TRIANGLE-MIDLINE-THEOREM-TRAPEZOID-KITE.pdf

Mathematics 9

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

2. COLUMN A COLUMN 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. HONESTY STREET
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 �� is parallel to �� and the length
of�� is half the length of ��
Consider △��� in which
�� cuts �� and �� at their midpoints.
��∥��
��=
�
�
��

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: △??????��
�� cuts ??????� and �� as their midpoints
A. If ��=48??????�, find ??????�.
B. If PG=2�+8 and AI=11,
what is x?
C. If AI=3�−4 and
PG=5�+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. �� and �� are the
bases since ��∥��
�� and �� are the legs
∠� and ∠� are base
angles with respect to ��.
∠� and ∠� are base angles
with respect to ��.

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.
�� and �� are the diagonals
�� 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.
��∥��∥��
����??????�=
�
�
(�??????��
�+ �??????��
�)
��=
�
�
(��+��)

17. Proof of Midsegment Theorem of Trapezoid

18. Given: ��??????��??????��� ��??????� ??????��� ����??????���
A. If �??????=5, and SR=
13, what is ��?
B. If AM=�+2, SR=
10 and HI=9, what is x?
C. If AM=2�−5,
SR=�+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
CD
With respect to base ��, ∠�≅∠�
With respect to base ��, ∠�≅∠�

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

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

24. 2.Opposite angles of an isosceles trapezoid
are supplementary.
�∠�+�∠�=�??????�°
�∠??????+�∠�=�??????�°

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

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

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

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

29. Given: ��??????��??????��� ���� is an isosceles trapezoid
A. If MT=10�+7 and
��=8�+15, what is x?
B. If ��=6�+7??????� and
MT=7�−9??????�, 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:
•��≅��
•��≅��
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.
-�� 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
�� ??????���� are diagonals.
��⊥�� at P.

35. 2. It has one pair of opposite angles congruent.
∠�≅∠�
�∠�=�∠�

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

37. 4. It has one diagonal that bisects the other
diagonal.
�� is the perpendicular
bisector of ��
�??????≅�??????
| |

38. 5. The area of the kite is half the product of the
lengths of its diagonals.
���??????=
�
�
(�
�)(�
�)
���??????=
�
�
(��)(��)

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

40. 7. It has one diagonal that bisects a pair of opposite
angles.
Diagonal �� bisects ∠��� and
∠���
∠���≅∠���
∠�????????????≅∠���