Question

10-17. fill in the proof using the options below. given: \\(\overline{de}\\) is a midsegment of \\(\triangle abc\\) \\(m\angle deb = 2x + 6,\\) \\(m\angle fce = 5x - 3\\) prove: \\(3 = x\\) possible statements: a. \\(2x + 6 = 5x - 3\\) b. \\(3 = 3x\\) c. \\(\overline{fd}\parallel\overline{cb}\\) d. \\(9 = 3x\\) e. \\(\overline{de}\parallel\overline{ac}\\) possible reasons: a. corresponding angles are congruent when lines are parallel b. division property of equality c. given d. subtraction property of equality e. midsegment is equal to half its base statements reasons \\(\overline{de}\\) is a midsegment of \\(\triangle abc\\) 10. given 11. midsegment of a triangle is parallel to the third side \\(m\angle deb = m\angle fce\\) 12. \\(m\angle deb = 2x + 6; m\angle fce = 5x - 3\\) 13. 14. substitution property \\(6 = 3x - 3\\) 15. 16. addition property 17. \\(3 = x\\)

Explanation:

Step 10: Identify the reason for the first statement

The first statement is " $\overline{DE}$ is a midsegment of $\triangle ABC$", and the reason should be "Given" (option C from possible reasons).

Step 11: Determine the statement from the reason

The reason is "Midsegment of a triangle is parallel to the third side", so the statement should be "$\overline{DE} \parallel \overline{AC}$" (option E from possible statements).

Step 12: Find the reason for $m\angle DEB = m\angle FCE$

Since $\overline{DE} \parallel \overline{AC}$, corresponding angles are congruent. So the reason is "Corresponding angles are congruent when lines are parallel" (option A from possible reasons).

Step 13: Identify the reason for $m\angle DEB = 2x + 6; m\angle FCE = 5x - 3$

These angle measures are given in the problem, so the reason is "Given".

Step 14: Determine the statement from the reason (Substitution Property)

We substitute $m\angle DEB$ and $m\angle FCE$ with their expressions, so the statement is "$2x + 6 = 5x - 3$" (option A from possible statements).

Step 15: Find the reason for $6 = 3x - 3$

We get this by subtracting $2x$ from both sides of $2x + 6 = 5x - 3$, so the reason is "Subtraction Property of Equality" (option D from possible reasons).

Step 16: Determine the statement from the reason (Addition Property)

We add 3 to both sides of $6 = 3x - 3$, so the statement is "$9 = 3x$" (option D from possible statements).

Step 17: Find the reason for $3 = x$

We divide both sides of $9 = 3x$ by 3, so the reason is "Division Property of Equality" (option B from possible reasons).

Answer:

1. C (Given)
2. E ($\overline{DE} \parallel \overline{AC}$)
3. A (Corresponding angles are congruent when lines are parallel)
4. Given
5. A ($2x + 6 = 5x - 3$)
6. D (Subtraction Property of Equality)
7. D ($9 = 3x$)
8. B (Division Property of Equality)