Question

suppose that the two parts of the ladder, ac and bd, are the same length (that is, ac = bd).
complete the following two statements.

1. because a - b - c, ac = ?.
2. because b - c - d, bd = ?.
3. why is ab + bc = bc + cd?
4. why is ab = cd?

Explanation:

Step1: Use segment - addition postulate for 38

Since points A, B, and C are collinear with A - B - C, by the segment - addition postulate, $AC=AB + BC$.

Step2: Use segment - addition postulate for 39

Since points B, C, and D are collinear with B - C - D, by the segment - addition postulate, $BD=BC + CD$.

Step3: Analyze the equation in 40

We know that $AC = BD$ (given). From step1, $AC=AB + BC$ and from step2, $BD=BC + CD$. So, $AB + BC=BC + CD$ because $AC = BD$.

Step4: Solve for 41

Starting with $AB + BC=BC + CD$, subtract $BC$ from both sides of the equation. Using the subtraction property of equality, $(AB + BC)-BC=(BC + CD)-BC$. Simplifying the left - hand side gives $AB+(BC - BC)=AB$, and simplifying the right - hand side gives $(BC - BC)+CD = CD$. So, $AB = CD$.

Answer:

1. $AC=AB + BC$
2. $BD=BC + CD$
3. Because $AC = BD$, and $AC=AB + BC$, $BD=BC + CD$.
4. By the subtraction property of equality, subtracting $BC$ from both sides of $AB + BC=BC + CD$.