Parallel Lines and Angle Theorems
Look down a railroad track. The two rails never meet, and every tie crosses both rails at the same angle. A land surveyor faces the same question with two property lines: how can she be sure they are parallel using only a protractor, without measuring the distance between them? By the end of this page you will know which angles to measure, what to compare, and which theorem justifies your answer.
Vertical angles and linear pairs
Every angle theorem about parallel lines starts with two intersecting lines. They form four angles.
Why does the Vertical Angles Theorem work? In the figure, $\angle 1$ and $\angle 2$ are a linear pair, so $m\angle 1 + m\angle 2 = 180^\circ$. So are $\angle 2$ and $\angle 3$, so $m\angle 2 + m\angle 3 = 180^\circ$. Both $\angle 1$ and $\angle 3$ equal $180^\circ - m\angle 2$, so they are equal.
In the intersection of two straight walking paths shown below, $\angle 1$ and $\angle 2$ form a linear pair, while $\angle 1$ and $\angle 3$ are vertical angles. If…
In the intersection of two straight walking paths shown below, $\angle 1$ and $\angle 2$ form a linear pair, while $\angle 1$ and $\angle 3$ are vertical angles. If $m\angle 1 = 115^\circ$, find the measures of $\angle 2$ and $\angle 3$.
Two intersecting lines form a pair of vertical angles. The measure of one angle is given by the expression $(4x - 12)^\circ$ and the measure of the other is…
Two intersecting lines form a pair of vertical angles. The measure of one angle is given by the expression $(4x - 12)^\circ$ and the measure of the other is $(2x + 28)^\circ$. Find the value of $x$ and the exact measure of each angle.
Parallel lines and a transversal
A transversal is a line that crosses two or more other lines at different points. When it crosses two lines, it forms eight angles. Their positions have names.
The small arrowheads on the lines mean "these lines are parallel." Angles 3, 4, 5, 6 lie between the two lines, so they are interior. Angles 1, 2, 7, 8 are exterior.
| Angle pair | Example | Where they are |
|---|---|---|
| Corresponding | $\angle 1$ and $\angle 5$ | same position at each intersection |
| Alternate interior | $\angle 3$ and $\angle 6$ | between the lines, opposite sides of the transversal |
| Alternate exterior | $\angle 1$ and $\angle 8$ | outside the lines, opposite sides of the transversal |
| Same-side interior | $\angle 3$ and $\angle 5$ | between the lines, same side of the transversal |
The first statement is a postulate: we accept it, because sliding the top intersection down along the transversal lands it exactly on the bottom one. The other three are theorems, and they can be proved from it. Here is the proof for alternate interior angles.
Given: $a \parallel b$, cut by transversal $t$. Prove: $\angle 3 \cong \angle 6$.
| Statement | Reason |
|---|---|
| 1. $a \parallel b$ | Given |
| 2. $\angle 3 \cong \angle 7$ | Corresponding Angles Postulate |
| 3. $\angle 7 \cong \angle 6$ | Vertical Angles Theorem |
| 4. $\angle 3 \cong \angle 6$ | Transitive Property of Congruence |
The same-side interior theorem follows in one more step. $\angle 5$ and $\angle 7$ are a linear pair, so $m\angle 5 + m\angle 7 = 180^\circ$. Since $\angle 3 \cong \angle 7$, we get $m\angle 3 + m\angle 5 = 180^\circ$.
In the diagram below, the two horizontal lines are parallel and are intersected by a transversal. The alternate interior angles are given as expressions in terms of $x$.…
In the diagram below, the two horizontal lines are parallel and are intersected by a transversal. The alternate interior angles are given as expressions in terms of $x$. Determine the value of $x$ and the measure of the angles.
The diagram shows a transversal beam cutting across two parallel steel girders. The same-side interior angles are given as algebraic expressions. Calculate the value of…
The diagram shows a transversal beam cutting across two parallel steel girders. The same-side interior angles are given as algebraic expressions. Calculate the value of $x$ and the measure of both angles.
A wooden ladder leans against a wall, and its rungs are built parallel to the ground. A diagonal support beam intersects the rungs, acting as a transversal. If the angle…
A wooden ladder leans against a wall, and its rungs are built parallel to the ground. A diagonal support beam intersects the rungs, acting as a transversal. If the angle between the support beam and the first rung is $68^\circ$, find the measure of the corresponding angle at the second rung. Name the angle relationship and the postulate used.
Proving lines are parallel
Section 2 started with "the lines are parallel, so the angles are equal." The surveyor needs the opposite direction: the angles are equal, so the lines are parallel. That is the converse, and it is just as true.
You only need one pair that passes the test. The converses also give a quick "no": if a pair fails (unequal alternate or corresponding angles, or same-side interior angles that do not add to $180^\circ$), the lines are not parallel.
A land surveyor is marking two property lines that are both intersected by a straight county road. The road forms an angle of $72^\circ$ with the first property line.…
A land surveyor is marking two property lines that are both intersected by a straight county road. The road forms an angle of $72^\circ$ with the first property line. What must the corresponding angle at the second property line be to ensure the two property boundaries are perfectly parallel? State the geometric theorem that justifies your answer.
An architect draws two lines on a blueprint intersected by a transversal. The alternate exterior angles measure $105^\circ$ and $(4x + 25)^\circ$. For what value of $x$…
An architect draws two lines on a blueprint intersected by a transversal. The alternate exterior angles measure $105^\circ$ and $(4x + 25)^\circ$. For what value of $x$ will the two lines be parallel?
In the diagram below, line $a$ is parallel to line $b$. The measure of $\angle 1$ on line $a$ is $112^\circ$. On line $c$, the alternate exterior angle to $\angle 1$…
In the diagram below, line $a$ is parallel to line $b$. The measure of $\angle 1$ on line $a$ is $112^\circ$. On line $c$, the alternate exterior angle to $\angle 1$ measures $(3x + 22)^\circ$. If $x = 30$, prove whether line $c$ is parallel to line $b$ by calculating the angles and applying the converse of the corresponding angles postulate.
Angle relationships with algebra
Most problems hand you angle measures as expressions in $x$. The method is always the same:
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Name the relationship between the two angles.
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Decide whether it gives equal angles (vertical, corresponding, alternate interior or exterior) or angles that add to $180^\circ$ (linear pair, same-side interior).
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Write the equation, then solve it.
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Substitute back to find the angle measure. That is usually the actual answer.
Oak Avenue and Maple Avenue are parallel streets intersected by Diagonal Road. The angle formed at the intersection of Oak Avenue and Diagonal Road is given by the…
Oak Avenue and Maple Avenue are parallel streets intersected by Diagonal Road. The angle formed at the intersection of Oak Avenue and Diagonal Road is given by the expression $(6x - 18)^\circ$. The alternate interior angle at Maple Avenue and Diagonal Road is $(4x + 12)^\circ$. Find the value of $x$ and the actual angle measure of the intersection.
A carpenter is building a custom bookcase. To ensure the shelves are parallel to the base, she measures the angles formed by the vertical side panel, which acts as a…
A carpenter is building a custom bookcase. To ensure the shelves are parallel to the base, she measures the angles formed by the vertical side panel, which acts as a transversal. The same-side interior angles between the base and the first shelf measure $(2x + 15)^\circ$ and $(4x - 9)^\circ$. If the shelves are indeed parallel, what must be the value of $x$ and the measure of each angle?
Lines $m$ and $n$ are parallel and are cut by a transversal line $t$. Angle $A$ and Angle $B$ are same-side interior angles. Angle $C$ and Angle $A$ are vertical angles.…
Lines $m$ and $n$ are parallel and are cut by a transversal line $t$. Angle $A$ and Angle $B$ are same-side interior angles. Angle $C$ and Angle $A$ are vertical angles. If $m\angle B = (5x - 10)^\circ$ and $m\angle C = (3x + 30)^\circ$, find the value of $x$ and the exact measure of all three angles.
When the equation is quadratic
Sometimes an angle expression contains $x^2$. The angle relationship still gives an equation, but now it is a quadratic equation and may have two solutions. A solution is valid only if every angle it produces is a real angle: greater than $0^\circ$ and less than $180^\circ$.
Solve by moving everything to one side, then factor (or use the quadratic formula). Then test each solution in the angle expressions.
Two parallel lines are intersected by a transversal. A pair of corresponding angles have measures of $(x^2 - 10)^\circ$ and $(3x + 30)^\circ$. Find all possible values…
Two parallel lines are intersected by a transversal. A pair of corresponding angles have measures of $(x^2 - 10)^\circ$ and $(3x + 30)^\circ$. Find all possible values for $x$ and determine the valid angle measures for each case.
A civil engineer is designing a parking lot with angled spaces. To ensure the painted lines are exactly parallel, she uses the converse of the corresponding angles…
A civil engineer is designing a parking lot with angled spaces. To ensure the painted lines are exactly parallel, she uses the converse of the corresponding angles postulate. The angle between the straight curb and the first painted line is represented by $(y^2 - 5y)^\circ$. The corresponding angle for the adjacent painted line is $(2y + 18)^\circ$. Find all possible values of $y$. If the parking spaces are designed to have an angle greater than $20^\circ$ for easy turning, which value of $y$ must be used, and what is the resulting angle?
A surveyor is checking the boundaries of a large rectangular plot of land. The north and south property lines are intersected by a straight diagonal fence. The surveyor…
A surveyor is checking the boundaries of a large rectangular plot of land. The north and south property lines are intersected by a straight diagonal fence. The surveyor measures one pair of alternate interior angles as $(2x^2 - 8x)^\circ$ and $(x^2 + 2x + 24)^\circ$. If the north and south lines are truly parallel, find the valid measure(s) of the angles. (Note: The angles must be positive and strictly less than $180^\circ$, so you must reject any extraneous solutions).
Two variables, and "not parallel"
Some problems give you two unknowns, $x$ and $y$. Each angle relationship gives you one equation. Two relationships give a system of linear equations, which you can solve by substitution or elimination.
Now the reverse question: for which values of $x$ are the lines not parallel? By the converse, lines are parallel exactly when same-side interior angles add to $180^\circ$. So they are not parallel whenever the sum is anything else. Add the requirement that both angles must really exist, between $0^\circ$ and $180^\circ$.
Two parallel streets are crossed by a straight railway line. A pair of same-side interior angles are given as $(4x + 2y)^\circ$ and $(3x + 2y)^\circ$. A pair of…
Two parallel streets are crossed by a straight railway line. A pair of same-side interior angles are given as $(4x + 2y)^\circ$ and $(3x + 2y)^\circ$. A pair of alternate interior angles are given as $(3x + 2y)^\circ$ and $(5x - 2y)^\circ$. Write and solve a system of linear equations to find the values of $x$ and $y$, then state the measure of the obtuse angle formed at the intersection.
Line $p$ and line $q$ are cut by transversal $r$. A pair of consecutive (same-side) interior angles have measures $m\angle 1 = (4x - 20)^\circ$ and…
Line $p$ and line $q$ are cut by transversal $r$. A pair of consecutive (same-side) interior angles have measures $m\angle 1 = (4x - 20)^\circ$ and $m\angle 2 = (3x + 25)^\circ$. For what range of values for $x$ are the lines guaranteed NOT to be parallel? Assume that both angle measures must be strictly between $0^\circ$ and $180^\circ$ to physically exist in the diagram.
Auxiliary lines and chains of parallels
Sometimes a figure does not have the transversal you need. You can draw an auxiliary line: a new line added to the diagram that helps you use a theorem. The most useful one is a line through a point parallel to the given parallel lines.
Two parallel lines are bent by a path $T \to V \to B$. Draw the auxiliary line through $V$ parallel to both. Then each of the two angles at $V$ is alternate interior to an angle at the ends: $35^\circ$ at $T$ and $50^\circ$ at $B$. The whole angle at $V$ is their sum.
The same idea shows that the angles of a triangle add to $180^\circ$. Put the base of a triangle on the lower line and draw the upper line through the top vertex parallel to it. The two base angles reappear at the top vertex as alternate interior angles. Together with the triangle's own top angle they fill a straight angle, which is $180^\circ$.
Parallel lines can also be chained together.
This means a transversal makes the same corresponding angle with all the lines in a chain of parallels. If a transversal crosses $L_1$, $L_2$, $L_3$ and makes $70^\circ$ with $L_1$, it makes $70^\circ$ with $L_2$ and $L_3$ too, in the same position.
Line $L_1$ is parallel to $L_2$, and $L_2$ is parallel to $L_3$. A transversal $t_1$ intersects all three lines, creating a corresponding angle of $85^\circ$ at $L_1$. A…
Line $L_1$ is parallel to $L_2$, and $L_2$ is parallel to $L_3$. A transversal $t_1$ intersects all three lines, creating a corresponding angle of $85^\circ$ at $L_1$. A second transversal $t_2$ intersects $L_2$ and $L_3$, creating same-side interior angles of $(5y)^\circ$ and $(4y + 9)^\circ$. Find the value of $y$ and explain how the transitive property of parallel lines applies to $L_1$ and $L_3$.
In the diagram below, the top and bottom horizontal lines are parallel. Two transversals intersect at a single point on the top line, forming a triangle between the…
In the diagram below, the top and bottom horizontal lines are parallel. Two transversals intersect at a single point on the top line, forming a triangle between the parallel lines. The interior alternate angles on the bottom line are $45^\circ$ and $65^\circ$. An exterior angle on the top line is marked as $(5x + 15)^\circ$. Determine the value of $x$ by applying alternate interior angles and the linear pair postulate.
The diagram shows two parallel horizontal pipelines. A connecting pipe has a bend in the middle, forming an angle of $x^\circ$. The acute interior angles formed with the…
The diagram shows two parallel horizontal pipelines. A connecting pipe has a bend in the middle, forming an angle of $x^\circ$. The acute interior angles formed with the parallel pipelines are $38^\circ$ and $54^\circ$. By drawing an auxiliary line parallel to the pipelines through the bend, determine the value of $x$. Explain your reasoning step by step.
The Perpendicular Bisector Theorem
A perpendicular bisector of a segment is a line that is perpendicular to the segment at its midpoint. It cuts the segment into two equal halves at a right angle.
The reason is a congruent-triangle argument. $M$ is the midpoint, so $AM = MB$. The angles at $M$ are right angles, and $PM$ is shared. So $\triangle PMA \cong \triangle PMB$ by SAS, and the matching sides $PA$ and $PB$ are equal.
Drag the point along the bisector in this tool and watch the two distances stay equal.
A new cell phone tower is being built exactly on the perpendicular bisector of the straight highway segment connecting two neighboring towns, Springfield and…
A new cell phone tower is being built exactly on the perpendicular bisector of the straight highway segment connecting two neighboring towns, Springfield and Shelbyville. If the tower is located $15.5 \text{ miles}$ away from Springfield in a straight line, how far is it from Shelbyville? Explain the geometric theorem that supports your conclusion.
Points $A$ and $B$ are located at opposite ends of a state park. A straight hiking trail represents the perpendicular bisector of the segment connecting $A$ and $B$. A…
Points $A$ and $B$ are located at opposite ends of a state park. A straight hiking trail represents the perpendicular bisector of the segment connecting $A$ and $B$. A hiker standing at a point $P$ on the trail walks directly to $A$, covering a distance of $3.2 \text{ miles}$. She then plans to walk from $A$ to $B$, and finally from $B$ back to $P$. If the distance from $A$ to $B$ is $4.5 \text{ miles}$, what is the total distance of her trip? State the geometric theorem that justifies your calculation.
Bisector problems with algebra
When distances are written as expressions, the theorem gives the equation: set the two distances equal.
If the distances involve $x^2$, you get a quadratic equation. Solve it as in the quadratic-angle section, then test each solution: a length must be positive. Also remember that the endpoints $A$ and $B$ are $AB = 2 \cdot AM$ apart. The midpoint satisfies $AM = MB$, but $PA = PB$ is a different pair of equal segments.
A wooden roof truss is designed so that the central vertical support beam is the perpendicular bisector of the bottom horizontal chord. The bottom chord has a total…
A wooden roof truss is designed so that the central vertical support beam is the perpendicular bisector of the bottom horizontal chord. The bottom chord has a total length of $24 \text{ ft}$. The distance from the left edge of the bottom chord to the top of the central support is given as $(3y + 2) \text{ ft}$, and the distance from the right edge to the top is $(5y - 8) \text{ ft}$. Find the value of $y$ and determine the actual length of these angled roof beams.
On a city park map, a new drinking fountain is placed on the perpendicular bisector of the straight path connecting the playground at point $P$ and the restroom at point…
On a city park map, a new drinking fountain is placed on the perpendicular bisector of the straight path connecting the playground at point $P$ and the restroom at point $R$. The distance from the fountain to $P$ is given by the expression $(4x - 7) \text{ yards}$, and the distance to $R$ is $(2x + 19) \text{ yards}$. Calculate the value of $x$ and the exact distance from the fountain to the playground.
A steel bridge features a bottom horizontal chord and a top horizontal chord that are designed to be parallel. A vertical strut acts as the perpendicular bisector of the…
A steel bridge features a bottom horizontal chord and a top horizontal chord that are designed to be parallel. A vertical strut acts as the perpendicular bisector of the bottom chord section $AB$. Diagonal beams connect the top of the strut $M$ (located on the top chord) to points $A$ and $B$. The lengths of the diagonal beams are given by $MA = (4x^2 - 10x) \text{ ft}$ and $MB = (x^2 + 11x + 24) \text{ ft}$.
Determine all mathematically possible values of $x$. If engineering constraints require the diagonal beams to be shorter than $50 \text{ ft}$, which value of $x$ is valid? Using this valid scenario, if the angle between the top chord and beam $MA$ is $42^\circ$, what is the exact measure of $\angle MBA$ on the bottom chord? Justify your answer using parallel line theorems and properties of the perpendicular bisector.
Back to the start: the surveyor measures one pair of corresponding angles where the road crosses both property lines. If they are congruent, the lines are parallel by the converse of the Corresponding Angles Postulate. No distance measurement is needed.
Summary
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Vertical angles are congruent. A linear pair adds to $180^\circ$.
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For parallel lines cut by a transversal: corresponding, alternate interior, and alternate exterior angles are congruent. Same-side interior angles are supplementary.
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The converses prove lines parallel: one pair of congruent corresponding (or alternate) angles, or supplementary same-side interior angles, is enough. A pair that fails shows the lines are not parallel.
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With algebra: name the relationship, write "equal" or "sum is $180$," solve, and substitute back. For quadratics, reject any solution whose angle (or length) is not strictly between $0^\circ$ and $180^\circ$ (or not positive).
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An auxiliary line through a point parallel to the given lines turns a bend into alternate interior angles you can add. Lines parallel to the same line are parallel to each other.
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Perpendicular Bisector Theorem: a point on the perpendicular bisector is equidistant from the endpoints, and a point equidistant from the endpoints is on the bisector.