5. if two parallel lines are cut by a transversal, what is the relationship between interior angles that are on the same side of the transversal: supplementary, complementary, or vertical?
6. in $\triangle abc$, the measures of the angles $a$, $b$, and $c$, respectively, are in the ratio $2:4:9$.

Question

Accepted Answer

### For Question 5:
# Brief Explanations:
When two parallel lines are cut by a transversal, same - side interior angles are supplementary. This is a fundamental theorem in geometry. Supplementary angles are two angles whose sum is $180^{\circ}$. When a transversal intersects two parallel lines, the same - side interior angles form a linear - pair - like relationship in terms of their sum, and their sum is $180^{\circ}$. Complementary angles sum to $90^{\circ}$, and vertical angles are equal and are formed by the intersection of two lines (not related to same - side interior angles in this context).
# Answer:
supplementary

### For Question 6 (assuming we need to find the measures of the angles or something related, let's assume we need to find the measures of angles $A$, $B$, and $C$):
# Explanation:
## Step 1: Recall the sum of angles in a triangle
The sum of the interior angles of a triangle is $180^{\circ}$. So, $\angle A+\angle B+\angle C = 180^{\circ}$.
## Step 2: Let the common ratio be $x$
Given that the ratio of the measures of angles $A$, $B$, and $C$ is $2:4:9$. So, we can represent the measures of the angles as $\angle A = 2x$, $\angle B=4x$, and $\angle C = 9x$.
## Step 3: Set up the equation and solve for $x$
Substitute the expressions for the angles into the angle - sum formula of a triangle:
$$2x + 4x+9x=180^{\circ}$$
Combine like terms:
$$15x = 180^{\circ}$$
Divide both sides by 15:
$$x=\frac{180^{\circ}}{15}=12^{\circ}$$
## Step 4: Find the measures of each angle
$\angle A = 2x=2\times12^{\circ} = 24^{\circ}$
$\angle B = 4x=4\times12^{\circ}=48^{\circ}$
$\angle C=9x = 9\times12^{\circ}=108^{\circ}$
# Answer:
$\angle A = 24^{\circ}$, $\angle B = 48^{\circ}$, $\angle C = 108^{\circ}$ (If the question was different, for example, if it was to find the largest angle, the answer would be $108^{\circ}$, etc. But based on the common follow - up for such ratio - based triangle angle problems, this is the solution for finding the angle measures)

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QUESTION IMAGE

Question

Explanation:

For Question 5:

Step 1: Recall the sum of angles in a triangle

Step 2: Let the common ratio be \(x\)

Step 3: Set up the equation and solve for \(x\)

Step 4: Find the measures of each angle

Answer:

For Question 6 (assuming we need to find the measures of the angles or something related, let's assume we need to find the measures of angles \(A\), \(B\), and \(C\)):