Worksheet: Points, Lines, and Planes in Space

In this worksheet, we will practice identifying and modeling geometric concepts like points, lines, and planes in space along with their properties.

Q1:

State whether the following is true or false: Two coplanar lines perpendicular to the same line will always intersect each other.

  • Afalse
  • Btrue

Q2:

Complete the following: Two skew lines are two lines that .

  • Alie in the same plane and do not have a common point
  • Bintersect but do not lie in the same plane
  • Care parallel
  • Ddo not lie in the same plane and do not intersect
  • Elie in the same plane and have a common point

Q3:

How many planes can pass through three non-collinear points?

Q4:

In what line do planes 𝛼 and 𝑄𝑅𝑉 intersect?

  • A⃖⃗𝑉𝑁
  • B⃖⃗𝑆𝑅
  • C⃖⃗𝑄𝑅
  • D⃖⃗𝑁𝑄
  • E⃖⃗𝑅𝑉

Q5:

What is the intersection of the plane through 𝐴𝐡𝐡′𝐴′ and the plane through 𝐡𝐢𝐢′𝐡′?

  • A⃖⃗𝐡𝐡′
  • Bβˆ…
  • C⃖⃗𝐴𝐡
  • D⃖⃗𝐴′𝐡′

Q6:

Complete the following: Two intersecting lines are two lines that .

  • Ado not lie in the same plane
  • Blie in the same plane and do not have a common point
  • Clie in the same plane and have a common point
  • Dare perpendicular
  • Eare parallel

Q7:

What is the intersection of planes 𝑀𝐴𝐡, 𝑀𝐡𝐢, and 𝑀𝐴𝐢?

  • Aβˆ…
  • B𝑀𝐡
  • C{𝑀}
  • D{𝐡}

Q8:

What is the intersection between segment 𝐡𝐡′ and the plane 𝐴𝐡𝐢?

  • A𝐡𝐡′
  • B{𝐡}
  • C{𝐴}
  • D𝐴𝐡

Q9:

What is the intersection of the planes 𝑀𝐴𝐡 and 𝑀𝐡𝐢?

  • A⃖⃗𝐴𝐢
  • B⃖⃗𝑀𝐡
  • Cthe plane 𝐴𝐡𝐢
  • Dthe plane 𝑀𝐴𝐢

Q10:

Determine the intersection of plane 𝐷𝐢𝐢′𝐷′ and plane 𝐴𝐡𝐢𝐷 and plane 𝐴𝐷𝐷′𝐴′.

  • A⃖⃗𝐴𝐷
  • B{𝐷}
  • C⃖⃗𝐢𝐷
  • D{𝐢}

Q11:

Observe the following figure and choose.

  • AThe straight line is contained within the plane.
  • BThe straight line is intersected with the plane.
  • CThe straight line is parallel to the plane.

Q12:

Observe the following figure and choose.

  • AThe straight line is parallel to the plane.
  • BThe straight line is contained within the plane.
  • CThe straight line is intersected with the plane.

Q13:

𝑋 and π‘Œ are planes that intersect at line 𝐿. 𝐴 is a point on line 𝐿 and 𝐢 is a point on plane 𝑋. Which of the following are true?

  • AπΆβˆ‰π‘Œ
  • BπΆβˆˆπ‘Œ
  • CπΆβŠ‚π‘Œ
  • DπΆβˆ‰π‘‹

Q14:

𝑋 and π‘Œ are planes that intersect at a line 𝐿. 𝐴 and 𝐡 are points on line 𝐿 and 𝐢 is a point on plane 𝑋. Which of the following statements is true?

  • Aβƒ–οƒ©οƒ©οƒ©οƒ©βƒ—π΅πΆβˆˆπ‘Œ
  • Bβƒ–οƒ©οƒ©οƒ©οƒ©βƒ—π΅πΆβˆ‰π‘‹
  • Cβƒ–οƒ©οƒ©οƒ©οƒ©βƒ—π΅πΆβŠ‚π‘Œ
  • Dβƒ–οƒ©οƒ©οƒ©οƒ©βƒ—π΅πΆβŠ„π‘Œ

Q15:

𝑋 and π‘Œ are planes that intersect at line 𝐿. 𝐡 is a point on plane 𝑋 and 𝐢 is a point on plane π‘Œ. Determine the intersection of plane π‘Œ with plane 𝐴𝐡𝐢.

  • A⃖⃗𝐴𝐡
  • B{𝐴}
  • C⃖⃗𝐴𝐢
  • D⃖⃗𝐿

Q16:

Complete: If ⃖⃗𝐿 is parallel to the plane 𝑋, then βƒ–βƒ—πΏβˆ©π‘‹=.

  • A𝑋
  • B⃖⃗𝐿
  • Cβˆ…

Q17:

Describe what is shown in the figure.

  • AA plane 𝐴𝐡𝐢 intersected by 𝐷𝐸
  • BA plane 𝐴𝐡𝐢 intersected by 𝐢𝐡
  • CA plane 𝐡𝐢𝐷 intersected by 𝐴𝐸

Q18:

There are two parallel planes, 𝐴𝐡𝐷 and 𝐸𝐹𝐺. A line 𝑙 intersects the planes 𝐴𝐡𝐷 and 𝐸𝐹𝐺 at points 𝐴 and 𝐺 respectively. The plane 𝐴𝐡𝐷 contains 𝐴𝐢. Which figure matches the description?

  • A
  • B
  • C

Q19:

Which of the following is a line containing the point π‘Œ?

  • A⃖⃗𝑇𝑋
  • Bβƒ–οƒ©οƒ©οƒ©οƒ©οƒ©οƒ©βƒ—π‘‡π‘Š
  • Cβƒ–οƒ©οƒ©οƒ©οƒ©οƒ©οƒ©βƒ—π‘‰π‘Š
  • D⃖⃗𝑋𝑍
  • E⃖⃗𝑉𝑍

Q20:

Which of the following is a line containing the point 𝑇?

  • Aβƒ–οƒ©οƒ©οƒ©οƒ©οƒ©οƒ©βƒ—π‘Šπ‘‹
  • Bβƒ–οƒ©οƒ©οƒ©οƒ©οƒ©οƒ©βƒ—π‘‰π‘Š
  • C⃖⃗𝑇𝑋
  • Dβƒ–οƒ©οƒ©οƒ©οƒ©βƒ—π‘Œπ‘
  • Eβƒ–οƒ©οƒ©οƒ©οƒ©οƒ©οƒ©βƒ—π‘Šπ‘Œ

Q21:

Which of the following is a line containing the point 𝑋?

  • Aβƒ–οƒ©οƒ©οƒ©οƒ©βƒ—π‘‡π‘Œ
  • Bβƒ–οƒ©οƒ©οƒ©οƒ©οƒ©οƒ©βƒ—π‘‹π‘Š
  • Cβƒ–οƒ©οƒ©οƒ©οƒ©οƒ©οƒ©βƒ—π‘‡π‘Š
  • Dβƒ–οƒ©οƒ©οƒ©οƒ©οƒ©οƒ©βƒ—π‘Šπ‘
  • Eβƒ–οƒ©οƒ©οƒ©οƒ©βƒ—π‘Œπ‘

Q22:

Which of the following is a line containing the point π‘Š?

  • Aβƒ–οƒ©οƒ©οƒ©οƒ©βƒ—π‘‡π‘Œ
  • B⃖⃗𝑉𝑍
  • Cβƒ–οƒ©οƒ©οƒ©οƒ©οƒ©οƒ©βƒ—π‘‡π‘Š
  • Dβƒ–οƒ©οƒ©οƒ©οƒ©βƒ—π‘Œπ‘
  • Eβƒ–οƒ©οƒ©οƒ©οƒ©οƒ©βƒ—π‘‰π‘Œ

Q23:

Which of the following lines contains the point 𝑉?

  • Aβƒ–οƒ©οƒ©οƒ©οƒ©βƒ—π‘Œπ‘
  • Bβƒ–οƒ©οƒ©οƒ©οƒ©οƒ©οƒ©βƒ—π‘‹π‘Š
  • Cβƒ–οƒ©οƒ©οƒ©οƒ©οƒ©οƒ©βƒ—π‘‡π‘Š
  • D⃖⃗𝑋𝑍
  • Eβƒ–οƒ©οƒ©οƒ©οƒ©βƒ—π‘‡π‘Œ

Q24:

Consider the following figure.

Write another name for the plane 𝒫.

  • APlane 𝐷𝐸𝐹
  • BPlane 𝐴𝐡𝐸𝐷
  • CPlane 𝐴𝐡𝐢
  • DPlane 𝐴𝐢𝐹𝐷
  • EPlane 𝐡𝐢𝐹𝐸

How many planes are shown in the figure?

Which two planes intersect in 𝐴𝐷?

  • APlane 𝐴𝐢𝐹𝐷 and plane 𝐴𝐡𝐸𝐷
  • BPlane 𝐴𝐢𝐹𝐷 and plane 𝐴𝐡𝐢
  • CPlane 𝐴𝐢𝐹𝐷 and plane 𝐷𝐸𝐹
  • DPlane 𝐴𝐡𝐸𝐷 and plane 𝐷𝐸𝐹
  • EPlane 𝐴𝐡𝐢 and plane 𝐴𝐡𝐸𝐷

Are planes 𝒫 and 𝐷𝐸𝐹 parallel or intersecting?

  • AIt cannot be determined without more information.
  • BThey are intersecting.
  • CThey are parallel.

Q25:

𝑋 and π‘Œ are planes which intersect at line 𝐿. Which of the following holds?

  • Aπ΄βˆˆπ‘‹
  • Bπ΄βŠ‚π‘‹
  • Cπ΄βˆ‰π‘‹
  • Dπ΄βŠ„π‘‹

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