Competition problem — Geometry — Maths Olympiad Prep

Library / /53 of 520

Geometry Difficulty 2.6 Junior Find the answer

Let $m$ , $n$ , and $l$ be three different lines, and $\alpha$ , $\beta$ , and $\gamma$ be three different planes. Then, the correct proposition is

Pick one

Solution

For option A, if $\alpha \perp \gamma$ and $\beta \perp \gamma$ , then it could be that $\alpha \parallel \beta$ or $\alpha$ intersects $\beta$ . Therefore, option A is incorrect.

For option B, if $m \parallel \beta$ and $l \perp m$ , then it could be that $l \perp \beta$ , $l \parallel \beta$ , or $l$ is contained in $\beta$ . Therefore, option B is incorrect.

For option C, if $m \parallel \alpha$ and $n \parallel \alpha$ , then it could be that $m \parallel n$ , $m$ intersects $n$ , or $m$ and $n$ are skew lines. Therefore, option C is incorrect.

For option D, if $m \perp \alpha$ and $n \perp \alpha$ , then $m \parallel n$ . Therefore, option D is correct.

Hence, the correct choice is $\boxed{\text{D}}$ .

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.