Question

examine the figure below. match each reason with the proper statement. knowing that $overline{ab}congoverline{xy}$ and $overline{ac}congoverline{xz}$, prove that $overline{bc}congoverline{yz}$.
proof: 2 - column
given: $overline{ab}congoverline{xy}$ and $overline{ac}congoverline{xz}$
prove: $overline{bc}congoverline{yz}$
statements

1. $overline{ac}congoverline{xz}$ and $overline{ab}congoverline{xy}$ choose
2. $ac = xz$ and $ab = xy$ choose
3. $ac=ab + bc$ and $xz=xy + yz$ choose
4. $ab + bc=xy + yz$ choose
5. $ab + bc=ab + yz$ choose
6. $ab - ab+bc=ab - ab + yz$ choose
7. $bc = yz$ choose
8. $overline{bc}congoverline{yz}$ choose

Explanation:

Step1: Given information

Given that $\overline{AB}\cong\overline{XY}$ and $\overline{AC}\cong\overline{XZ}$, which means $AB = XY$ and $AC=XZ$ by the definition of congruent segments.

1. Reason: Given
2. Reason: Definition of congruent segments (If two segments are congruent, their lengths are equal)

Step2: Segment - addition postulate

By the segment - addition postulate, for collinear points $A$, $B$, $C$ we have $AC = AB + BC$ and for collinear points $X$, $Y$, $Z$ we have $XZ=XY + YZ$.

1. Reason: Segment - addition postulate

Step3: Substitution

Since $AC = XZ$ and $AC = AB + BC$ and $XZ=XY + YZ$, we can substitute to get $AB + BC=XY + YZ$.

1. Reason: Substitution property of equality

Step4: Substitute $AB$ for $XY$

Because $AB = XY$, we substitute $XY$ with $AB$ in the equation $AB + BC=XY + YZ$ to get $AB + BC=AB + YZ$.

1. Reason: Substitution property of equality

Step5: Subtract $AB$ from both sides

Using the subtraction property of equality, if we subtract $AB$ from both sides of the equation $AB + BC=AB + YZ$, we have $AB - AB+BC=AB - AB + YZ$.

1. Reason: Subtraction property of equality

Step6: Simplify

Simplifying $AB - AB+BC=AB - AB + YZ$ gives $BC = YZ$.

1. Reason: Simplification

Step7: Definition of congruent segments

Since $BC = YZ$, by the definition of congruent segments, $\overline{BC}\cong\overline{YZ}$.

1. Reason: Definition of congruent segments

Answer:

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