This document contains a multiple choice mathematics test on geometry concepts. It has 52 questions testing topics like lines, angles, triangles, quadrilaterals, circles, transformations, and coordinate geometry. For each question, the student is to choose the best answer among 4 options labeled a, b, c, or d.

1. Republic of the Philippines
Department of Education
Region IV-A CALABARZON
MATHEMATICS III
(Geometry)
MULTIPLE CHOICE
Direction: Encircle the letter of the best answer.
1. Which undefined term has no width or thickness but has infinite length?
a. point c. plane
b. line d. space
2. In , which point is the vertex of the angle?
a. B c. X
b. O d. any of the three points
3. If a triangle has no two sides congruent, then the triangle is
a. scalene c. isosceles
b. equilateral d. equiangular
4. What is the measure of an acute angle?
a. Greater than 00
but less than 900
b. Greater than 900
but less than 1800
c. Exactly 900
d. Exactly 1800
5. Which of the following is a convex polygon?
a. c.
b. d.
6. If the measures of three consecutive angles of a quadrilateral are 600
, 1200
and 1100
respectively,
what is the measure of the fourth angle?
a. 1000
c. 800
b. 900
d. 700
7. Which condition must be true for a cylinder to be right?
a. The altitude and height are equal.
b. The axis is parallel to the side.
c. The axis is perpendicular to the base.
d. The axis makes an angle with the base.
8. What is the length of a rectangle if its width is 10 cm and its perimeter is 44 cm?
a. 8 cm c. 12 cm
b. 9 cm d. 14 cm
9. If the side of a cube is 10 cm, what is the total surface area?
a. 60 cm2
c. 600 cm2
b. 100 cm2
d. 1000 cm2
10. If P is between Q and R, and PQ = 9 and QR = 11, then
a. PQ – QR = 2 c. PR = 13
b. PR = 2 d. PR = 20
11. If the measure of an angle is 200
less than 4 times the measure of its supplement, then its measure
is
a. 1200
c. 1400
b. 1300
d. 1500

2. E
D
C
B
A
12. Given that two parallel lines p and q are cut by the transversal t and the illustration below, then
i.
ii.
iii. are supplementary
a. i and ii only c. ii and iii only
b. i and iii only d. i, ii and iii
13. Use the information given and the figure to find .
Given:
a. 420
b. 520
c. 620
d. 720
14. If and , then the value of x is
a. 200
b. 300
c. 400
d. 500
15. Which of the following can be the measures of the sides of a triangle?
a. 4, 9, 11 c. 5, 7, 12
b. 3, 6, 10 d. 2, 2, 4
16. If two angles and an included side of one triangle are congruent respectively to two angles and an
included side of another triangle, then the two triangles are congruent. By what congruence
theorem is this possible?
a. ASA c. SAS
b. SSS d. SAA
17. Given , which of the correspondences are congruences?
i.
ii.
iii.
a. i and ii only c. i and iii only
b. ii and iii only d. i, ii and iii
18. In the pair of triangles sketched below, like markings indicate congruent parts. Supply the
additional information to enable us to apply the specified congruence postulate or theorem.
t
8 7
6
5
4 3
2
1 p
q

3. T
H
S
A
M
6
5
4
3
2
1
E
R
A
C
R
Q
P
O
S
For SAS
a. c.
b. d. none of these
19. In the figure, , which other segments are congruent by
CPCTC?
a.
b.
c.
d.
20. Which group of congruent pairs show that by SAS congruence? a
a. c.
b. d.
21. If two angles of a triangle are not congruent, then the sides opposite these two angles are not
congruent and the longer side is opposite the _________________.
a. smaller angle c. larger angle
b. vertex angle d. base angle
22. Given , which of the following is true by the transitivity property of
inequality? a
a. c.
b. d.
23. Which of the following pairs of ratios form a proportion?
a. 4:5 and 5:4 c. 3:2 and 5:4
b. :2 and 3:12 d. 2:7 and 6:14
24. If , then the length of is derived by
a. Multiplying
b. Multiplying and and dividing the result by
c. Multiplying and and dividing the result by
d. Dividing by the product of and
25. Which of the following proportions is true if in ?
a. c.
b. d.
26. Which of the following conditions are necessary for two triangles to be similar?
i. Corresponding angles are congruent
ii. Corresponding sides are congruent
iii. Corresponding sides are proportional
a. i and ii b. ii and iii c. i and iii d. all of these
Q
P
O
N
M

4. S
E B
U
C
Y A
T
S
F E D
C
B
A
27. If , what is the value of x?
a. 18
b. 20
c. 27
d. 36
28. If , ST = 6 cm, AS = 4 cm and SE = 3 cm, how long is ?
a. 2 cm
b. 4 cm
c. 5 cm
d. 6 cm
29. The geometric mean between 2 and 8 is
a. 3 c. 5
b. 4 d. 6
30. In a proportion, the ratios are
a. constant c. equivalent
b. increasing d. decreasing
31. In a proportion, adding the numerator and denominator of the second ratio to the numerator and
denominator of the first ratio, respectively does not change the proportion. This statement is
a. always true c. sometimes true
b. never true d. none of these
32. Which of the conditions below justifies that quadrilateral CUBE is a parallelogram?
a.
b.
c.
d.
33. In rhombus STAY, . What is ?
a. 12o
b. 51o
c. 78o
d. 102o
34. In trapezoid ACDF, is the median. If and , what is ?
a. 11.75
b. 23.5
c. 26
d. 33
35. is a rectangle with diagonals and intersect at Q. If GE = 12 and GA = 16, what is
EQ?
a. 10 c. 15
b. 20 d. 25
36. Which of these is not a property of all trapezoids?
a. Diagonals are congruent.
b. Two pairs of opposite sides are parallel.
c. Diagonals intersect.
d. None of these
37. Which of the following is not sufficient to prove that a parallelogram is a rectangle?
a. One angle is a right angle.
b. Two adjacent angles are congruent.
c. The diagonals are congruent.
d. All angles are congruent.
x
12 8
Q
P
R
9
6
4
C
A
B
S
E
A
T
N

5. 38. The length of the median of a trapezoid is the __________ of the length of the bases.
a. average c. sum
b. difference d. square
39. Which point lies at the greatest distance from the origin?
a. (0, 9) c. (5, 8)
b. (-2, 9) d. (-6, -7)
40. The locus of the midpoints of all 6-cm chords in a circle of 5 cm is
a. a point c. a line
b. a circle d. a plane
41. For □ROSE, a pair of consecutive sides is
i.
ii.
iii.
a. I and iii only c. ii and iii only
b. I and iii only d. I, ii and iii
Answer: B
Solution:
RO and OS
𝑅𝑂
̅̅̅̅ 𝑎𝑛𝑑 𝐸𝑅
𝑂𝑆
̅̅̅̅ 𝑎𝑛𝑑 𝑆𝐸
̅̅̅̅
𝑆𝐸
̅̅̅̅ 𝑎𝑛𝑑 𝐸𝑅
̅̅̅̅
42. If a parallelogram has a pair of congruent adjacent sides, then it is a
a. rhombus c. trapezium
b. rectangle d. trapezoid
Answer: A
Solution:
43. In circle O, the is
a. 20o
b. 70o
c. 110o
d. 160o
Answer: Insufficient given
Solution:
44. Equilateral triangle ABC is inscribed in circle O. Find the degree measure of .
a. 60o
c. 75o
b. 90o
d. 120o
Answer: D
Solution:
200
E
D C
B
A
O
S
O
E
R
O
A
B
C
𝑚∠𝐴𝑂𝐶 𝑖𝑠 120 ° 𝑏𝑒𝑐𝑎𝑢𝑠𝑒 𝑡ℎ𝑒 𝑐𝑖𝑟𝑐𝑙𝑒 𝑖𝑠 𝑑𝑖𝑣𝑖𝑑𝑒𝑑 𝑖𝑛𝑡𝑜 𝑡ℎ𝑟𝑒𝑒 𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡𝑠 𝑎𝑟𝑐
𝑤ℎ𝑖𝑐 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑠 120°. 𝑡ℎ𝑖𝑠 𝑎𝑟𝑐 𝑖𝑠 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡𝑒𝑑 𝑎𝑟𝑐 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑒𝑛𝑡𝑟𝑎𝑙 𝑎𝑛𝑔𝑙𝑒 𝑎𝑛𝑑 𝑡ℎ𝑒
𝑚𝑒𝑎𝑠𝑢𝑟𝑒 𝑜𝑓 𝑐𝑒𝑛𝑡𝑟𝑎𝑙 𝑎𝑛𝑔𝑙𝑒 𝑖𝑠 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 𝑡ℎ𝑒 𝑚𝑒𝑎𝑠𝑢𝑟𝑒 𝑜𝑓 𝑖𝑡𝑠 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡𝑒𝑑 𝑎𝑟𝑐𝑠
𝑡ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒 𝑚∠𝐴𝑂𝐶 𝑖𝑠 120°

6. 45. is inscribed in circle Y. If the m and , what is the measure of
?
a. 45o
c. 55o
b. 80o
d. 120o
Answer: A
Solution:
𝑚∠𝑆𝐴𝑀 =
1
2
𝑚 𝑆𝑀
̅̅̅̅
=
1
2
(90)
= 45°
46. is a segment tangent to circle R at point S. If the radius of the circle R is 5 cm, and PS = 12
cm, find .
a. 8 cm
b. 10 cm
c. 11 cm
d. 13 cm
Answer: D
Solution:
𝑃𝑄
̅̅̅̅ = √𝑚𝑅𝑆2
̅̅̅̅̅ + 𝑀𝑆𝑃
̅̅̅̅
= √52+122
= √25 + 144
= √169
= 13 cm
47. and are tangent to circle K. If the measure of is of the circle, then the measure of
is
a. 60o
b. 120o
b. 180o
d. 240o
Answer: A
Solution:
𝑚∠𝐿 =
1
2
(240° − 120°)
𝑚∠𝐿 =
1
2
(120°)
𝑚∠𝐿 = 60°
48. and are chords of a circle O. if , and the measure of , find the
measure of .
a. 80o
b. 100o
c. 120o ●
d. 140o
Answer:B
Solution:
80 =
1
2
(60 + 𝑥 )
Q
P
S
R
S P
R
Q
O
1
Y
.
S
A
M
90
160
110

7. 160 = 60 + x
160 - 60 = x
100 = x
49. The intersection of the lines 2x – 6y = 2 and 3x + 5y = 17 is
a. (1, 4) c. (4, 1)
b. (-4, 1) d. (-1, -4)
Answer: C
Solution:
2x – 6y = 2
3x + 5y = 17
6x - 18y = 6
6x + 10y = 34
Subtracting the two equations give
- 28y = -28
y = 1; substitute to equation 1
2x - 6 (1) = 2
2x = 2 + 6
2x = 8
x = 4
Therefore the point of intersection is ( 4, 1)
50. The equation of the line parallel to 2x – y = 4 and passing through (- 1, 4) is
a. 2x + y = 6 c. 2x + y = -6
b. 2x – y = -6 d. 2x – y = 6
Answer: B
Solution:
2x - y = 4
- y = - 2x + 4
y = 2x - 4
m = 2 ; ( -1, 4)
y - y1 = m ( x - x1)
y - 4 = 2 ( x + 1)
y - 4 = 2x + 2
2x - y = - 6
51. How long is the segment joining the points (-3, 1) and (3, -2)?
a. c.
b. d.
Answer : D
Solution:
d = √(−2 − 1)2 + (3 + 3)2
= √(−3)2 + (6)2
= √9 + 36
= √45
= 3√5
52. Find the center and radius of the circle whose equation is 4x2
+ 4y2
– 8x + 4y + 1 = 0.
a. c.
b. d.
Answer : B

8. Solution:
4x2 + 4y2– 8x + 4y + 1 = 0.
4
x2
+ y2
– 2x + y = −
1
4
x2
– 2x + 1 + y2
+ y +
1
4
= −
1
4
+
1
4
+ 1
(𝑥 − 1)2
+ (𝑦 +
1
2
)2
= 1
𝐶 (1, −
1
2
) , 𝑟 = 1
53. Find the equation of the circle of radius 4 and center at (3, -2).
a. (x – 3)2
+ (y – 2)2
= 16 c. (x – 3)2
+ (y + 2)2
= 16
b. (x + 3)2
+ (y + 2)2
= 16 d. (x + 3)2
+ (y – 2)2
= 16
Answer : C
Solution:
(x –h)2
+ (y - k)2
= r2
(x – 3)2
+ (y + 2)2
= 42
(x – 3)2
+ (y + 2)2
= 16
54. Find the slope of the line that passes through (5, 5) and (-5, 8).
a. 0 c.
b. d.
Answer : C
Solution:
m =
𝑦2−𝑦1
𝑥2−𝑥1
=
8−5
−5−5
m =
3
−10
55. Write an equation in slope-intercept form of the line that is parallel to and has y-intercept
6.
a. c.
b. d.
Answer : A
Solution:
m =
1
2
; b = 6
y = mx + b
y =
1
2
x + 6
56. Find the slope and y-intercept of the line whose equation is .
a. c.
b. d.
Answer : D
Solution:
y = mx + b
y =
3
4
𝑥 − 3

9. m =
3
4
; b = -3
57. A line segment has one endpoint at (4, -2) and its midpoint at (3, -6). What are the coordinates of
the other end point?
a. (2, -8) c. (-2, -4)
b. (2, -10) d. (-2, -10)
Answer : B ( 2, -10)
Solution:
M = (
𝑥1+𝑥2
2
,
𝑦1+𝑦2
2
)
3
1
=
𝑥1+𝑥2
2
4 + 𝑥2 = 6
𝑥2 = 6 - 4
𝑥2 = 2
𝑦1+𝑦2
2
= −6
-2 + 𝑦2 = -12
𝑦2 = - 12 + 2
𝑦2 = - 10
Coordinates of the point is ( 2, -10)
58. How long is the segment joining the points whose coordinates are (-2, -5) and (2, -3)?
a. 2 c.
b. 5 d.
Answer : C
Solution:
d = √(−3 + 5)2 + (2 + 2)2
= √(2)2 + (4)2
= √4 + 16
= √20
= 2√5
59. An equation of the line with slope -4 and passing through the point P(3, -5) is
a. c.
b. d.
Answer : A
Solution:
m = 4; ( 3, -5 )
y - y1 = m ( x - x1)
y + 5 = -4 ( x - 3)
y + 5 = -4x + 12
4x + y + 5 - 12 = 0
4x + y - 7 = 0
60. An equation of the line parallel to the x-axis and passing through (5, 3) is
a. c.
b. d.
Answer : B
Solution:
m = 0
y - y1 = m ( x - x1)
y - 3 = 0 ( x - 5 )
y = 3