\circ \) itself) form a straight line. Wait, maybe the two angles \( (10x + 26)^\circ \) and \( (18x - 2)^\circ \) are equal to each other plus or minus \( 21^\circ \). Wait, perhaps the correct approach is that the angle \( (10x + 26)^\circ \) is equal to \( (18x - 2)^\circ+21^\circ \) (vertical angles with a \( 21^\circ \) angle in between). So:

\( 10x + 26=18x - 2+21 \)

\( 10x + 26=18x + 19 \)

Subtract \( 10x \) from both sides:

\( 26=8x + 19 \)

Subtract 19 from both sides:

\( 7 = 8x \)

\( x=\frac{7}{8}=0.875 \) (This is incorrect. Maybe the diagram is such that \( (10x + 26)+21=(18x - 2) \) is wrong. Wait, maybe the sum of \( (10x + 26) \), \( 21 \), and \( (18x - 2) \) is \( 180 \). Let's try that again:

\( 10x+26 + 21+18x - 2=180 \)

\( 28x+45 = 180 \)

\( 28x=180 - 45=135 \)

\( x=\frac{135}{28}\approx4.82 \)

Now, if we want to find \( y \), since \( y \) is supplementary to \( (10x + 26)^\circ \), \( y = 180-(10x + 26) \). Let's substitute \( x=\frac{135}{28} \):

\( 10x=\frac{1350}{28}=\frac{675}{14}\approx48.21 \)

\( 10x + 26=\frac{675}{14}+\frac{364}{14}=\frac{1039}{14}\approx74.21 \)

\( y = 180-\frac{1039}{14}=\frac{2520 - 1039}{14}=\frac{1481}{14}\approx105.79 \)

But this seems messy. Maybe there is a mistake in the problem interpretation. Alternatively, maybe the angle \( (10x + 26)^\circ \) and \( (18x - 2)^\circ \) are vertical angles, and the \( 21^\circ \) angle is related to them. Wait, if \( 10x + 26=18x - 2 \), then:

\( 26 + 2=18x-10x \)

\( 28 = 8x \)

\( x = 3.5 \)

Ah! Maybe that's the case. I made a mistake earlier. If \( (10x + 26)^\circ \) and \( (18x - 2)^\circ \) are vertical angles (opposite angles when two lines intersect), then they are equal. So:

Step1: Set vertical angles equal

\( 10x + 26=18x - 2 \)

Step2: Solve for x

Subtract \( 10x \) from both sides:
\( 26=8x - 2 \)

Add 2 to both sides:
\( 28 = 8x \)

Divide both sides by 8:
\( x=\frac{28}{8}=\frac{7}{2}=3.5 \)

Step3: Find the measure of the angle

Now, substitute \( x = 3.5 \) into \( 10x + 26 \):
\( 10\times3.5+26=35 + 26=61^\circ \)

Then, to find \( y \), we know that the sum of \( y \), \( 61^\circ \), and \( 21^\circ \) is \( 180^\circ \) (since they form a straight line):

\( y+61 + 21=180 \)

\( y+82 = 180 \)

\( y=180 - 82=98^\circ \)

Answer:

If we are finding \( x \), \( x = \frac{7}{2}=3.5 \); if we are finding \( y \), \( y = 98^\circ \) (assuming the correct angle relationship where \( (10x + 26)^\circ \) and \( (18x - 2)^\circ \) are vertical angles).