1. Section 4.3 -A Right Angle Theorem Michael Smertz H Geometry- 8 30 May 2008

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 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.” 2 1 NOTE: You can now assume that whenever two angles form a straight line, they are supplementary.

4. Sample Problems Problem #1 –Given: Ray CD bisects LACB Segment AC is congruent to segment CB –Prove: LCDA and LCDB are right angles Solution A B C D 1. Ray CD bisects LACB 2. AC congruent to CB 3. LACD congruent LBCD 4. CD congruent CD 5. ∆ACD congruent ∆BCD 6. LCDA congruent LCDB 7. LCDA and LCDB are right angles 1.Given 2.Given 3.If a ray bisects an angle, then it divides the angle into 2 congruent angles 4.Reflexive 5.SAS (2.3.4) 6.CPCTC 7. If two angles are both supplementary and congruent, then they are right angles.

5. Sample Problems Problem #2 –Given: DA congruent to DC AB congruent to BC –Prove: DB altitude of AC Solution D A B C E 1.DA congruent to DC 2.AB congruent to BC 3.DB congruent to DB 4.∆DAB congruent to ∆DCB 5.LABE congruent to LCBE 6.EB congruent to EB 7.∆ABE congruent to ∆CBE 8.LAEB congruent to LCEB 9.LAEB and LCEB rt. L’s 10.DB alt. of AC 1.Given 2.Given 3.Reflexive 4.SSS (1,2,3) 5.CPCTC 6.Reflexive 7.SAS (2,5,6) 8.CPCTC 9.If two angles are both supplementary and congruent, then they are rt. L’s 10.An altitude of a ∆ forms right angles with the side to which it is drawn. Note: This is a detour problem.

6. Sample Problems Problem #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. Solution –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. ABC 12 22

7. Practice Problem #1 Given: – P –AB congruent to BC Prove: –LDBC and LDBA are right angles ABC D

8. Practice Problem #2 M O 3x+142x+22 Is M perpendicular to O? Justify your answer.

9. Practice Problem #3 Given: –XY congruent to XZ –XQ bisects LYXZ Prove: –XQ is perpendicular to YZ X Y Q Z

10. 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. (1,6) (5,8)

11. Answer Sheet Practice Problem #1 1.P, AB congruent to BC 2.Draw DC 3.AD congruent to DC 4.DB congruent to DB 5.∆ADB congruent to ∆CDB 6.LDBA congruent to LDBC 7. LDBC and LDBA are right angles 1.Given 2.Two points determine a segment 3.All radii of a circle are congruent 4.Reflexive 5.SSS (1,3,4) 6.CPCTC 7.If two angles are both supplementary and congruent, then they are rt. L’s

12. 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.

13. 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 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 7. XQ is perpendicular to YZ

14. Answer Sheet Practice Problem #4 1+5 2 And 6+8 2 Answer: (3,7)

15. Works Cited "Chapter 2 Notes." 18 Oct. 2007. 29 May 2008. Rhoad, Richard, George Milauskas, and Robert Whipple. Geometry for Enjoyment and Challenge. New ed. Evanston: McDougal, Littell & Company, 1997. 180-183.