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Section 4.3 -A Right Angle Theorem Michael Smertz H Geometry- 8 30 May 2008 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. 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 at right. It is given that L1 is congruent to L2 and you must prove both angles are right angles. 1. Since L1 and L2 form 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.” 1 2 NOTE: You can now assume that whenever two angles form a straight line, they are supplementary. Sample Problems Solution • Problem #1 – Given: 1. Ray CD bisects LACB 1. Given 2. Given 3. If a ray bisects an angle, then it divides the angle into 2 congruent angles 4. Reflexive 6. LCDA congruent LCDB 5. SAS (2.3.4) 6. CPCTC 7. LCDA and LCDB are right angles 7. If two angles are both supplementary and congruent, then they are right angles. • Ray CD bisects LACB 2. AC congruent to CB • Segment AC is congruent 3. LACD congruent to segment CB LBCD – Prove: • LCDA and LCDB are right angles C A 4. CD congruent CD 5. ∆ACD congruent ∆BCD B D Sample Problems • Problem #2 – Given: • DA congruent to DC • AB congruent to BC – Prove: • DB altitude of AC A E D C B 1. • Solution DA congruent to DC 1. Given 2. AB congruent to BC 2. Given 3. Reflexive 3. DB congruent to DB 4. SSS (1,2,3) 5. CPCTC 4. ∆DAB congruent to ∆DCB 6. Reflexive 5. LABE congruent to LCBE 7. SAS (2,5,6) 8. CPCTC 6. EB congruent to EB 9. 7. ∆ABE congruent to ∆CBE If two angles are both supplementary and congruent, then they are rt. L’s 8. LAEB congruent to LCEB 9. LAEB and LCEB rt. L’s 10. DB alt. of AC 10. An altitude of a ∆ forms right angles with the side to which it is drawn. Note: This is a detour problem. Sample Problems • Problem #3 • Solution – 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. 12 2 A B C 2 – 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. Practice Problem #1 • Given: – P – AB congruent to BC • Prove: – LDBC and LDBA are right angles A B D C Practice Problem #2 M 3x+14 2x+22 O Is M perpendicular to O? Justify your answer. Practice Problem #3 • Given: – XY congruent to XZ – XQ bisects LYXZ • Prove: – XQ is perpendicular to YZ X Y Q Z Practice Problem #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) Answer Sheet Practice Problem #1 1. P, AB congruent to BC 1. Given 2. Draw DC 2. Two points determine a segment 3. AD congruent to DC 3. All radii of a circle are congruent 4. DB congruent to DB 4. Reflexive 5. ∆ADB congruent to ∆CDB 5. SSS (1,3,4) 6. LDBA congruent to LDBC 7. LDBC and LDBA are right angles 6. CPCTC 7. If two angles are both supplementary and congruent, then they are rt. L’s Answer Sheet Practice Problem #2 • YES 3x+14=2x+22 • • – X=8 38=38 This means the angles are congruent. Theorem #23 states, “If two angles are both supplementary and congruent, then they are rt. L’s.” The answer is yes because right angles are formed by perpendicular lines. Answer Sheet Practice Problem #3 1. XY congruent to XZ and XQ bisects LYXZ 2. LYXQ congruent to LZXQ 3. LY congruent to LZ 4. ∆YXQ congruent to ∆ ZXQ 5. LXQY congruent to LXQZ 6. LXQY and LXQZ are rt. L’s 7. XQ is perpendicular to YZ 1. Given 2. If a ray bisects an L, then it divides the L into 2 congruent L’s 3. If sides, then angles 4. ASA (1,2,3) 5. CPCTC 6. If two angles are both supplementary and congruent, then they are rt. L’s 7. Rt. L’s are formed by perpendicular lines Answer Sheet Practice Problem #4 1+5 2 And 6+8 2 Answer: (3,7) Works Cited "Chapter 2 Notes." 18 Oct. 2007. 29 May 2008 <home.cvc.org/math/dgeom/Chapter_2_notes/2_8_2.pdf>. Rhoad, Richard, George Milauskas, and Robert Whipple. Geometry for Enjoyment and Challenge. New ed. Evanston: McDougal, Littell & Company, 1997. 180-183.