91Ó°ÊÓ

Write (a) the contrapositive and (b) the inverse of the following statement.

If $n=17$ then $4n=68$.

Some information about the diagram is given. Tell whether the other statements can be deduced from what is given. (Write Yes or No).

Given: Point Y lies between points X and Z

a.$XY=\frac{1}{2}XZ$

b.$XZ=XY+YZ$

c.$XZ>XY$

d.$YZ>XY$

e.$XZ>YZ$

f.$XZ>2XY$

Write an indirect proof in paragraph form.

Given: $\stackrel{¯}{OJ}â‰\dots \stackrel{¯}{OK};\stackrel{¯}{JE}≠\stackrel{¯}{KE}$

Prove: $\stackrel{→}{OE}$doesn't bisect $∠JOK$.

Arrange sentences (a)-(e) in an order that completes an indirect proof of the following statement: In a plane, two lines perpendicular to a third line are parallel to each other.

Given: Lines a, b and t lie in a plane; $aâŠ\yen t;bâŠ\yen t$

Prove: $a‖b$

(a) Then a intersects b in some point Z.

(b) But this contradicts the theorem which says that there is exactly one line perpendicular to a given line through a point outside the line.

(c) It is false that a is not parallel to b, and it follows that $a‖b$

(d) Assume temporarily that a is not parallel to b.

(e) Then there are two lines through Z and perpendicular to t.

Write an indirect proof in paragraph form.

Given: $\stackrel{¯}{OJ}â‰\dots \stackrel{¯}{OK};\stackrel{¯}{JE}≠\stackrel{¯}{KE}$

Prove: $\stackrel{→}{OE}$doesn't bisect $∠JOK$

For each statement in Ex 5-10 copy and complete a table like the one shown below.

If $x^2>1$, then $x>1$.

Write the letters (a)-(d) in an order that completes an indirect proof of the statement: Through a point outside a line, there is at most one line perpendicular to the given line.

Given: Point P not on the line k.

Prove: There is at most one line through P perpendicular to k.

(a) But this contradicts corollary3 of theorem 3-11: In a triangle, there can be at most one right angle or obtuse angle.

(b) Then $∠PAB$and $∠PBA$are right angles, and $\mathrm{Δ}PAB$has two right angles.

(c) Thus, our temporary assumption is false, and there is at most one line through P perpendicular to k.

(d) Assume temporarily that there are two lines through P perpendicular to k at A, B.

Express the following statement in if-then form.

No trapezoids are equiangular.

Write proofs in two-column form.

Given: $\stackrel{¯}{VY}âŠ\yen \stackrel{¯}{YZ}$. Prove: $∠VXZ$ is an obtuse angle.

For the following conditional statement, classify the statement as true or false.

If $d>e$and $f>e$, then $d>f$.