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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)
since angle a = angle b (anglea and angle b are alternate exterior angles)
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
If two angles are supplementary to the same angle, or to the equal angles, then the two angles are equal to each other.
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 above, 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 l
_{1}
// l
_{2}
and m
_{1}
// m
_{2}
. Lines l
_{1}
and m
_{1}
intersect at point A. Lines l
_{2}
and m
_{1}
intersect at point B. Lines l
_{1}
and m
_{2}
intersect at point C and lines l
_{2}
and m
_{2}
intersect at point D.
Prove angle 3 = angle 4.
Proof:
Since l
_{1}
// l
_{2}
(given), so angle 1 = angle 4 (alternate interior angles are congruent)
Since m
_{1}
// m
_{2}
(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.