1. AIMS Math Prep Circles & Geometry Potpourri March 27, 2009 Friday

2. Period 1 Find the measurement of x. (Use formulas above) 1. 2. 3. 4. 5. x x x x x 4 3 or a*d = b*c 8 90° 130° 80° 100° 10 3 5 4 * x = ___ * ____ x = ½ (__ + 130) 5x = 10*__ find the missing arc 1 st then x=½(arc) x = ½ (___)

3. Period 1 ANSWERS Find the measurement of x. 1. 2. 3. 4. 5. x x x x x 4 3 or a*d = b*c 8 90° 130° 80° 100° 10 3 5 8*3=4x x = 6 x = ½ (90 + 130) = 110 5x = 10*3 x = 6 find the missing arc (180) 1 st then x=½(180) = 90 x = ½ (80) = 40

4. Period 2 Find x. Remember that a circle has 360°. 1. 2. 3. 4. 5. x x x x x 160° 70° 100° 60° 120° 100° 150° 80° 220° x = ½ ( 100 – __) x = ½ (220 - __) x = ½ ( ___- 70) x = ½ (__-100) x = ½ (___ - __)

5. Period 2 ANSWERS Find x. Remember that a circle has 360°. 1. 2. 3. 4. 5. x x x x x 160° 70° 100° 60° 120° 100° 150° 80° 220° x = ½ ( 100 – 30) = 35 x =½(220-140)=40 x = ½ ( 160- 70) = 45 x = ½ (260-100) =80 x = ½ (120-60) = 30

6. Period 3 Sketch or define the different types of triangles. 1.Scalene 2.Isosceles 3.Equilateral 4.Equiangular 5.Right Triangle 6.Acute Triangle 7.Obtuse Triangle

7. Period 3 ANSWERS Sketch or define the different types of triangles. 1.Scalene – triangle with no equal sides 2.Isosceles – triangle with 2 or more equal sides 3.Equilateral – triangle with 3 equal sides 4.Equiangular – triangle with 3 equal angles (all = 60°) 5.Right Triangle – triangle with a 90° (right) angle 6.Acute Triangle – triangle with all angles < 90° 7.Obtuse Triangle – triangle with one angle > 90°

8. Period 4 1.5, 5, 8 2.6, 3, 8 3.4, 4, 4 4.6, 10, 8 5.2, 4, 3 6.2, 1, 1 7.3, 4, 8 8.5, 12, 13 Check to see if each the sides given can make a triangle. (A simple trick is to cover one number and ask yourself if the two exposed numbers have a sum larger than the one covered.) Example: Given the sides of length 5, 8, and 4. Cover 5, 8 + 4 > 5. Cover 8, 5 + 4 > 8. Cover 4, 5 + 8 > 4. It’s a triangle.

9. Period 4 ANSWERS 1.5, 5, 8 Yes…5 + 5 > 8, 5 + 8 > 5, 8 + 5 > 5 2.6, 3, 8 Yes…6 + 3 > 8, 3 + 8 > 6, 8 + 6 > 3 3.4, 4, 4 Yes…4 + 4 > 4 4.6, 10, 8 Yes… 6 + 10 > 8, 10 + 8 > 6, 8 + 6 > 10 5.2, 4, 3 Yes…2 + 4 > 3, 4 + 3 > 2, 3 + 2 > 4 6.2, 1, 1 No, 1 + 1 is not greater than 2 7.3, 4, 8 No, 3 + 4 is not greater than 8 8.5, 12, 13 Yes, 5 + 12 > 13, 12 + 13 > 13, 5 + 13 > 12 Check to see if each the sides given can make a triangle. (A simple trick is to cover one number and ask yourself if the two exposed numbers have a sum larger than the one covered.) Example: Given the sides of length 5, 8, and 4. Cover 5, 8 + 4 > 5. Cover 8, 5 + 4 > 8. Cover 4, 5 + 8 > 4. It’s a triangle.

10. Period 5 What type of triangle does each form? If it is a triangle, square each side. The largest square larger than the other two squares added together…it’s an obtuse triangle. Is the largest square smaller than the sum of the other two? Then it’s an acute triangle. If the largest square equals the sum of the other two, then it’s a right triangle. Example: Given the sides of length 5, 8, and 4. Square the numbers…25, 64, 16. The largest 64, is larger than 25 + 16 so it’s an obtuse triangle. 1.5, 5, 8 2.6, 9, 8 3.4, 4, 4 4.6, 10, 8 5.2, 4, 3 6.5, 12, 13 7.6, 5, 4 8.3, 4, 5

11. Period 5 - ANSWERS What type of triangle does each form? If it is a triangle, square each side. The largest square larger than the other two squares added together…it’s an obtuse triangle. Is the largest square smaller than the sum of the other two? Then it’s an acute triangle. If the largest square equals the sum of the other two, then it’s a right triangle. Example: Given the sides of length 5, 8, and 4. Square the numbers…25, 64, 16. The largest 64, is larger than 25 + 16 so it’s an obtuse triangle. 1.5, 5, 8 Obtuse 64 > 25 + 25 2.6, 9, 8 Acute 81 < 36 + 64 3.4, 4, 4 Acute, 16 < 16 + 16 4.6, 10, 8 Right, 100 = 36 + 64 5.2, 4, 3 Acute, 16 > 4 + 9 6.5, 12, 13 Right, 169 = 144 + 25 7.6, 5, 4 Obtuse, 36 > 25 + 16 8.3, 4, 5 Right, 25 > 16 + 9

12. Period 6 Determine the principle of congruency used to prove the two triangles congruent. (SAS, AAS, ASA, SSS, or HL) (Hint: Mark each arc (angle congruency) with “A” and tic mark (side congruency) with “S” then read the letters written. 1. Example 2. 3. 4.  1   2;  3   4 5. S S S S S S SSS

13. Period 6 ANSWERS Determine the principle of congruency used to prove the two triangles congruent. (SAS, AAS, ASA, SSS, or HL) (Hint: Mark each arc (angle congruency) with “A” and tic mark (side congruency) with “S” then read the letters written. 1. 2. 3. 4.  1   2;  3   4 5. S SS S S S S S S S S SS S A SSSSS S S SAA A A A A A A A A A A A A A AA AA A