Question

prove: $overline{ac}congoverline{bd}$ statements reasons 1. $overline{ab}congoverline{cd}$ 1. 2. $ab = cd$ 2. 3. $ac + cd = ad$ 3. 4. $ab + bd = ad$ 4. 5. $cd + bd = ad$ 5. 6. $ac + cd = cd + bd$ 6. 7. $ac = bd$ 7. 8. $overline{ac}congoverline{bd}$ 8.

Explanation:

Step1: Given

Given that $\overline{AB}\cong\overline{CD}$

Step2: Definition of congruent segments

If $\overline{AB}\cong\overline{CD}$, then $AB = CD$ (by the definition of congruent - length segments)

Step3: Segment - addition postulate

$AC + CD=AD$ (by the segment - addition postulate which states that if C is between A and D, then $AC + CD = AD$)

Step4: Segment - addition postulate

$AB + BD=AD$ (by the segment - addition postulate which states that if B is between A and D, then $AB + BD = AD$)

Step5: Substitution

Since $AB = CD$ (from step 2), we can substitute $AB$ with $CD$ in the equation $AB + BD=AD$, getting $CD + BD=AD$

Step6: Transitive property of equality

Since $AC + CD=AD$ and $CD + BD=AD$, then $AC + CD=CD + BD$ (by the transitive property of equality: if $a = c$ and $b = c$, then $a = b$)

Step7: Subtraction property of equality

Subtract $CD$ from both sides of the equation $AC + CD=CD + BD$. We get $AC=BD$ (by the subtraction property of equality: if $a + c=b + c$, then $a = b$)

Step8: Definition of congruent segments

Since $AC = BD$, then $\overline{AC}\cong\overline{BD}$ (by the definition of congruent - length segments)

Answer:

1. Given
2. Definition of congruent segments
3. Segment - addition postulate
4. Segment - addition postulate
5. Substitution
6. Transitive property of equality
7. Subtraction property of equality
8. Definition of congruent segments