Question

assume all given information to be true.

1. if m∠a is less than 90°, then ∠a is an acute angle. the measure of ∠a = 75°. what can you logically conclude?

a ∠a is acute.
b ∠a is right.
c ∠a is obtuse.
d not enough information

1. what conclusion can you draw from the following statements?

• if a figure is a square, then it has four sides.
• if a figure has four sides, then it is not a triangle.

a if a figure is not a triangle, then it is a square.
b if a figure has four sides, then it is a square.
c if a figure is a square, then it is not a triangle.
d if a figure is not a triangle, then it has four sides.

1. determine whether the hypothesis of each given conditional is true from the given information.

if a, b, c, and d are collinear, then they lie in the same plane. a, b, c, and d lie in the same plane. yes no
if (overrightarrow{bd}) bisects ∠abc, then d lies in the interior of ∠abc. d lies in the interior of abc. yes no

1. in the law of syllogism, suppose (p

ightarrow q) and (q
ightarrow r) are true, where p is “a polygon has three sides”, q is “a polygon is a triangle”, and r is “the interior angles of a polygon sum to 180°”. complete the statement.
if p is true, then r is true false.

1. if c lies between a and b on (overline{ab}), then (ac + cb=ab). choose the option you can logically conclude about (overline{ab}).

a (ab = 2(ac))
b (ab = 11)
c (ab = 10.7)
d none of the above

Explanation:

Step1: Recall angle - type definitions

An acute angle has measure less than 90°. Given \(m\angle A = 75^{\circ}<90^{\circ}\), so \(\angle A\) is acute.

Step2: Analyze conditional statements

Let \(p\): a figure is a square, \(q\): a figure has four sides, \(r\): a figure is not a triangle. We know \(p
ightarrow q\) and \(q
ightarrow r\). By the law of syllogism \(p
ightarrow r\), which means if a figure is a square, then it is not a triangle.

Step3: Check hypotheses

For “If \(A,B,C,D\) are collinear, then they lie in the same plane. \(A,B,C,D\) lie in the same plane.” We only know the conclusion is true, not enough to say if the hypothesis ( \(A,B,C,D\) are collinear) is true. For “If \(\overrightarrow{BD}\) bisects \(\angle ABC\), then \(D\) lies in the interior of \(\angle ABC\). \(D\) lies in the interior of \(\angle ABC\)”, we only know the conclusion is true, not enough to say if the hypothesis (\(\overrightarrow{BD}\) bisects \(\angle ABC\)) is true.

Step4: Apply law of syllogism

If \(p
ightarrow q\) and \(q
ightarrow r\) are true, by the law of syllogism, if \(p\) is true, then \(r\) is true.

Step5: Use segment - addition postulate

If \(C\) lies between \(A\) and \(B\) on \(\overline{AB}\), then \(AC + CB=AB\), but we have no information about the lengths of \(AC\) and \(CB\) to determine the length of \(AB\).

Answer:

1. A. \(\angle A\) is acute.
2. C. If a figure is a square, then it is not a triangle.
3. No, No
4. true
5. D. none of the above