When two lines intersect they form two pairs of opposite angles, A + C and B + D. Another word for opposite angles are vertical angles.
Vertical angles are always congruent, which means that they are equal.
Adjacent angles are angles that come out of the same vertex. Adjacent angles share a common ray and do not overlap.
The size of the angle xzy in the picture above is the sum of the angles A and B.
Two angles are said to be complementary when the sum of the two angles is 90°.
Two angles are said to be supplementary when the sum of the two angles is 180°.
If we have two parallel lines and have a third line that crosses them as in the ficture below - the crossing line is called a transversal
When a transversal intersects with two parallel lines eight angles are produced.
The eight angles will together form four pairs of corresponding angles. Angles 1 and 5 constitutes one of the pairs. Corresponding angles are congruent. All angles that have the same position with regards to the parallel lines and the transversal are corresponding pairs e.g. 3 + 7, 4 + 8 and 2 + 6.
Angles that are in the area between the parallel lines like angle 2 and 8 above are called interior angles whereas the angles that are on the outside of the two parallel lines like 1 and 6 are called exterior angles.
Angles that are on the opposite sides of the transversal are called alternate angles e.g. 1 + 8.
All angles that are either exterior angles, interior angles, alternate angles or corresponding angles are all congruent.
Example
The picture above shows two parallel lines with a transversal. The angle 6 is 65°. Is there any other angle that also measures 65°?
6 and 8 are vertical angles and are thus congruent which means angle 8 is also 65°.
6 and 2 are corresponding angles and are thus congruent which means angle 2 is 65°.
6 and 4 are alternate exterior angles and thus congruent which means angle 4 is 65°.
Video lesson
Find the measure of all the angles in the figure
In geometry, a transversal is a line that intersects two or more other (often parallel ) lines.
In the figure below, line n is a transversal cutting lines l and m .
When two or more lines are cut by a transversal, the angles which occupy the same relative position are called corresponding angles .
In the figure the pairs of corresponding angles are:
∠ 1 and ∠ 5 ∠ 2 and ∠ 6 ∠ 3 and ∠ 7 ∠ 4 and ∠ 8
When the lines are parallel, the corresponding angles are congruent .
When two lines are cut by a transversal, the pairs of angles on one side of the transversal and inside the two lines are called the consecutive interior angles .
In the above figure, the consecutive interior angles are:
∠ 3 and ∠ 6 ∠ 4 and ∠ 5
If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles formed are supplementary .
When two lines are cut by a transversal, the pairs of angles on either side of the transversal and inside the two lines are called the alternate interior angles .
In the above figure, the alternate interior angles are:
∠ 3 and ∠ 5 ∠ 4 and ∠ 6
If two parallel lines are cut by a transversal, then the alternate interior angles formed are congruent .
When two lines are cut by a transversal, the pairs of angles on either side of the transversal and outside the two lines are called the alternate exterior angles .
In the above figure, the alternate exterior angles are:
∠ 2 and ∠ 8 ∠ 1 and ∠ 7
If two parallel lines are cut by a transversal, then the alternate exterior angles formed are congruent .
Example 1:
In the above diagram, the lines j and k are cut by the transversal l . The angles ∠ c and ∠ e are…
A. Corresponding Angles
B. Consecutive Interior Angles
C. Alternate Interior Angles
D. Alternate Exterior Angles
The angles ∠ c and ∠ e lie on either side of the transversal l and inside the two lines j and k .
Therefore, they are alternate interior angles.
The correct choice is C .
Example 2:
In the above figure if lines A B ↔ and C D ↔ are parallel and m ∠ A X F = 140 ° then what is the measure of ∠ C Y E ?
The angles ∠ A X F and ∠ C Y E lie on one side of the transversal E F ↔ and inside the two lines A B ↔ and C D ↔ . So, they are consecutive interior angles.
Since the lines A B ↔ and C D ↔ are parallel, by the consecutive interior angles theorem , ∠ A X F and ∠ C Y E are supplementary.
That is, m ∠ A X F + m ∠ C Y E = 180 ° .
But, m ∠ A X F = 140 ° .
Substitute and solve.
140 ° + m ∠ C Y E = 180 ° 140 ° + m ∠ C Y E − 140 ° = 180 ° − 140 ° m ∠ C Y E = 40 °