Question

given: (m\angle b = 30^\circ), (m\angle p = 30^\circ), (bc = 3.5\text{ cm}), (ab = 3\text{ cm}), (qp = 6\text{ cm}), (pr = 7\text{ cm})
prove: (\delta abc \sim \delta qpr)

statement | reason
--- | ---
(m\angle b = 30^\circ), (m\angle p = 30^\circ), (bc = 3.5\text{ cm}), (ab = 3\text{ cm}), (qp = 6\text{ cm}), (pr = 7\text{ cm}) | given
(\angle b \cong \angle p) | definition of congruent angles
(\frac{ab}{qp} = \frac{3}{6} = 1/2) | common ratio property
(\frac{bc}{pr} = \frac{3.5}{7} = 1/2) | dropdown
(\frac{ab}{qp} = \frac{bc}{pr}) |
(\delta abc \sim \delta qpr) |

Explanation:

Identify the missing reason in the proof

We need to find the reason for the fourth statement in the two-column proof.
The statement is:

$$ \frac{BC}{PR} = \frac{3.5}{7} = \frac{1}{2} $$

This statement calculates the ratio of the lengths of the second pair of corresponding sides, \(BC\) and \(PR\), and simplifies it to \(\frac{1}{2}\).

Analyze the preceding steps and options

The third statement is:

$$ \frac{AB}{QP} = \frac{3}{6} = \frac{1}{2} $$

Its reason is given as "Common Ratio Property" (or a similar property establishing the ratio of corresponding sides).
The fourth statement performs the exact same mathematical operation of finding a simplified ratio for another pair of sides:

$$ \frac{BC}{PR} = \frac{3.5}{7} = \frac{1}{2} $$

Therefore, the reason for this step must be the same as the reason for the third step.

Select the correct reason from the dropdown

Looking at the dropdown options:

• Common Ratio Property
• Reflexive Property
• Division Property
• Multiplication Property

The correct reason that matches the calculation of the side ratio is "Common Ratio Property".

Answer:

The missing reason for the statement \(\frac{BC}{PR} = \frac{3.5}{7} = 1/2\) is <blank>Common Ratio Property</blank>.