Exercise 22 Page 377

We are given the following diagram.

We want to determine which of the given statements is not true concerning $\angle A,$ $\angle B,$ and $\angle C.$ Let's analyze the statements one at a time.

Statement F

First, we will consider statement F.

Recall that obtuse angles are angles that measure more than $90^{\circ }$ but less than $180^{\circ }.$ Let's look at $\angle B$ and $\angle C.$

We can see that $\angle B$ and $\angle C$ are right angles. Since these angles have a measure of $90^{\circ },$ they are not obtuse. Therefore, statement F is not true.

Statement G

Now we will analyze statement G.

When two lines or line segments intersect, vertical angles are formed on opposite sides of the point of intersection. To check whether $\angle A$ and $\angle C$ are vertical, let's take another look at the diagram.

Notice that $\angle A$ and $\angle C$ are on opposite sides of the point of intersection of two lines. This means that these angles are vertical, so statement G is true.

Statement H

Next, we will consider statement H.

Notice that in the diagram, we have two parallel lines that are cut by a transversal. Recall that alternate interior angles are interior angles that lie on opposite sides of the transversal. Looking at the diagram, we can determine whether $\angle A$ and $\angle B$ are alternate interior angles.

Indeed, $\angle A$ and $\angle B$ are interior angles that lie on opposite sides of the transversal. Therefore, these angles are alternate interior angles, and statement H is true.

Statement I

Finally, we will analyze statement I.

Since statement G is true, we know that $\angle A$ and $\angle C$ are vertical angles. Recall that vertical angles are always congruent. This means that $\angle A$ and $\angle C$ are congruent and statement I is also true.

Conclusion

We found that only statement F is not true.