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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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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

Property 5

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

In the figure above,

Property 6

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

In the figure above,

If lines are parallel?

Property 7

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

In the figure above,

Property 8

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

In the figure above,

Property 9

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

In the figure above,

Example 1

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

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.

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

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 = 90^o, prove AB // CD.

Proof

Since angle CED = 90^o (Given)

So angle 2 + angle 3 = 90^o

Since angle 1 and angle 2 are complementary (Given)

So angle 1 + angle 2 = 90^o

angle 1 = 90^o - angle 2

angle 3 = 90^o - 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.

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.