Track / Stage 3 / 59 of 260 #59 of 1964

Problem 59

AMC 10/12, early questions

Geometry Difficulty 3.2 Find the answer

Let $m$ and $n$ be two different lines, and $\alpha$ , $\beta$ , and $\gamma$ be three different planes. Given the following four propositions:

$①$ If $m \subset \alpha$ and $n \parallel \alpha$ , then $m \parallel n$ ;

$②$ If $\alpha \parallel \beta$ , $\beta \parallel \gamma$ , and $m \perp \alpha$ , then $m \perp \gamma$ ;

$③$ If $\alpha \cap \beta = n$ , $m \parallel n$ , and $m \parallel \alpha$ , then $m \parallel \beta$ ;

$④$ If $m \parallel \alpha$ , $n \parallel \beta$ , and $m \parallel n$ , then $\alpha \parallel \beta$ .

Among these, the true proposition(s) is/are __________ (Fill in the correct proposition number(s)).

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

Analysis

This question examines the spatial relationship between lines and planes. By referring to the relevant definitions, we can judge each case accordingly.

Solution

If $m \subset \alpha$ and $n \parallel \alpha$ , then $m \parallel n$ or $m$ and $n$ are skew lines, hence proposition $①$ is incorrect;

If $\alpha \cap \beta = n$ , $m \parallel n$ , and $m \parallel \alpha$ , then $m \parallel \beta$ or $m \subset \beta$ , hence proposition $③$ is incorrect;

If $m \parallel \alpha$ , $n \parallel \beta$ , and $m \parallel n$ , then $\alpha \parallel \beta$ or $\alpha$ intersects $\beta$ , hence proposition $④$ is incorrect.

Therefore, the answer is $\boxed{②}$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.