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Teacher Notes Angle Relationships Objective: To identify and use adjacent, vertical, complementary, supplementary, and linear pairs of angles and perpendicular lines. To determine what information can and can not be assumed from a diagram Definitions: Perpendicular lines – special intersecting lines that form right angles. Adjacent angles – angles in the same plane that have a common vertex and common side, but no common interior points. Vertical angles – two nonadjacent angles formed by two intersecting lines. Linear pair – adjacent angles whose noncommon sides are opposite rays. Supplementary angles – two angles whose measures have a sum of 180˚. Complementary angles – two angles whose measures have a sum of 90˚. Notes: Vertical angles are congruent. (This is the symbol for congruent, .) The sum of the measures of the angles in a linear pair is 180˚. means perpendicular m n is read m is perpendicular to n. lines intersect to form four right angles. If a line is perpendicular to a plane then that line is perpendicular to every line in the plane that it intersects. M N O Q From the picture above, You CAN assume: All points shown are coplanar. L, P, and Q are collinear. PM, PN, PO and LQ intersect at P. P is between L and Q. N is in the interior of angle MPO. Angle LPQ is a straight angle. P L You CAN NOT assume: PN is perpendicular to PM Angle QPO is congruent to angle LPM LP is congruent to PQ PQ is congruent to PO Angle QPO is congruent to angle OPN Angle OPN is congruent to angle LPM Angle LPM and angle MPN are adjacent angles Angle LPN and angle NPQ are a linear pair. Angle QPO and angle OPL are supplementary PO is congruent to PN PN is congruent to PL M N O Q P L With these additional markings, you can now assume a few more things that could not be assumed from the first picture. PN is perpendicular to PM QP is congruent to PL Angle QPO is congruent to angle MPL Examples: J G I K H Find the value of x. Angle GIJ = 9x – 4 and angle JIH = 4x – 11 Angle GIJ = 16x – 20 and angle KIH = 13x + 7 1 2 3 5 4 Identify each pair of angles as adjacent, vertical, complementary, supplementary, and/or as a linear pair. Angle 1 and angle 2 Angle 1 and angle 4 Angle 3 and angle 4 Angle 1 and angle 5