Mathematics 9

1. REVIEW
YOU ARE PERFECTLY MATCHED!!!
Match MATH WORD from Column A to its
corresponding DEFINITION in Column B.

2. C O L U M N A C O L U M N B
_______1. Parallelogram
_______2. Parallel Lines
_______3. CPCTC
_______4. Midpoint
_______5. SAS Congruence
Postulate
a) The point on the line segment that divides
the same segment in two congruent parts.
b) Two or more coplanar lines that never
intersect.
c) A quadrilateral in which two pairs of
opposite sides are parallel.
d) It states that if two sides and the included
angle of one triangle are equal to the
corresponding sides and the included angle
of another triangle, the triangles are
congruent.
e) Corresponding parts of congruent
triangles are congruent.
C
B
E
A
E

3. H O N E S T Y S T R E E T
H U M I L I T Y S T R E E T
In Mathematics,
particularly in
Geometry, there is a
certain theorem that
relates these roads,
that is, the Triangle
Midline Theorem.

4. In a triangle, a segment can be formed
by joining the midpoints of any of its two
sides.
This segment is called the
midline or midsegment of
the triangle.

5. TRIANGLE MIDLINE
THEOREM

6. TRIANGLE MIDLINE THEOREM
The Triangle Midline Theorem states that
the segment that joins the midpoints of two
sides of a triangle is parallel to the third side
and is half of its length.

7. B
A C
M L
The Triangle Midline Theorem states
that 𝑴𝑳 is parallel to 𝑨𝑪 and the length
of 𝑴𝑳 is half the length of 𝑨𝑪
Consider △ 𝐴𝐵𝐶 in which
𝑴𝑳 cuts 𝑩𝑨 and 𝑩𝑪 at their midpoints.
𝑴𝑳 ∥ 𝑨𝑪
𝑴𝑳 =
𝟏
𝟐
𝑨𝑪

9. _____________1. Using the roads in the neighborhood in
Barangay LNHS, what road is parallel to Humility Street?
_____________2. If the length of Honesty Street bounded by its
intersections with Peace Street and Love Street is 1km, what do
you think is the length of Humility Street?
_____________3. If the length of Humility Street from its
intersections with Peace Street to its intersection with Love
Street is 850m, can you tell the length of Honesty Street?

10. Use the triangle below to solve for the missing parts.
Given: △𝑃𝑆𝐺
𝑨𝑰 cuts 𝑷𝑺 and 𝑮𝑺 as their midpoints
A. If 𝐴𝐼 = 48𝑐𝑚, find 𝑃𝐺.
B. If PG = 2𝑥 + 8 and AI = 11,
what is x?
C. If AI = 3𝑥 − 4 and
PG = 5𝑥 + 1, find x?

11. THEOREMS ON
TRAPEZOID

12. TRAPEZOID
A trapezoid is a quadrilateral with exactly
one pair of parallel sides.
The parallel sides of a trapezoid are called
bases while the non-parallel sides are
called legs.
.

13. 𝑩𝑪 and 𝑨𝑫 are the
bases since 𝑩𝑪 ∥ 𝑨𝑫
𝑨𝑩 and 𝑫𝑪 are the legs
∠𝑩 and ∠𝑪 are base
angles with respect to 𝑩𝑪.
∠𝑨 and ∠𝑫 are base angles
with respect to 𝑨𝑫.

14. M S
A diagonal is a segment that
joins two non-adjacent vertices
of a trapezoid.
Moreover, the median or
midsegment is the segment
that connects the midpoints of
the two legs of the trapezoid.
𝑩𝑫 and 𝑪𝑨 are the diagonals
𝑴𝑺 is the median or midsegment.

15. Midsegment Theorem of Trapezoid
The median/midsegment
of a trapezoid is parallel
to each of the bases and
its length is half the sum
of the lengths of the two
bases

16. Midsegment Theorem of Trapezoid
The median/midsegment of a
trapezoid is parallel to each of the
bases and its length is half the sum
of the lengths of the two bases.
𝑴𝑺 ∥ 𝑩𝑪 ∥ 𝑨𝑫
𝑴𝒆𝒅𝒊𝒂𝒏 =
𝟏
𝟐
(𝑩𝒂𝒔𝒆𝟏 + 𝑩𝒂𝒔𝒆𝟐)
𝑴𝑺 =
𝟏
𝟐
(𝑩𝑪 + 𝑨𝑫)

17. Proof of Midsegment Theorem of Trapezoid

18. Given: 𝑻𝒓𝒂𝒑𝒆𝒛𝒐𝒊𝒅 𝑆𝐴𝑀𝑅 𝒘𝒊𝒕𝒉 𝒎𝒆𝒅𝒊𝒂𝒏 𝑯𝑰
A. If 𝐴𝑀 = 5, and SR =
13, what is 𝐻𝐼?
B. If AM = 𝑥 + 2, SR =
10 and HI = 9, what is x?
C. If AM = 2𝑦 − 5,
SR = 𝑦 + 8 and HI =
15, what is y?

19. ISOSCELES TRAPEZOID
A special type of trapezoid in which two
pairs of its angles are congruent.
An isosceles trapezoid is a trapezoid in
which the legs are congruent and the
base angles are congruent.

20. Theorems related to isosceles trapezoid
1. The base angles of an isosceles trapezoid
are congruent.
2. Opposite angles of an isosceles trapezoid
are supplementary.
3. The diagonals of an isosceles trapezoid
are congruent.

21. 1. The base angles of an isosceles trapezoid
are congruent. A B
C
D
With respect to base 𝑨𝑩, ∠𝑨 ≅ ∠𝑩
With respect to base 𝑫𝑪, ∠𝑫 ≅ ∠𝑪

22. 1. The base angles of an isosceles trapezoid
are congruent.

23. Given: 𝑻𝒓𝒂𝒑𝒆𝒛𝒐𝒊𝒅 𝑴𝑨𝑻𝑯 is an isosceles trapezoid
A. If m∠𝐴𝑀𝐻 = 75°, what is
∠𝑇𝐻𝑀?
B. If m∠𝐴 = 100° and
m∠𝑇 = 3𝑥 + 10 °, what
is x?
C. If m∠𝑇𝐻𝑀 = 4𝑥 + 55 °,
m∠𝐴𝑀𝐻 = 9𝑥 + 15 °, what
is m∠𝑇𝐻𝑀?

24. 2. Opposite angles of an isosceles trapezoid
are supplementary.
𝒎∠𝑨 + 𝒎∠𝑪 = 𝟏𝟖𝟎°
𝒎∠𝐃 + 𝒎∠𝑩 = 𝟏𝟖𝟎°

25. 2. Opposite angles of an isosceles trapezoid
are supplementary.

26. Given: 𝑻𝒓𝒂𝒑𝒆𝒛𝒐𝒊𝒅 𝑴𝑨𝑻𝑯 is an isosceles trapezoid
A. If m∠𝑀 = 75°, what is m∠𝑇?
B. If m∠𝐴 = 100° and
m∠𝐻 = 3𝑥 − 10 °, what
is x?
C. If m∠𝐻 = 10𝑥 + 15 °,
m∠𝐴 = 111 − 𝑥 °, what is
m∠𝐴?

27. 3. The diagonals of an isosceles trapezoid
are congruent.
𝑨𝑪 ≅ 𝑩𝑫

28. 3. The diagonals of an isosceles trapezoid
are congruent.

29. Given: 𝑻𝒓𝒂𝒑𝒆𝒛𝒐𝒊𝒅 𝑴𝑨𝑻𝑯 is an isosceles trapezoid
A. If MT = 10𝑥 + 7 and
𝐻𝐴 = 8𝑥 + 15, what is x?
B. If 𝐻𝐴 = 6𝑥 + 7 𝑖𝑛 and
MT = 7𝑥 − 9 𝑖𝑛, what is HA?

30. THEOREMS ON KITE

31. KITE
A kite is a quadrilateral with two pairs of
adjacent sides that are congruent, and no
opposite sides are congruent.
In other words, it has two distinct sets of
congruent adjacent sides.

32. A
B
C
D
/
The two distinct sets of congruent adjacent
sides are:
• 𝑨𝑩 ≅ 𝑪𝑩
• 𝑨𝑫 ≅ 𝑪𝑫
The common vertices of the congruent sides
of the kite are called the ends of the kite.
- B and D are the ends of Kite ABCD
Moreover, the line containing the ends of the
kite is a symmetry line for the kite.
- 𝑩𝑫 is the segment contained in the
symmetry line.

33. Theorems related to kite
1. The diagonals of a kite are perpendicular.
2. It has one pair of opposite angles congruent.
3. It has one diagonal that forms two congruent triangles.
4. It has one diagonal that bisects the other diagonal
5. The area of the kite is half the product of the lengths of
its diagonals.
6. It has one diagonal that forms two isosceles triangles
7. It has one diagonal that bisects a pair of opposite
angles

34. 1. The diagonals of a kite are perpendicular.
P
𝑨𝑪 𝒂𝒏𝒅 𝑩𝑫 are diagonals.
𝑨𝑪 ⊥ 𝑩𝑫 at P.

35. 2. It has one pair of opposite angles congruent.
∠𝑨 ≅ ∠𝑪
𝒎∠𝑨 = 𝒎∠𝑪

36. 3. It has one diagonal that forms two congruent
triangles.
Diagonal 𝑩𝑫 cuts KITE ABCD
into two congruent triangles
∆𝑫𝑨𝑩 ≅ ∆𝑫𝑪𝑩

37. 4. It has one diagonal that bisects the other
diagonal.
𝑩𝑫 is the perpendicular
bisector of 𝑨𝑪
𝑨𝑷 ≅ 𝑪𝑷
| |

38. 5. The area of the kite is half the product of the
lengths of its diagonals.
𝑨𝒓𝒆𝒂 =
𝟏
𝟐
(𝑫𝟏)(𝑫𝟐)
𝑨𝒓𝒆𝒂 =
𝟏
𝟐
(𝑨𝑪)(𝑩𝑫)

39. 6. It has one diagonal that forms two isosceles
triangles.
Diagonal 𝑨𝑪 cuts Kite ABCD
and two isosceles triangles are
formed.
∆𝑨𝑩𝑪 and ∆𝑪𝑫𝑨

40. 7. It has one diagonal that bisects a pair of opposite
angles.
Diagonal 𝑩𝑫 bisects ∠𝑨𝑩𝑪 and
∠𝑨𝑫𝑪
∠𝑨𝑩𝑫 ≅ ∠𝑪𝑩𝑫
∠𝑨𝐃𝐁 ≅ ∠𝑪𝑫𝑩