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Section 4.3 A Right Angle Theorem

Section 4.3 A Right Angle Theorem. Objective: *Apply one way of proving that two angles are right angles. The Theorem. In order to prove that lines are perpendicular, you must first prove that they form right angles. * For this reason, it is necessary to know the following theorem:

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Section 4.3 A Right Angle Theorem

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  1. Section 4.3A Right Angle Theorem Objective: *Apply one way of proving that two angles are right angles.

  2. The Theorem In order to prove that lines are perpendicular, you must first prove that they form right angles. *For this reason, it is necessary to know the following theorem: Theorem 23: If two angles are both supplementary and congruent, then they are right angles.

  3. How The Theorem Works Most of the problems dealing with this theorem will be proofs. Here is how you would use it in a proof if you were given the diagram to the right. It is given that 1 is congruent to 2 and you must prove both angles are right angles. 1. Since 1 and 2form a straight line, then they are supplementary. 2. Then, since the angles are congruent, you know that each must equal 90°. 3. Therefore, you now know that both of the angles are supplementary and congruent. You can now use the theorem that “If two angles are both supplementary and congruent, then they are right angles.” 2 1 NOTE: You can now assume that whenever two angles form a straight line, they are supplementary. No formal statement of this fact will be necessary.

  4. C Example 1: A B D Given mACD = mBCD Definition of bisect ACD  BCD Definition of congruent angles Given Reflexive Property ∆ACD  ∆BCD SAS CPCTC CDA CDB

  5. A E B D Example 2: C Given Given Reflexive Property SSS ∆ADB  ∆CDB CPCTC ABE  CBE Reflexive Property ∆ABE  ∆CBE SAS AEB  CEB CPCTC If 2 s are supp. and , then they are rt. s AEB and CEB are right angles Definition of altitude

  6. Example 3: If squares A and C are folded across the dotted segments onto B, find the area of B that will not be covered by either square. In order to solve this problem, you first have to find that the top part of B is eight. Then, fold over squares A and C to get the top part of B to be 4. Next, you know that the side of B will be two because A is a square when it is folded over. Lastly, you multiply two and four to find the area of B that will not covered by either square. The final answer is eight. 12 2 A B C 2

  7. Example 4: A diameter of a circle has endpoints with coordinates (1, 6) and (5, 8). Find the coordinates of the center of the circle. (5, 8) (1, 6)

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