Question

pretest: polygons\
statement\treason\

1. \tdraw $\overline{bd} \perp \overline{ac}$.\tconstruction\
2. \\t$\angle abc \cong \angle bdc$\tangles with the same measure are congruent.\
3. \\t$\angle bca \cong \angle dcb$\treflexive property of congruence\
4. \\t$\delta abc \sim \delta bdc$\taa criterion for similarity\
5. \\t$\frac{bc}{dc} = \frac{ac}{bc}$\tcorresponding sides of similar triangles are proportional.\
6. \\t$bc^2 = ac \times dc$\tcross multiplication\
7. \\t$\angle abc \cong \angle adb$\tangles with the same measure are congruent.\
8. \\t$\angle bac \cong \angle dab$\treflexive property of congruence\
9. \\t$\delta abc \sim \delta adb$\taa criterion for similarity\
10. \\t$\frac{ab}{ad} = \frac{ac}{ab}$\tcorresponding sides of similar triangles are proportional.\
11. \\t\\t\
12. \\t\\t\
13. \\t\\t\
14. \\t\\t\
15. \\t\\t\

\
$ab^2 + bc^2 = ac^2$\
reason: multiplication\
$ab^2 + bc^2 = ac \times ac$\
reason: segment addition\
$ab^2 = ac \times ad$\
reason: cross multiplication\
$ab^2 + bc^2 = ac(ad + dc)$\
reason: distributive property\
$ab^2 + bc^2 = ac \times ad + ac \times dc$\
reason: addition

Explanation:

Step1: Cross multiply Step10

Starting from $\frac{AB}{AD} = \frac{AC}{AB}$, cross multiply to get the next statement.
$AB^2 = AC \times AD$

Step2: Add Step6 and Step11

Add the equations from Step6 ($BC^2 = AC \times DC$) and Step11.
$AB^2 + BC^2 = AC \times AD + AC \times DC$

Step3: Factor right-hand side

Use distributive property on the right-hand side.
$AB^2 + BC^2 = AC(AD + DC)$

Step4: Replace $AD+DC$ with $AC$

Use segment addition: $AD + DC = AC$.
$AB^2 + BC^2 = AC \times AC$

Step5: Simplify right-hand side

Multiply $AC \times AC$ to get $AC^2$.
$AB^2 + BC^2 = AC^2$

Answer:

1. $AB^2 = AC \times AD$

Reason: cross multiplication

1. $AB^2 + BC^2 = AC \times AD + AC \times DC$

Reason: addition

1. $AB^2 + BC^2 = AC(AD + DC)$

Reason: Distributive Property

1. $AB^2 + BC^2 = AC \times AC$

Reason: segment addition

1. $AB^2 + BC^2 = AC^2$

Reason: multiplication