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Math Session 3 Geometry

Points, Lines, Planes, Angles

Exercises on: Area, Perimeter, Circumference, Volume

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- 1. GEOMETRY
- 2. The 3 undefined terms in geometry • POINT
- 3. • LINE
- 4. • PLANE
- 5. Exercises - Answers• Name a point. T, E, L, U, S, Y, or Z• Name three points that are collinear. T-L-S, or E-L-U, or Y-L-Z• Name three points that are non-coplanar. T- L-Z, T-L-Y, S-L-Y, others.• Name two rays. Ray LY, others. P• Name two coplanar lines. Line TS and T Eline EU. Y L7. Name two non-coplanar lines. Z Line YZ and line EU. U S
- 6. 7. If line YZ is perpendicular to line TS, does it mean it is also perpendicular to line EU? YES.• Why or why not?
- 7. 8. Name two pairs of adjacent angles. • Pair 1:∠bac and ∠cad • Pair 2: ∠dae and ∠eaf 9. Based on the figure on the left, give an example of: c d • Acute angle:b e ∠eaf , others • Obtuse angle: a ∠dae , others f
- 8. 10. Complementary Angles 10. (a.) What is the complement of 40o? • COMPLEMENTARY ANGLES • - Two angles whose sum is 90o. • Answer to 10.(a.): = 90° − 40° = 50° 10.(b.) What is the complement of ( 14 − y ) °? = 90° − ( 14 − y ) ° = 90° − 14 + y = ( 76 + y ) °
- 9. 11. Supplementary Angles• 11.(a.) Find the supplement of 78.6°• Supplement: = 180° − 78.6° = 101.4°• SUPPLEMENTARY ANGLES• - Angles whose sum is 180o.11.(b.) Find the supplement of ( x + 56 ) ° = 180° − ( x + 56 ) ° = 180° − x − 56° = ( 124 − x ) °
- 10. 12. Solve for x. 5xo 2xo • One full revolution 3x o is 360o. 2 x + 5x + 3x + 90 = 360 10 x + 90 = 360 10 x = 270 x = 27°
- 11. 13. Calculate the values of y and z. • 2y and 7x are 2y o VERTICAL ANGLES. 7x o Formed when two lines intersect. 140o • 2y = 7xPCA Theorem:Given two lines Parallel, the Corresponding Anglesare congruent. 140 = 7x
- 12. 2y = 7 x140 = 7 x 2y = 7 ( 20 )140 7 x = 2y = 140 7 7 20 = x 2y 140 = 2 2 y = 70
- 13. 13. (b.) Solve for y and z. • Angle y and 316 make one cycle. y + 316 = 360 • So, y = 44o. • PAI Theorem: z • Given parallel lines, the Alternate Interior Angles 58 are congruent. z = 58o
- 14. 14. The ratio of the angles of a triangle is 2 : 5 : 8. What are the measures of the angles of the triangle?• What is the sum of the measures of the angles of a triangle?• 180o• How do we calculate proportion? 15 squares •2 : 5 : 8
- 15. • If the sum of the 3 angles was 15 degrees, then the measures would be 2°,5°, and 8.°• 2 : 5 : 8 15 15 × ? = 180 ? = 12 24 : ____ : _____ 180•___ 60 96• Therefore , the angle measures are 24, 60, and 96 degrees.
- 16. 15. What is the value of x?• CONCEPT:The sum of the measures of the angles of a triangle is 180 degrees.
- 17. 16. If the three angles of a quadrilateral were: y, ( 50 + y ), ( y – 75 ), what is the fourth angle? CONCEPT: The sum of the measures of the angles of a quadrilateral is 360 degrees.
- 18. 17. In a parallelogram PQRS, ∠QPR = 56° and ∠QRS = 70°. Calculate . ∠RQP ________ and∠PRQ _________.• PARALLELOGRAM - A quadrilateral with 2 pairs of opposite sides parallel.• Drawings: Be careful in labeling.
- 19. Q R 110° 14° 110° ∠RQP ________ 70° Adjacent angles of a parallelogram are 56° SUPPLEMENTARY. 14° ∠PRQ _________P S ∠PRQ = 180 − ( 56 + 110 ) = 180 − 166 = 14°
- 20. 18. In the figure below, x = 60 deg. Howmuch more is the perimeter of the triangleDEF compared to that of triangle ABC?
- 21. 19. Square ABCD is inscribed in a circle.The square has a side of length 6cm. What is the area of the shaded region? • Area of a circle A =πr 2 • Area of a square A = side = s 2 2 • Pythagorean Theorem a +b =c 2 2 2
- 22. 20. Daniel has a square piece of paper of side 4 inches. If he rolls up the paper tomake a cylinder, what is the volume of the cylinder formed?
- 23. 21. A rectangle and a triangle share thesame base. If the area of the triangle is 6times the area of the rectangle, and theheight of the rectangle is 4, what is theheight of the triangle?
- 24. 22. In the figure on the right, BCDE is asquare and AB = 12. What is the area ofsquare BCDE? • 30-60-90 Theorem »What is the measure of side BE?
- 25. 23. How many circles of radius10cm can be cut from arectangular board 1.2m by 0.8m?

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