Question 30: Solve for $x$ and Name the Angle Pair

For question 30, the angle pair shown in the diagram consists of two angles labeled $3x+5$ and $8x-15$. These angles form a linear pair, which means their sum is $180^\circ$ because they are supplementary (they add up to form a straight line). Let's solve for $x$:

$3x + 5 + 8x - 15 = 180$ $11x - 10 = 180$ $11x = 190$ $x = \frac{190}{11} \approx 17.27$

Now, using $x \approx 17.27$, we calculate the measures of the angles: $3x + 5 = 3(17.27) + 5 = 51.81 + 5 = 56.81^\circ$ $8x - 15 = 8(17.27) - 15 = 138.16 - 15 = 123.16^\circ$

Since the angles form a linear pair, each of the other two angles opposite them are equal to these values (by the Vertical Angles Theorem):

• The angle opposite $56.81^\circ$ is also $56.81^\circ$.
• The angle opposite $123.16^\circ$ is also $123.16^\circ$.

Question 31: Measure of Angle CGE

For question 31, given that line $AD$ is parallel to line $BE$ and $AC$ is a transversal, angle $ACD$ (which is $138^\circ$) corresponds to angle $CGE$ due to the Alternate Interior Angles Theorem. Thus:

$\angle CGE = 138^\circ$

Conclusions and Further Inquiry

1. Do you understand how the concept of supplementary and vertical angles was applied in question 30?
2. Would you like to know more about how parallel lines and a transversal create congruent and supplementary angles, as used in question 31?
3. Is there any step in the calculation of $x$ for question 30 that you need further explanation on?
4. Would you like to explore more problems involving linear pairs and vertical angles?
5. Are there any other types of angle pairs or geometric concepts you're curious about?

Tip: When solving for an unknown in equations arising from geometry problems, always double-check that the computed angle measures are sensible (e.g., non-negative and feasible within the context of the diagram).