back to Geometry

Angle pairs, perpendicular and parallel lines

Transversal

A transversal is a line that intesects two or more lines in the same plane but at different points.

What is a transversal? What property does the transversal has?

In the figure above, line l interests line m and line n at point A and point B. Line l is a transversal.

Corresponding angles

Corresponding angles are the angles that appear to be in the same relative position in each group of four angles.

What are the corresponding angles? How to determine that two angles are corresponding angles?

In the figure above, the four pairs of corresponding angles are angle 1 and angle 5, angle 2 and angle 6, angle 3 and angle 7, angle 4 and angle 8.

Alternate interior angles

Alternate interior angles are the angles within the lines being intersected, on opposite side of the transversal.

What are the alternate interior angles? How to determine that two angles are alternate interior angles?

In the figure above, the two pairs of alternate interior angles are angle 3 and angle 5, angle 4 ang angle 6.

Same-side interior angles

Same-side interior angles are interior angles on the same side of the transversal.

What are the same side interior angles? How to determine that two angles are the same side interior angles?

In the figure above, the two pairs same-side interior angles are angle 3 and angle 6, angle 4 and angle 5.

Alternate exterior angles

Alternate exterior angles are angles outside the line being intersected, on opposite side of the transversal, and are not adjacent.

What are the alternate exterior angles? How to determine that two angles are alternate exterior angles?

In the figure above, the two pairs of the alternate exterior angles are angle 2 and angle 8, angle 1 ang angle 7.

Same-side exterior angles

Same side exterior angles are exterior angles on the same side of transversal.

What are the same side exterior angles How to determine that two angles are same side exterior angles?

In the figure above, the two pairs of the same-side exterior angles are angle 2 and angle 7, angle 1 and angle 8.

Properties of Parallel Lines

Property 1

If two lines are parallel, then their alternate interior angles are equal.

How to find that two angles are alternate interior angles?

In the figure above, if m // n, then angle 1 = angle 2

Property 2

If two lines are parallel, then their corresponding angles are equal.

How to find that two angles are corresponding angles?

In the figure above, if m // n, then angle 3 = angle 4

Property 3

If two lines are parallel, then their alternate exterior angles are equal.

How to find that two angles are alternate exterior angles?

In the figure above, if m // n, then angle 5 = angle 6

Property 4

If two lines are parallel, then their interior angles on the same side of the traversal are supplementary.

How to find that two angles are supplementary? What is the value of the sum of these two ngles?

In the figure above, if m // n, then angle a + angle b = 180o

Property 5

In a plane, if two lines are perpendicular to the same line, then the two lines are parallel.

How to determine that two line are parallel?

In the figure above,

How to find whether two line are parallel.

Property 6

In a plane, if two lines are parallel to the third line, then the two lines are parallel.

How to determine that two line are parallel?

In the figure above,

How to determine that two lines are parallel?

If lines are parallel?

Property 7

When two lines cut by a transversal, if corresponding angles are equal, then the two lines are parallel.

How to finf that two lines are parallel? How to find two corresponding angles?

In the figure above,

Example of finding two corresponding angles. Example of determine that two lines are parallel.

Property 8

When two lines cut by a transversal, if alternate interior angles are equal, then the two lines are parallel.

How to find the alternate interior angles? How to determine that two lines are parallel?

In the figure above,

If alternate interior angles are equal, then the two lines are parallel.

Property 9

When two lines cut by a transversal, if the same side of interior angles are supplementary, then the two lines are parallel.

How to determine that two angles are supplementary? How to determine that two lines are parallel?

In the figure above,

 Example that find two angles are supplementary. Example of proving two lines parallel.

Example 1

Prove that when two lines cut by a transversal, if the alternate exterior angles are equal, then the two lines are parallel.

Example 1 of proving that two lines are parallel by using equal alternate exterior angles.
Proof
lines m and n cut by l, then
angle a = angle c (vertical angles are equal)
angle d = angle b (vertical angles are equal)
So angle c = angle d
since angle c = angle d, so lines m // n (Property 8)

Equal angles

Property 10

If two angles are complementary to the same angle, or to the equal angles, then the two angles are equal to each other.

How to determine that two angles are complementary? What is the value of the sum of two complementary angles?

In the figure above, if angle b is complementary to angle a and angle c is complementary to angle a, then angle c = angle b.

Property 11

How to determine that two angles are supplementary? What is the value of the sum of two supplementary angles?

In the figure above, if angle 2 is supplementary to angle 1 and angle 3 is supplementary to angle 1, then angle 3 = angle 2.

Example 2

In the figure below, point E lies on line AB. If angle 1 and angle 2 are complementary. The angle CED = 90o, prove AB // CD.

How to determine that two angles are complementary? What will be if two angles are complementary?
Proof
Since angle CED = 90o (Given)
So angle 2 + angle 3 = 90o
Since angle 1 and angle 2 are complementary (Given)
So angle 1 + angle 2 = 90o
angle 1 = 90o - angle 2
angle 3 = 90o - angle 2
So angle 1 = angle 3
So AB // CD (Property 8)

Example 3

In the figure below, lines l1 // l2 and m1 // m2. Lines l1 and m1 intersect at point A. Lines l2 and m1 intersect at point B. Lines l1 and m2 intersect at point C and lines l2 and m2 intersect at point D. Prove angle 3 = angle 4.

parallel lines example 3
Proof
Since l1 // l2 (given), so angle 1 = angle 4 (alternate interior angles are congruent)
Since m1 // m2 (given), so angle 1 = angle 3 (corresponding angles are congruent)
Since angle 1 = angle 4 and angle 1 = angle 3, so
angle 3 = angle 4 (substitution)

note: in the figure above, ABCD is a quadrilateral. When AB // CD and AC // BD, there is angle 3 = angle 4. Which means that in a quadrilateral when two pairs of opposite sides are parallel, its opposite angles are equal. It is a property of a parallelogram.