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1 GEOMETRY – ADVANCED Proofs Code Name: ___________________ Period: ______ Point B is between A and C C B If a point is a on a segment, the segment is separated into two smaller parts which add up to equal the whole segment A Segment Addition Postulate AB + BC = AC E is the Midpoint of FD D E F BD is in the interior of ∠ABC E Midpoint Theorem FE ≅ ED If a segment has a midpoint, then it is divided into 2 congruent segments. D F D A B C Angle Addition Postulate m∠ABD + m∠DBC = m∠ABC If 2 small angles share a ray, then they can be added together to get one large angle 2 AB ≅ CD B D A If 2 segments are congruent, then they have equal measures. C Definition of Congruent Segments AB = CD ∠1 and ∠2 are complementary 1 2 Definition of Complementary Angles m∠1 + m∠2 = 90 If 2 angles are complementary, then their measures have a sum of 90 degrees. ∠3 and ∠4 are supplementary 3 m∠3 + m∠4 = 180 4 Definition of Supplementary Angles If 2 angles are complementary, then their measures have a sum of 180 degrees. ∠5 and ∠6 are Vertical Angles 5 6 If 2 are vertical angles, then they are congruent. ∠5 ≅ ∠6 Vertical Angles Are Congruent 3 ∠3 and ∠4 are In a Linear Pair 3 If two angles are in a linear pair, then the sums of their measures is 180 4 Sum of Angles in a Linear Pair Equals 180 m∠3 + m∠4 = 180 E is the Midpoint of FD D E A midpoint divides a segment into two equal parts. F FE = ED Definition of Midpoint Line l is a Segment Bisector B M l A Definition of Segment Bisector If a segment, line, or a plane is a segment bisector then it intersects the segment at its midpoint. M is a Midpoint ∠1 ≅ ∠2 2 1 If two angles are congruent, then they have equal measures. m∠1 = m∠2 Definition of Congruent Angles 4 BD is an Angle Bisector D A B C If a ray bisects an angle, it creates two congruent angles. Definition of Congruent Angles ∠ABD ≅ ∠DBC ∠T is a Right Angle T If an angle is a right angle, then it measures 90 degrees. Definition of Right Angles m∠T = 90 ST ⊥ TW S W T ∠STW is a right angle Definition of Perpendicular Lines If two lines are perpendicular (symbol ⊥ ) Then the four angles formed are right angles. ∠1 and ∠2 are Right angles 2 1 ∠1 ≅ ∠2 All Right Angles are Congruent If two angles are right angles, then they are congruent. 5 ∠1 is complementary to ∠2 ∠3 is complementary to ∠2 1 2 3 ∠1 ≅ ∠3 2 Angles are complementary to the same angle are congruent ∠1 is supplementary to ∠2 ∠3 is supplementary to ∠2 ∠1 ≅ ∠3 If two angles are supplementary to the same angle (or congruent angles), then they are congruent to each other. 2 3 1 2 Angles are supplementary to the same angle are congruent If two angles are supplementary to the same angle (or congruent angles), then they are congruent to each other. ∠1 and ∠2 Are a linear pair 1 ∠1 and ∠2 Are supplementary 2 If two angles are in a linear pair, then they are supplementary angles. Supplements Theorem *2 angles in a linear pair are supplementary angles*