Question

1. complete the following proof. reasons to be chosen from the bank provided on the last page.

given: cd = fg
prove: cd = fg
f is the mid - point of eg
statements\treasons

1. cd = fg\t1. given
2. cd = fg\t2. given
3. ef = fg\t3.
4. f is the mid - point of eg\t4.
5. provide a counterexample for each statement.

if a figure has 4 sides then it is a rectangle.
if x² + x = 30, then x = 5.
proof reason word bank
definition of complementary angles\tdefinition of bisector\tdistributive property
given\tdefinition of right angle\tcongruent supplements theorem
right angles congruence theorem\tdefinition of congruent segments\tcongruent complements theorem
division property of equality\tangle addition postulate\tvertical angles congruence theorem
definition of supplementary angles\tsubstitution\tsegment addition postulate
definition of midpoint\taddition property of equality\tsubtraction property of equality
combine like terms\tmultiplication property of equality\ttransitive property

Explanation:

Step1: Analyze the first proof statement

We are given $\overline{CD}\cong\overline{EF}$ and $\overline{CD}\cong\overline{FG}$. By the transitive property of congruence, if $a = b$ and $a = c$, then $b = c$. Here, since $\overline{CD}$ is congruent to both $\overline{EF}$ and $\overline{FG}$, we have $\overline{EF}\cong\overline{FG}$. The reason for the step $\overline{EF}=\overline{FG}$ (assuming equality in length sense) is the transitive property.

Step2: Analyze the counter - example for the first statement

A figure with 4 sides is not necessarily a rectangle. A counter - example is a rhombus. A rhombus has 4 sides but its angles are not all right angles like in a rectangle.

Step3: Analyze the counter - example for the second statement

We have the equation $x^{2}+x = 30$. Rearranging it gives $x^{2}+x - 30=0$. Factoring, we get $(x + 6)(x - 5)=0$. So the solutions are $x=-6$ or $x = 5$. So $x = 6$ is not a solution, and a counter - example to the statement "If $x^{2}+x = 30$, then $x = 5$" is $x=-6$ since when $x=-6$, $(-6)^{2}+(-6)=36 - 6=30$.

Answer:

1. Reason for $\overline{EF}=\overline{FG}$: Transitive Property
2. Counter - example for "If a figure has 4 sides then it is a rectangle": Rhombus
3. Counter - example for "If $x^{2}+x = 30$, then $x = 5$": $x=-6$