m103

1. A D V A N C E D
E N H A N C E M E N T
P R O G R A M F O R
T E A C H E R S

2. Perpendicular
Vocabulary Check
Match each word to its meaning.
Equal
Bisect
Adjacent
Right-angle
Parallel
Interior angle
Diagonal
next to
lines that have the same distance
between them and will never meet
a line or plane at an angle of 90° to
another line or plane
the same as
a straight line joining two opposite
corners of a straight-sided shape,
such as a square or rectangle
divide something into two equal
parts
an angle of 90°
an angle inside a

3. 1. Two angles are supplementary. If the measure
of one of the angles is represented by 8x-4 and
the other one is represented by 2x + 14, what is
the measurement of the larger side?
a. 17
b.132
c. 48
d.156
Sol’n: (8x- 4) + (2x + 14) = 180
10x=170
x= 17

4. 2. Which of the following statement/s is/are
TRUE when two coplanar lines are being cut by
a transversal line?
i. Vertical angles are congruent
ii. Alternate Exterior angles are congruent
iii. Corresponding angles are congruent
iv. Same- side interior angles are congruent
a. i only
b. ii only
c. i , ii, iii only
d. i, ii, iii and iv

5. 3. What would be the measurement of the
exterior angle adjacent to the base angle of
an isosceles right triangle?
a.90°
b.45 °
c.145 °
d.135 °
180 ° − 45°=135 °

6. 4. In the diagram, B is a point of
tangency. Find the radius r of C.
a.93 ft.
b. 33 ft.
c.25 ft
d.39 ft
AC2 = BC2 + AB2
(r + 50)2 = r2 + 802
r2 + 100r + 2500 = r2 + 6400
100r = 3900
r = 39 ft .

7. 5. RS is tangent to C at S and
RT is tangent to C at T. Find the
value of x.
a.15
b.8
c.17
d.19
Sol’n: Properties of Tangent Segment:
Tangent segments from a common
external point are congruent.
RS = RT
28 = 3x + 4
8 = x

8. 6. Find the length
of the midsegment
RT.
a. 10
b. 45. 5
c. 3
d. 25
Answer: Midsegment Theorem
for trapezoids : Midsegment is parallel
to each base and its length is one half
the sum of the lengths of the bases.
MN=
1
2
(𝑏1 + 𝑏2)
MN=
1
2
7 + 13 = 10

9. 7. Find the measure of ∠ A in the
diagram.
C 136°
E
142°
Solution
The polygon has 6 sides, so the sum of the measures of the interior
angles is:
(n — 2) • 180° = (6 — 2) • 180° = 4 • 180° = 720°.
Add the measures of the interior angles and set the sum equal to 720°.
136° + 136° + 88° + 142° + 105° +m∠A = 720°607° + m ∠ A = 720°
m ∠ A = 113°
ANSWER } The measure of ∠ A is 113°.
136°
B A
88°
D
a. 121°
b. 45 °
c. 113 °
d. 171 °

10. 8. The following statements are
true except for:
a. All linear pairs are supplementary angle but not all supplementary
angles are linear pair
b. The diagonals of a rhombus are perpendicular
c. If all angles of quadrilateral are congruent then it is a parallelogram
d. All parallelogram are quadrilaterals
If all angles of quadrilateral are
congruent, then it is a rectangle

11. 9. Find the volume of the cylinder.
a. 401.92 𝑚3
b. 132 𝑚3
c. 32 𝑚3
d.256 𝑚3
The volume of a right circular cylinder is the product of
the area of the base and the altitude.
𝑉 = 𝜋𝑟2
ℎ = 3.14 x 16 x 8 = 401.92 𝑚3

12. 10. A square has an area of 400 sq. cm.
What is the perimeter?
a. 20 cm
b. 80 cm
c. 100 cm
d. 40 cm
A= 𝑠2
s= 20 cm
P= 4s = 80 cm

13. 11. Find the measure of ∠ G in the
diagram.
Solution
The polygon has 4 sides, so the sum of the measures of the interior
angles is:
(n — 2) • 180° = (4 — 2) • 180° = 4 • 180° = 360°.
103°+33°+58+x = 360°.
x=166°
a. 116°
b. 166°
c. 161°
d. 216 °
G
H
J
K
103°
33 °
58 °
166°

14. 12.ABCD is a rectangle, what is
x?
Solution
Diagonals are congruent
4x+ 2 = 5x – 1
x = 3
a. 1
b. 2
c. 3
d. 4
JL = 4X + 2
MK = 5X - 1

15. 12.The figure below is a
parallelogram, what is the value
of unknown variable y?
Solution
Opposite sides are
congruent
x = 12
7y-3 = 3(12)+3
7y = 42
y = 6
a. 4
b. 5
c. 6
d. 7
7y-3
x
3x+3
12

16. 13.The figure below is a parallelogram,
what is the measurement of the longer
diagonal?
Solution
Diagonals bisect each
other
3x+ 5 = 4x + 1
x = 4
y-1 = x
y - 1= 4
y = 5
a. 4
b. 17
c. 34
d. 5
y-1
4x+1
x
3x + 5

17. 14.In circle P, find the value/s of x
Solution
Tangent segments from
the same point are ≅
𝒙𝟐
= 𝟗
𝒙 = ±𝟑
a. 4
b. -3
c. 3
d. ±3

18. 15.Find the volume of the pyramid
Solution
V =
𝟏
𝟑
𝒃𝒉
b = 36 ft
V=
𝟏
𝟑
𝟑𝟔𝒙𝟒
V 48 cu. ft.
a. 420 cu. Ft.
b. 720 cu.ft.
c. 48 cu. ft.
d. 84 𝑐𝑢. 𝑓𝑡.

19. 16. You own a Rubik’s cube with a
volume of 343 𝑐𝑚3
. What is the length
of the cube?
Solution
V =𝒔𝟑
343= 𝒔𝟑
𝟑
𝟑𝟒𝟑 =
𝟑
𝒔𝟑 = 𝟕 𝒄𝒎
a. 14 cm
b. 41 cm
c. 7 cm
d. 49𝑐𝑚

20. 17. In the diagram of QR = ST . Find
CU.
a. 2
b. 4
c. 6
d. 8
CU = CV
2x = 5x – 9
x = 3
So, CU = 2x = 2(3) = 6.

21. Statement
1.
−3 + 3 = 0
2.
9 (−1) = −9
3.
(−6 + 1) + 7 = −6 + (1 + 7)
4.
(0.3) (−3) = (−3) (0.3)
5.
5(𝑥 + 3) = 5𝑥 + 5 3
6.
−8 + 0 = −8
1 (𝑎𝑏) = 𝑎𝑏
8.
(𝑥 + 2) + 3 = 𝑥 + (2 + 3)
9.
(−6) 𝑦 = 𝑦 (−6)
10.
9 + (−1) = (−1) + 9
11.
(−2 ∗ 5) 4 = −2 (5 ∗ 4)
12𝑥 = 𝟏2𝑥
13.
6 ∗
1
6
= 1
14.
3(4 + −7) = 3(4) + 3(−7)
15.
(𝑥 ∗ 7) 0.5 = 𝑥 (7 ∗ 0.5)
16.
𝑦 + −4 = −4 + 𝑦
Additive Inverse Property
Closure Property
Commutative Property of Addition
Commutative Property of Multiplication
Distributive Property
Identity Property of Addition
Multiplicative Identity Property
Associative Property of Addition
Commutative Property of Multiplication
Commutative Property of Addition
Associative Property of Multiplication
Reflexive Property of Equality
Multiplicative Inverse Property
Distributive Property of Equality
Associative Property of Multiplication
Commutative Property of Addition

24. If the volume of a cube is 216 cubic units, then what is its surface area
units?
Solution:
The volume of a cube is given by the
formula V = s3, where V is the volume, and s is
the length of each side. We can set V to 216
and then solve for s.
That is, 𝑠 =
3
216 then s = 6
S.A= 6𝑠2
= 6 62
= 216 𝑠𝑞𝑢𝑎𝑟𝑒 𝑢𝑛𝑖𝑡𝑠
a. 36
b. 64
c. 216
d. 54

25. A right rectangular prism has dimensions of 3 x 5 x 20. What is its surface area?
Solution:
S.A= 2 (l x h) + 2 (l x w) + 2 (w x h)
S.A = 2 (3 x 5 ) + 2 ( 5 x 20) + 2 ( 3 x 20 )
= 350
a. 175
b. 300
c. 112
d. 350

26. SOME DEFINITIONS
Angles- Two rays emerging from a single point makes an
angle.
Linear Pair- made by such rays are called linear pairs.
Vertically opposite angles- If there are two lines crossing
from one particular point then the opposite angles made in
such a condition are equals.
Alternate Exterior Angles Theorem-When two parallel
lines are cut by a transversal then resulting alternate
exterior angles are congruent.

27. SOME DEFINITIONS
Alternate Interior Angles Theorem - When two parallel
lines are cut by a transversal then resulting alternate interior
angles are congruent.
Right Angles Theorem- If two angles are both supplement
and congruent then they are right angles.
Same-Side Interior Angles Theorem-
If two parallel lines are cut by a transversal,
then the interior angles on the same side
of the transversal are supplementary.

28. SOME DEFINITIONS
Vertical Angles Theorem- Angles that are
opposite to each other and are formed by
two intersecting lines are congruent.
Circle Theorems 1- Angles in the same
segment and on the same chord are always
equal.

29. SOME DEFINITIONS
Circle Theorems 3
The angle at the center of a circle is twice
the angle at the circumference.
The angle between the tangent and the side
of the triangle is equal to the interior opposite
angle.

30. Parallelogram Theorems
If both pairs of opposite sides of a
quadrilateral are congruent, then the
quadrilateral is a parallelogram.
If both pairs of opposite angles of a
quadrilateral are congruent, then the
quadrilateral is a parallelogram.
If one pair of opposite sides of a quadrilateral
is both parallel and congruent, then the
quadrilateral is a parallelogram.