College Entrance Test Review Math Session 3 Geometry Points, Lines, Planes, Angles Exercises on: Area, Perimeter, Circumference, Volume

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
E
line EU. Y
L
7. 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 = 7x
PCA Theorem:
Given two lines Parallel, the Corresponding Angles
are congruent.
 140 = 7x

12. 2y = 7 x
140 = 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. How
much more is the perimeter of the triangle
DEF 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 to
make a cylinder, what is the volume of the
cylinder formed?

23. 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?

24. 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?

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