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Definitions  If . . . Then Statements

Definitions  If . . . Then Statements. then it divides the angle into two congruent angles. If a ray bisects an angle,. then the ray bisects the angle. If a ray divides an angle into two congruent angles,. then it divides the segment into two congruent segments.

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Definitions  If . . . Then Statements

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  1. Definitions  If . . . Then Statements then it divides the angle into two congruent angles. If a ray bisects an angle, . . . then the ray bisects the angle. If a ray divides an angle into two congruent angles, . . . then it divides the segment into two congruent segments. If a line ( ray/segment/point) bisects a segment, . . . then the line (ray/segment/point) bisects the segment. If a line (ray/segment/point) divides a segment into two congruent segments, . . . then it is the midpoint of the segment. If a point divides a segment into two congruent segments, . . . then it divides the segment into two congruent segments. If a point is the midpoint of a segment, . . .

  2. Definitions  If . . . Then (cont.) Statements then they divides the angle into three congruent angles. If two rays trisect an angle, . . . then the rays trisect the angle. If two rays divide an angle into three congruent angles, . . .

  3. 2-5 Proving Angles Congruent Angle Pairs Vertical Anglestwo angles whose sides form two pairs of opposite rays. 1 4 3 2 Adjacent Anglestwo coplanar angles with a common side, a common vertex, and no common interior points. 5 6

  4. 5 6 120 D 2-5 Proving Angles Congruent Angle Pairs 60 30 Complementary Anglestwo angles whose measures add to 90. (Not necessarily adjacent.) Each is the complement of the other. A 1 B 2 Supplementary Anglestwo angles whose measures add to 180. (Not necessarily adjacent.) Each is the supplement of the other. 60 C

  5. 1 4 3 2 2-5 Five Angle Theorems Vertical Angles TheoremVertical angles are congruent.

  6. B 3 C A 1 4 D 2-5 Five Angle Theorems Congruent Supplements TheoremIf two angles are supplements of the same angle, then the two angles are congruent. Congruent Supplements TheoremIf two angles are supplements of congruent angles, then the two angles are congruent. C and 3 are supplementary.D and 4 are supplementary.3  4Therefore, C  D. A and 1 are supplementary.B and 1 are supplementary.Therefore, A  B.

  7. 3 1 C A 4 B D 2-5 Five Angle Theorems, cont. Congruent Complements TheoremIf two angles are complements of the same angle, then the two angles are congruent. Congruent Complements TheoremIf two angles are complements of congruent angles, then the two angles are congruent. C and 3 are complementary.D and 4 are complementary.3  4Therefore, C  D. A and 1 are complementary.B and 1 are complementary.Therefore, A  B.

  8. 2-5 Five Angle Theorems, cont. Right Angle TheoremAll right angles are congruent. Congruent and Supplementary TheoremIf two angles are congruent and supplementary, then each is a right angle.

  9. 1 3 2 Proving the Five Theorems Vertical Angles Theorem

  10. 2 Congruent Supplements Theorem(Same Angle) 1 3

  11. 1 Congruent Supplements Theorem(Congruent Angles) 2 3 4

  12. B A All Right Angles Congruent Given: A is a right angle. B is a right angle. Prove:A  B Reasons 1. Given 2. Def. rt.  3. Substitution POE 4. Def. congruence • Statements • 1. A is a right angle;B is a right angle • 2. mA = 90; mB = 90 • 3. mA = mB • A z B

  13. Y X Angles Both Congruent and Supplementary are Right Angles Given: X zY ; X supp.Y Prove:X and Y are right s • Statements • 1. X zY ; X supp.Y • 2. mX= mY • 3. mX+ mY=180 • mX+ mX=180 • 2mX=180 • mX=90 • mY=90 • X is a rt. ; Y is a rt.  • Reasons • Given • Definition of congruence • 3. Def. supp. s • Substitution POE • Combine like terms (simplify) • Division POE • Substitution POE • Def. rt. 

  14. Using the Theorems In the REASON column you can now write the short form abbreviations. Now you can use these five theorems as part of other proofs.

  15. Complementary/Supplementary Proof C E D 3 F 2 4 1 A B J G H Reasons Statements 1. Diagram; 3  1 • Given 2. FJD is a straight angle 2. Assumed from diagram 3. 2 and  3 are supp. 3. If two s form a straight ,then they are supp. 4. 2 and  1 are supp. 4. Substitution POC

  16. X B A O Y More Practice Reasons Statements 1. Given • Diagram; XOB  YOB 2. AOB is a straight . 2. Assumed from diagram. 3. AOX and XOB are supp. 3. If two s form a straight, then they are supp. 4. AOX and  YOB are supp. 4. Substitution POC

  17. X Y 4 3 1 2 A B Using the Theorems

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