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Section 2.5 Addition and Subtraction Properties Quick Review Define complementary Define supplementary Define congruent Define congruent angles angles angles segments Two angles are complementary if their sum is 90° Two angles are supplementary if their sum is 180° Two segments are congruent if their measures are equal. Two angles are congruent if they have the same measure. Six New Properties and Theorems If a segment is added to two congruent segments, the sums are congruent. (Addition Property) 1. A Note that we first need to know that two segments are congruent, and then that we are adding the same segment to both of them. 7 cm B 3cm C 7cm D AB + BC = CD + BC AC = BD, so AC BD Cont… If an angle is added to two congruent angles, the sums are congruent (Addition Property) 2. Note that we first need to know that we have 2 congruent angles, then that we are adding the A mABC = 50.03 same angle to both C mABC + mCBD = mDBE + mCBD D mABD = mCBE, so B ABD CBE mDBE = 50.03 E Cont… If congruent segments are added to congruent segments, the sums are congruent. (Addition Property) 3. Note that first we need 2 congruent segments, then we need 2 different congruent segments to add C F CF + FG = HE + ED CG = HD , so D E CG HD H G Cont… If congruent angles are added to congruent angles, the sums are congruent. (Addition Property) 4. Note that first we need 2 congruent angles, then we need to add two different congruent angles J L I mJIL + mLIK = mJKL + mLKI JIK JKI K Cont… If a segment (or angle) is subtracted from congruent segments (or angles), the differences are congruent. (Subtraction Property) 5. Note that we need to start with congruent angles or segments and then subtract the same angle or segment from both. 10 Q B R A 10 QR - BR = BA - BR QB BR Cont… If a segment (or angle) is subtracted from congruent segments (or angles), the differences are congruent. (Subtraction Property) 5. Note that we need to start with congruent angles or segments and then subtract the same angle or segment from both. A mABD = 78 C mABD - mCBD = mCBE - mCBD D mCBE = 78 B E ABC DBE Cont… If congruent segments (or angles) are subtracted from congruent segments (or angles), the differences are congruent. (Subtraction Property) 5. S Note that we start with congruent segments or angles, and then subtract congruent segments or angles. U W T V mSTV = mUVT = 130 mWTV =mWVT = 30 mSTV - mWTV = mUVT - mWVT STW UVW Using the Addition and Subtraction Properties An addition property is used when the segments or angles in the conclusion are greater than those in the given information A subtraction property is used when the segments or angles in the conclusion are smaller than those in the given information. Theorem 8: If a segment is added to two congruent segments, the sums are congruent. (Addition Property) Given: PQ RS P Q R S Conclusion: PR QS Statements Reasons 1. 1. Given PQ RS 2. PQ = RS 2. If two segments are congruent, then they have the same measure 3. PQ + QR = RS + QR 3. Additive Property of Equality 4. PR = QS 4. Addition of Segments 5. 5. If two segments have the same measure then they are congruent PR QS How to use this theorem in a proof: M Given: GJ HK Conclusion: GH JK G H Statements Reasons 1. 1. 2. GJ HK GH JK J K Given 2. If a segment is subtracted from congruent segments, then the resulting segments are congruent. (Subtraction)
