Question

if a polygon has 6 sides then it is a hexagon. match column a with column b. a. inverse 1. if it is not a hexagon then it is not a polygon with 6 sides. b. converse 2. if it is a hexagon, then a polygon has 7 sides. c. contrapositive 3. a polygon has 6 sides, if and only if, it is a hexagon. d. biconditional 4. if a polygon does not have 6 sides, then it is not a hexagon. 2. parallel lines sy and hz are shown below. fa is a transversal line. m∠rch=(8x - 4)°, m∠yrc=(64)°. find x, and the m∠hca. show work. no work, no credit. 3. line g is a perpendicular bisector of pw at point n. show work. no work, no credit. a. if pn = 4x + 15 and wn = 9x - 20, find the value of x. hint: pn = wn b. how long is line segment pn? hint: plug - in x. c. how long is line segment pw?

Explanation:

1. Matching Logical Statements

Step1: Recall inverse definition

The inverse of "If \(p\), then \(q\)" is "If not \(p\), then not \(q\)". Given the statement "If a polygon has 6 sides then it is a hexagon", the inverse is "If a polygon does not have 6 sides, then it is not a hexagon". So A matches 4.

Step2: Recall converse definition

The converse of "If \(p\), then \(q\)" is "If \(q\), then \(p\)". Here, the converse should be "If it is a hexagon, then a polygon has 6 sides", but the option "If it is a hexagon, then a polygon has 7 sides" is incorrect and does not match.

Step3: Recall contra - positive definition

The contra - positive of "If \(p\), then \(q\)" is "If not \(q\), then not \(p\)". So the contra - positive of the given statement is "If it is not a hexagon then it is not a polygon with 6 sides". So C matches 1.

Step4: Recall biconditional definition

The biconditional is " \(p\) if and only if \(q\)". So "A polygon has 6 sides, if and only if, it is a hexagon" is the biconditional. So D matches 3.

2. Finding \(x\) and angle measure for parallel lines

Step1: Use corresponding angles property

Since \(SY\parallel HZ\) and \(FA\) is a transversal, \(\angle RCH\) and \(\angle YRC\) are corresponding angles and are equal. So we set up the equation \(8x - 4=64\).

Step2: Solve for \(x\)

Add 4 to both sides of the equation: \(8x-4 + 4=64 + 4\), which gives \(8x=68\). Then divide both sides by 8: \(x=\frac{68}{8}=\frac{17}{2}=8.5\).

Step3: Find \(\angle HCA\)

\(\angle HCA\) and \(\angle YRC\) are vertical angles. Vertical angles are equal. So \(m\angle HCA = 64^{\circ}\).

3. Using perpendicular bisector property

Step3a: Solve for \(x\)

Since line \(g\) is a perpendicular bisector of \(\overline{PW}\) at point \(N\), \(PN = WN\). Set up the equation \(4x + 15=9x-20\).
Subtract \(4x\) from both sides: \(4x+15-4x=9x - 20-4x\), which gives \(15 = 5x-20\).
Add 20 to both sides: \(15 + 20=5x-20 + 20\), so \(35 = 5x\).
Divide both sides by 5: \(x = 7\).

Step3b: Find length of \(PN\)

Substitute \(x = 7\) into the expression for \(PN\): \(PN=4x + 15=4\times7+15=28 + 15=43\).

Step3c: Find length of \(PW\)

Since \(PN = WN\) and \(PN = 43\), \(PW=PN + WN=43+43 = 86\).

Answer:

1. A. 4, C. 1, D. 3
2. \(x = 8.5\), \(m\angle HCA=64^{\circ}\)

3a. \(x = 7\)
3b. \(PN = 43\)
3c. \(PW = 86\)