Question

1. (1.23) if the two angles in a linear pair have equal measures, then each is a(n) ____ angle.

a. acute
b. complementary
c. supplementary
d. adjacent
e. right

1. (1.26) if a statement is true, then its converse is ____.

a. true
b. false
c. either a or b
d. not existing
e. none of the above

1. (1.2) the following argument is a(n) ____ deduction.

if two parallel lines are cut by a transversal, then their alternate interior angles are equal.
if ∠a and ∠b are equal, then they are alternate interior angles.
a. valid
b. invalid
c. not enough data to determine
d. neither a nor b
e. none of the above

1. (1.2) the following argument is a(n) ____ deduction.

a transversal is a line that intersects two or more lines at different points.
line t intersects lines n and m at different points, so line t is a transversal.
a. valid
b. invalid
c. not enough data to determine
d. neither a nor b
e. none of the above

Explanation:

Step1: Recall linear - pair property

A linear pair of angles is supplementary, i.e., their sum is 180°. If the two angles in a linear pair are equal, let each angle be \(x\). Then \(x + x=180^{\circ}\), so \(2x = 180^{\circ}\), and \(x = 90^{\circ}\). A 90 - degree angle is a right - angle.

Step2: Recall the concept of converse statements

The truth of a statement does not imply the truth of its converse. For example, the statement "If a number is even, then it is divisible by 2" is true, but its converse "If a number is divisible by 2, then it is even" is also true, while the statement "If a shape is a square, then it is a rectangle" is true, but its converse "If a shape is a rectangle, then it is a square" is false. So if a statement is true, its converse can be either true or false.

Step3: Analyze the logical deduction about alternate - interior angles

The first statement is a correct property: if two parallel lines are cut by a transversal, then their alternate - interior angles are equal. But the second statement "If \(\angle A\) and \(\angle B\) are equal, then they are alternate - interior angles" is wrong. Just because two angles are equal does not mean they are alternate - interior angles. So the deduction is invalid.

Step4: Analyze the logical deduction about transversals

The first statement gives the definition of a transversal: a line that intersects two or more lines at different points. The second statement says that since line \(t\) intersects lines \(n\) and \(m\) at different points, line \(t\) is a transversal. This is a valid deduction based on the given definition.

Answer:

1. E. right
2. C. Either A or B
3. B. invalid
4. A. valid