Question

literacy and vocabulary
proving triangles similar
the column on the left shows the steps used to solve a proportion.
use the column on the left to answer each question in the column on the right.
problem sas - theorem
use a proportion to find x.
diagram of two triangles

1. look at the diagram. what measure are you trying to find?

verify that the triangles are similar.
\\(\frac{4}{8} = \frac{8}{16}\\)
\\(\angle a \cong \angle d\\)
\\(\triangle abc \sim \triangle def\\) by sas - theorem.

1. what is the sas - theorem?

write a proportion.
\\(\frac{4}{8} = \frac{7}{x}\\)

1. what is a proportion?

solve the proportion.
multiply both sides by x.
\\(\frac{4}{8} \cdot x = \frac{7}{x} \cdot x\\)
simplify.
\\(\frac{1}{2}x = 7\\)

1. what does it mean to solve the proportion?

multiply both sides by 2.
\\(x = 14\\)

1. why do you multiply both sides by 2? what do you know when you know the value of x?

Explanation:

1. Identify the unknown side in the diagram.
2. Define the SAS Similarity Theorem for triangles.
3. Explain the mathematical definition of a proportion.
4. Describe the goal of solving a proportion.
5. Justify the algebraic step and state the result.

Answer:

1. The length of side $\overline{EF}$ (or the value of $x$)
2. The SAS (Side-Angle-Side) Similarity Theorem states that if two sides of one triangle are proportional to two sides of another triangle, and the included angles are congruent, then the two triangles are similar.
3. A proportion is an equation that states that two ratios are equal, e.g., $\frac{a}{b}=\frac{c}{d}$.
4. To solve a proportion means to find the value of the unknown variable that makes the equation (the equal ratios) true.
5. Multiplying both sides by 2 isolates the variable $x$ (since $\frac{1}{2}x \times 2 = x$). When we know the value of $x$, we know the length of the corresponding side in the similar triangle, $\overline{EF}$.