GEOMETRY

The 3 undefined terms in geometry

• POINT

• LINE

• PLANE

Exercises - Answers1. Name a point. T, E, L, U, S, Y, or Z2. Name three points that are collinear. T-L-S,

or E-L-U, or Y-L-Z3. Name three points that are non-coplanar. T-

L-Z, T-L-Y, S-L-Y, others.4. Name two rays. Ray LY, others.5. Name two coplanar lines. Line TS and line EU.6. Name two non-coplanar lines.

Line YZ and line EU.

P

E

L

S

U

T

Y

Z

7. If line YZ is perpendicular to line TS, does it mean it is also perpendicular

to line EU? YES.

• Why or why not?

8. Name two pairs of adjacent angles.• Pair 1: • Pair 2: 9. Based on the figure on the left, give an example of:• Acute angle:

• Obtuse angle:

f

e

d

c

b

a

and bac cad and dae eaf

, otherseaf

, othersdae

10. Complementary Angles10. (a.) What is the complement of 40o?• COMPLEMENTARY ANGLES• - Two angles whose sum is 90o.• Answer to 10.(a.):

10.(b.) What is the complement of ? 14 y

90 4050

90 14

90 14

76

y

y

y

11. Supplementary Angles• 11.(a.) Find the supplement of • Supplement:

• SUPPLEMENTARY ANGLES• - Angles whose sum is 180o.

11.(b.) Find the supplement of

78.6180 78.6101.4

56x

180 56

180 56

124

x

x

x

12. Solve for x.

• One full revolution is 360o.

2xo

3xo

5xo

2 5 3 90 360 10 90 360 10 270 27

x x xx

xx

13. Calculate the values of y and z.• 2y and 7x are

VERTICAL ANGLES. Formed when

two lines intersect.• 2y = 7x

7x o

2y o

140o

PCA Theorem:Given two lines Parallel, the Corresponding Angles are congruent. 140 = 7x

140 7140 7

7 7 20

xx

x

2 7

2 7 20

2 1402 1402 2

70

y x

y

yy

y

13. (b.) Solve for y and z.• Angle y and 316 make one cycle. y + 316 = 360• So, y = 44o.

• PAI Theorem:• Given parallel lines, the Alternate Interior

Angles are congruent. z = 58o

z

58

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?

• 2 : 5 : 815squares

• If the sum of the 3 angles was 15 degrees, then the measures would be

• 2 : 5 : 8 15

•___ : ____ : _____ 180• Therefore , the angle measures are 24, 60,

and 96 degrees.

2 ,5 , and 8.

15 ? 180 ? 12

24 60 96

15. What is the value of x?

• CONCEPT:The sum of the measures

of the angles of a triangle is 180 degrees.

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.

17. In a parallelogram PQRS, and . . Calculate

• PARALLELOGRAM- A quadrilateral with 2 pairs of opposite sides parallel.

• Drawings: Be careful in labeling.

56QPR 70QRS ________ and _________.RQP PRQ

Q R

P S

70

56

_________PRQ

110

180 56 110

180 16614

PRQ

________ RQP

Adjacent angles of a parallelogram are SUPPLEMENTARY.

14

110 14

18. In the figure below, x = 60 deg. How much more is the perimeter of the triangle DEF compared to that of triangle ABC?

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

• Area of a square

• Pythagorean Theorem

2A r2 2A side s

2 2 2a b c

20. Daniel has a square piece of paper of side 4 inches. If he rolls up the paper to

make a cylinder, what is the volume of the cylinder formed?

21. A rectangle and a triangle share the same base. If the area of the triangle is 6 times the area of the rectangle, and the height of the rectangle is 4, what is the height of the triangle?

22. In the figure on the right, BCDE is a square and AB = 12. What is the area of square BCDE?

• 30-60-90 Theorem

»What is the measure of side BE?

23. How many circles of radius 10cm can be cut from a rectangular board 1.2m by 0.8m?