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 đ›ŧ and 𝑄𝑅𝑉 intersect?

  • A⃖īƒŠīƒŠīƒŠīƒŠīƒŠâƒ—đ‘‰đ‘
  • B⃖īƒŠīƒŠīƒŠâƒ—đ‘†đ‘…
  • C⃖īƒŠīƒŠīƒŠīƒŠâƒ—đ‘„đ‘…
  • D⃖īƒŠīƒŠīƒŠīƒŠīƒŠâƒ—đ‘đ‘„
  • E⃖īƒŠīƒŠīƒŠīƒŠâƒ—đ‘…đ‘‰

Q5:

What is the intersection of the plane through 𝐴đĩđĩ′𝐴′ and the plane through đĩđļđļ′đĩ′?

  • A⃖īƒŠīƒŠīƒŠīƒŠīƒŠâƒ—đĩđĩ′
  • B∅
  • C⃖īƒŠīƒŠīƒŠīƒŠâƒ—𝐴đĩ
  • D⃖īƒŠīƒŠīƒŠīƒŠīƒŠīƒŠīƒŠâƒ—đ´â€˛đĩ′

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 𝑀𝐴đĩ, 𝑀đĩđļ, and 𝑀𝐴đļ?

  • A∅
  • B𝑀đĩ
  • C{𝑀}
  • D{đĩ}

Q8:

What is the intersection between segment đĩđĩ′ and the plane 𝐴đĩđļ?

  • Ađĩđĩ′
  • B{đĩ}
  • C{𝐴}
  • D𝐴đĩ

Q9:

What is the intersection of the planes 𝑀𝐴đĩ and 𝑀đĩđļ?

  • A⃖īƒŠīƒŠīƒŠīƒŠâƒ—𝐴đļ
  • B⃖īƒŠīƒŠīƒŠīƒŠīƒŠâƒ—đ‘€đĩ
  • Cthe plane 𝐴đĩđļ
  • Dthe plane 𝑀𝐴đļ

Q10:

Determine the intersection of plane 𝐷đļđļ′𝐷′ and plane 𝐴đĩđļ𝐷 and plane 𝐴𝐷𝐷′𝐴′.

  • A⃖īƒŠīƒŠīƒŠīƒŠīƒŠâƒ—𝐴𝐷
  • B{𝐷}
  • C⃖īƒŠīƒŠīƒŠīƒŠâƒ—đļ𝐷
  • D{đļ}

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 𝑌 are planes that intersect at line đŋ. 𝐴 is a point on line đŋ and đļ is a point on plane 𝑋. Which of the following are true?

  • Ađļ∉𝑌
  • Bđļ∈𝑌
  • Cđļ⊂𝑌
  • Dđļ∉𝑋

Q14:

𝑋 and 𝑌 are planes that intersect at line đŋ. 𝐴 and đĩ are points on line đŋ and đļ is a point on plane 𝑋. Which of the following statements hold?

  • A⃖īƒŠīƒŠīƒŠīƒŠâƒ—đĩđļ∈𝑌
  • B⃖īƒŠīƒŠīƒŠīƒŠâƒ—đĩđļ∉𝑋
  • C⃖īƒŠīƒŠīƒŠīƒŠâƒ—đĩđļ⊂𝑌
  • D⃖īƒŠīƒŠīƒŠīƒŠâƒ—đĩđļ⊄𝑌

Q15:

𝑋 and 𝑌 are planes that intersect at line đŋ. đĩ is a point on plane 𝑋 and đļ is a point on plane 𝑌. Determine the intersection of plane 𝑌 with plane 𝐴đĩđļ.

  • A⃖īƒŠīƒŠīƒŠīƒŠâƒ—𝐴đĩ
  • B{𝐴}
  • C⃖īƒŠīƒŠīƒŠīƒŠâƒ—𝐴đļ
  • D⃖⃗đŋ

Q16:

Complete: If ⃖⃗đŋ is parallel to the plane 𝑋, then ⃖⃗đŋ∩𝑋=.

  • A𝑋
  • B⃖⃗đŋ
  • C∅

Q17:

Describe what is shown in the figure.

  • AA plane 𝐴đĩđļ intersected by 𝐷𝐸
  • BA plane 𝐴đĩđļ intersected by đļđĩ
  • CA plane đĩđļ𝐷 intersected by 𝐴𝐸

Q18:

There are two parallel planes, 𝐴đĩ𝐷 and 𝐸𝐹đē. A line 𝑙 intersects the planes 𝐴đĩ𝐷 and 𝐸𝐹đē at points 𝐴 and đē respectively. The plane 𝐴đĩ𝐷 contains 𝐴đļ. Which figure matches the description?

  • A
  • B
  • C

Q19:

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

  • A⃖īƒŠīƒŠīƒŠīƒŠâƒ—đ‘‡đ‘‹
  • B⃖īƒŠīƒŠīƒŠīƒŠīƒŠīƒŠâƒ—đ‘‡đ‘Š
  • C⃖īƒŠīƒŠīƒŠīƒŠīƒŠīƒŠâƒ—đ‘‰đ‘Š
  • D⃖īƒŠīƒŠīƒŠīƒŠâƒ—đ‘‹đ‘
  • E⃖īƒŠīƒŠīƒŠīƒŠâƒ—đ‘‰đ‘

Q20:

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

  • A⃖īƒŠīƒŠīƒŠīƒŠīƒŠīƒŠâƒ—đ‘Šđ‘‹
  • B⃖īƒŠīƒŠīƒŠīƒŠīƒŠīƒŠâƒ—đ‘‰đ‘Š
  • C⃖īƒŠīƒŠīƒŠīƒŠâƒ—đ‘‡đ‘‹
  • D⃖īƒŠīƒŠīƒŠīƒŠâƒ—đ‘Œđ‘
  • E⃖īƒŠīƒŠīƒŠīƒŠīƒŠīƒŠâƒ—đ‘Šđ‘Œ

Q21:

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

  • A⃖īƒŠīƒŠīƒŠīƒŠâƒ—đ‘‡đ‘Œ
  • B⃖īƒŠīƒŠīƒŠīƒŠīƒŠīƒŠâƒ—đ‘‹đ‘Š
  • C⃖īƒŠīƒŠīƒŠīƒŠīƒŠīƒŠâƒ—đ‘‡đ‘Š
  • D⃖īƒŠīƒŠīƒŠīƒŠīƒŠīƒŠâƒ—đ‘Šđ‘
  • E⃖īƒŠīƒŠīƒŠīƒŠâƒ—đ‘Œđ‘

Q22:

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

  • A⃖īƒŠīƒŠīƒŠīƒŠâƒ—đ‘‡đ‘Œ
  • B⃖īƒŠīƒŠīƒŠīƒŠâƒ—đ‘‰đ‘
  • C⃖īƒŠīƒŠīƒŠīƒŠīƒŠīƒŠâƒ—đ‘‡đ‘Š
  • D⃖īƒŠīƒŠīƒŠīƒŠâƒ—đ‘Œđ‘
  • E⃖īƒŠīƒŠīƒŠīƒŠīƒŠâƒ—đ‘‰đ‘Œ

Q23:

Which of the following lines contains the point 𝑉?

  • A⃖īƒŠīƒŠīƒŠīƒŠâƒ—đ‘Œđ‘
  • B⃖īƒŠīƒŠīƒŠīƒŠīƒŠīƒŠâƒ—đ‘‹đ‘Š
  • C⃖īƒŠīƒŠīƒŠīƒŠīƒŠīƒŠâƒ—đ‘‡đ‘Š
  • D⃖īƒŠīƒŠīƒŠīƒŠâƒ—đ‘‹đ‘
  • E⃖īƒŠīƒŠīƒŠīƒŠâƒ—đ‘‡đ‘Œ

Q24:

Consider the following figure.

Write another name for the plane đ’Ģ.

  • APlane 𝐷𝐸𝐹
  • BPlane 𝐴đĩ𝐸𝐷
  • CPlane 𝐴đĩđļ
  • DPlane 𝐴đļ𝐹𝐷
  • EPlane đĩđļ𝐹𝐸

How many planes are shown in the figure?

Which two planes intersect in 𝐴𝐷?

  • APlane 𝐴đļ𝐹𝐷 and plane 𝐴đĩ𝐸𝐷
  • BPlane 𝐴đļ𝐹𝐷 and plane 𝐴đĩđļ
  • CPlane 𝐴đļ𝐹𝐷 and plane 𝐷𝐸𝐹
  • DPlane 𝐴đĩ𝐸𝐷 and plane 𝐷𝐸𝐹
  • EPlane 𝐴đĩđļ and plane 𝐴đĩ𝐸𝐷

Are planes đ’Ģ and 𝐷𝐸𝐹 parallel or intersecting?

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

Q25:

𝑋 and 𝑌 are planes which intersect at line đŋ. Which of the following holds?

  • A𝐴∈𝑋
  • B𝐴⊂𝑋
  • C𝐴∉𝑋
  • D𝐴⊄𝑋

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