Points, lines, segments, and ratios

Key idea

A line segment \(\overline{AB}\) is the straight path connecting two points \(A\) and \(B\). A ray \(\overrightarrow{AB}\) starts at \(A\) and passes through \(B\), extending forever. When a segment is divided in a ratio, the ratio tells you how the total length is split into parts.

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Summary

• A segment connects two endpoints; a ray has one endpoint; a line extends forever both ways.
• For a ratio \(a:b\), the whole has \(a+b\) equal parts.

Angle types, complements, and supplements

Key idea

Angles are measured in degrees. Key angle benchmarks: right \(90^\circ\), straight \(180^\circ\), and a full rotation \(360^\circ\). If two angles add to \(90^\circ\), they are complementary. If two angles add to \(180^\circ\), they are supplementary.

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Summary

• Complement: \(90^\circ-\theta\). Supplement: \(180^\circ-\theta\).
• Reflex angles are between \(180^\circ\) and \(360^\circ\).

Perpendicular lines, adjacent angles, and angle facts

Key idea

When two lines are perpendicular, they intersect to form four right angles. Angles that share a side are adjacent. A straight line forms a straight angle of \(180^\circ\). In a plane, through a point not on a given line, there is exactly one line perpendicular to the given line.

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Summary

• Perpendicular lines form four \(90^\circ\) angles.
• In a plane, the perpendicular to a line through a given point is unique.

Planes and how lines and planes intersect

Key idea

A plane is determined by three non-collinear points. Two distinct planes that intersect do so in a line. If a line meets a plane but is not contained in that plane, the intersection is a single point.

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Summary

• Three non-collinear points determine exactly one plane.
• Line–plane intersection is typically a point; plane–plane intersection is a line.

Parallel lines, skew lines, and angles between planes

Key idea

In 2D, two lines either intersect or are parallel. In 3D, you can also have skew lines: lines that do not intersect and are not parallel because they are not in the same plane. The angle between two skew lines is defined using two intersecting lines that are parallel to the skew lines. The dihedral angle between planes is the angle between the planes measured along their line of intersection.

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Summary

• Skew lines are non-parallel, non-intersecting lines in 3D (not coplanar).
• Angle between skew lines: use parallel intersecting lines.
• Perpendicular planes have a dihedral angle of \(90^\circ\).

Direction vectors, plane normals, and projection onto a plane

Key idea

A plane \(ax+by+cz=d\) has a normal vector \(\mathbf{n}=(a,b,c)\), which is perpendicular to the plane. To project a vector \(\mathbf{v}\) onto the plane, subtract the component of \(\mathbf{v}\) in the normal direction: \[ \mathbf{v}_{\text{plane}}=\mathbf{v}-\operatorname{proj}_{\mathbf{n}}(\mathbf{v}) =\mathbf{v}-\frac{\mathbf{v}\cdot \mathbf{n}}{\mathbf{n}\cdot \mathbf{n}}\,\mathbf{n}. \]

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Summary

• A plane \(ax+by+cz=d\) has normal vector \((a,b,c)\).
• Projection onto a plane removes the component in the normal direction.

Why points, lines, planes, and angles matter

Where these ideas show up

• Geometry proofs: using definitions (parallel, perpendicular) and angle facts (supplementary/complementary).
• 3D geometry: modeling walls, floors, and intersections of surfaces (planes).
• Coordinate geometry: using vectors and dot products to compute angles and projections.
• Real-world measurement: navigation, design, engineering, and architecture use angles and planes constantly.

Worked example: parallel lines

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Final recap

• Two points determine a unique line; three non-collinear points determine a unique plane.
• Line–plane intersection is usually a point; plane–plane intersection is a line.
• Complementary angles sum to \(90^\circ\); supplementary angles sum to \(180^\circ\).
• Skew lines are 3D lines that are not parallel and do not intersect; their angle is defined using parallel intersecting lines.
• Perpendicular planes have a dihedral angle of \(90^\circ\), and vector tools can support 3D angle work.