When two lines intersect they form two pairs of opposite angles, A + C and B + D. Another word for opposite angles are vertical angles.
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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.
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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°.
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Two angles are said to be supplementary when the sum of the two angles is 180°.
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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.
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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
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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°.