Unit 3 is about what happens when lines run side by side or cross at right angles, studied through angles, 3D space, and slopes on the coordinate plane. By the end, you should be able to find any angle formed by a transversal, prove lines parallel or perpendicular, classify lines in 3D, and write equations of parallel and perpendicular lines.
Read the big ideas first, then work through each topic note and answer its checkpoint before you look at the answer.
This unit takes the angle facts from earlier geometry and points them at one question. Are these two lines parallel, perpendicular, or neither? You'll answer it with angle relationships, with 3D reasoning, and with algebra.
Honors Geometry Unit 3 covers parallel and perpendicular lines. It starts with the angle pairs formed when a transversal cuts two lines (corresponding, alternate interior, alternate exterior, and same-side interior) and what those angles do when the lines are parallel. Next it uses the converses of those theorems to prove lines parallel or perpendicular in two-column and paragraph proofs. It then moves into 3D, where lines can be skew and lines can be parallel to, inside, or perpendicular to planes. It finishes on the coordinate plane, where distinct nonvertical parallel lines have equal slopes, and two vertical lines are parallel. Perpendicular lines have slopes whose product is when both slopes are defined; a horizontal line and a vertical line are also perpendicular. You write equations for both relationships.
When a transversal crosses two parallel lines, corresponding, alternate interior, and alternate exterior angles are congruent, and same-side interior angles add to . That means one known angle tells you all eight. Every angle in the figure either matches it or is its supplement.
If the angle relationships hold, the lines must be parallel. In proofs, citing the converse instead of the original theorem is how you show you're concluding parallel lines rather than assuming them. Perpendicular proofs work similarly, since one right angle at an intersection gives you all four.
On the coordinate plane, distinct nonvertical parallel lines share a slope, and two vertical lines are parallel. Perpendicular lines have negative reciprocal slopes when both slopes are defined; horizontal and vertical lines are also perpendicular. You can check two equations in seconds and write a new line through any point. The same checks let you prove a shape is a parallelogram, rectangle, or right triangle.
Two lines can look parallel and still meet far off the page, so geometry needs evidence you can measure. Angle pairs give that evidence in a diagram, and slopes give it on a grid. In 3D you also have to check that the lines share a plane. The whole unit is about choosing the right evidence and stating it correctly.
When a transversal cuts two lines, it makes eight angles, four at each intersection.
Topic 3.1 went from parallel lines to angle facts.
In a single plane, two distinct lines either intersect or are parallel.
Slope turns parallel and perpendicular into a number check.
Open this guide for a closer review of the topic.
When a transversal cuts two lines, it makes eight angles, four at each intersection. Naming the pairs comes down to two questions. Are the angles on the same side of the transversal or on opposite (alternate) sides? Are they between the two lines (interior) or outside them (exterior)? Corresponding angles sit in the same position at each intersection, like both upper-left.
If the two lines are parallel, corresponding, alternate interior, and alternate exterior angles are congruent. Same-side interior angles are supplementary, meaning they add to . Combine those facts with vertical angles and linear pairs, and every angle is either equal to a given angle or supplementary to it.
Worked example. Lines are cut by transversal . Two alternate interior angles measure and . Since they're congruent, set . That gives , so . Each angle is , and every other angle in the figure is either or .
These angle relationships hold when the lines are parallel. If the problem doesn't state or mark the lines as parallel, look for other given facts that prove they are parallel instead of assuming it.
Check yourself. Two parallel lines are cut by a transversal, and one same-side interior angle measures . What does the other one measure? (Answer: , because same-side interior angles are supplementary and .)
| Angle pair | Where the angles sit | Relationship when lines are parallel |
|---|---|---|
| Corresponding | Same position at each intersection | Congruent |
| Alternate interior | Opposite sides of transversal, between the lines | Congruent |
| Alternate exterior | Opposite sides of transversal, outside the lines | Congruent |
| Same-side interior | Same side of transversal, between the lines | Supplementary (sum to 180°) |
Topic 3.1 went from parallel lines to angle facts. Proving lines parallel goes the other way and uses the converses. If corresponding angles are congruent, the lines are parallel. The same is true if alternate interior angles are congruent, if alternate exterior angles are congruent, or if same-side interior angles are supplementary. You only need one of these, so pick the one that matches your given information.
In a two-column proof, the reason you cite shows which direction you're going. If congruent corresponding angles are the evidence for , cite the Converse of the Corresponding Angles Postulate. Citing the original postulate would assume the lines are parallel rather than prove it.
For perpendicular lines, showing one right angle at the intersection is enough, because vertical angles and linear pairs make the other three too. If a transversal is perpendicular to one of two parallel lines, it's perpendicular to the other. In a plane, two lines perpendicular to the same line are parallel to each other.
Worked example. Corresponding angles measure and . What value of makes the lines parallel? Set the angles equal: . That gives and . At that value, both angles equal , so the Corresponding Angles Converse proves the lines parallel.
Compass constructions use the same logic. Copying an angle to create congruent corresponding angles builds a parallel line through a point.
Check yourself. Same-side interior angles measure and . What value of makes the lines parallel? (Answer: The angles must be supplementary, so , which gives and .)
In a single plane, two distinct lines either intersect or are parallel. Two lines can also coincide, meaning they share every point. In 3D there's a third option. Skew lines don't intersect and aren't parallel, because they don't lie in the same plane. Picture a classroom as a box. The vertical edge where the front wall meets the left wall and the edge where the ceiling meets the back wall never touch, and they point in different directions, so they're skew.
To classify two lines, first ask whether they're coplanar. If they aren't, they're skew. If they are, look for shared points. No shared point means parallel, exactly one means intersecting, and more than one means the lines coincide.
A line and a plane relate in one of three ways. The line can lie in the plane, be parallel to it and never touch it, or cross it at exactly one point. Two different planes are either parallel or intersect in a line.
A line is perpendicular to a plane when it's perpendicular to every line in the plane that passes through the point where it hits, called the foot. You don't have to check every line. If a line is perpendicular to two different lines in the plane that pass through its foot, it's perpendicular to the plane. Two lines perpendicular to the same plane are parallel to each other, which is why the vertical edges of a box are parallel.
Check yourself. Two lines never intersect. Can you conclude they're parallel? (Answer: Only if you know they're coplanar. In 3D, two lines that never meet could be skew.)
| Relationship | Same plane? | Shared points | Classroom example |
|---|---|---|---|
| Intersecting | Yes | Exactly one | Two floor edges meeting at a corner |
| Parallel | Yes | None | The left and right vertical edges of the whiteboard |
| Skew | No | None | The front-left vertical edge and the back ceiling edge |
Slope turns parallel and perpendicular into a number check. Find slope with , or read it from . For standard form with , solve for to get . If , the line is vertical and its slope is undefined.
Nonvertical parallel lines have equal slopes. When both slopes are defined, perpendicular lines have negative reciprocal slopes, so their product is . To get the negative reciprocal, flip the fraction and change the sign. A slope of becomes , and a slope of becomes . Horizontal and vertical lines are the exception. A horizontal line () is perpendicular to a vertical line (undefined slope), even though you can't multiply those slopes.
Worked example. Write the line perpendicular to through . Solving for gives , so the slope is . The perpendicular slope is . Point-slope form gives , so . Plugging in gives , so the point checks out.
For a parallel line, keep the original slope and follow the same steps. These checks also prove shapes on the coordinate plane. A quadrilateral with both pairs of opposite sides parallel is a parallelogram. If that parallelogram also has a right angle, it is a rectangle. Slopes can establish these relationships, but vertical sides need separate checks because their slopes are undefined.
Check yourself. Are and parallel, perpendicular, or neither? (Answer: The second line is , and , so they're perpendicular.)
| Term | Definition |
|---|---|
| Transversal | A line that intersects two or more other lines at different points. |
| Parallel lines | Coplanar lines that never intersect, which on the coordinate plane means they have equal slopes or are both vertical. |
| Perpendicular lines | Lines that intersect to form right angles, which on the coordinate plane means their slopes multiply to $-1$ or one is horizontal and the other vertical. |
| Corresponding angles | Angles in the same relative position at each intersection of a transversal with two lines. |
| Alternate interior angles | Angles on opposite sides of a transversal and between the two lines it crosses. |
| Same-side interior angles | Angles on the same side of a transversal and between the two lines, which are supplementary when the lines are parallel. |
| Converse | An if-then statement with its hypothesis and conclusion swapped, used in this unit to prove lines parallel from angle facts. |
| Skew lines | Noncoplanar lines that never intersect and are not parallel. |
| Negative reciprocal | For a nonzero slope, the slope of a perpendicular line, found by flipping the original slope and changing its sign. A line with slope $0$ is perpendicular to a vertical line, whose slope is undefined. |
Most angle pairs in this unit are congruent, so it's easy to set every pair equal out of habit. When the lines are parallel, same-side interior angles are supplementary instead. Before writing an equation, name the pair, then decide whether it calls for or .
If your proof uses congruent alternate interior angles to conclude that lines are parallel, cite the Alternate Interior Angles Converse. The original theorem starts by assuming the lines are parallel, so using it there is circular. Ask yourself whether parallel is your starting fact or your conclusion.
A perpendicular slope needs both a flip and a sign change. Turning into or into gives a line that isn't perpendicular. Check by multiplying the two slopes, which should give , since .
That rule only works in a plane. In 3D, two lines that never intersect can be skew, like a wall edge and a ceiling edge on opposite sides of a room. Confirm the lines are coplanar before calling them parallel.