Activity 8: THEOREMS ON RECTANGLE
O
H
P
E
Questions and Answers:
1. Measure ∠OHE and ∠PEH. What did you find? ∠𝑶𝑯𝑬 𝒂𝒏𝒅 ∠𝑷𝑬𝑯 are
both 90◦.
2. What can you say about the four angles of the figure? They formed
right angles.
3. Measure the diagonals. What did you find? The diagonals have the
same length which are both equal to 11.5 cm therefore ̅̅̅̅
𝑶𝑬 ≅ ̅̅̅̅̅
𝑯𝑷.
4. Does quadrilateral HOPE appear to be a parallelogram? Why? Yes,
because the opposite angles are congruent and consecutive angles
are supplementary.
5. What specific parallelogram does it represent? A parallelogram
represents a Rectangle.
Activity 9: Theorems on Rhombus
N
I
R
E
Angle
Measure
C
∠NIC
70°
∠NIE
35°
∠INE
110°
∠INC
55°
∠NRE
90°
∠CRE
90°
Questions and Answers:
1. Compare the measures of ∠NIC and ∠NIE. What did you observe? The measure of
∠NIE is one half ∠NIC.
2. What does IE do to ∠NIC? Why? ̅̅̅
𝐼𝐸 bisects ∠NIC, because bisector cuts the angle
in to two equal parts.
3. Compare the measures of ∠INE and ∠INC. What did you observe? The measure of
∠INC is one half ∠INE.
4. What does NC do to ∠INE? Why? ̅̅̅̅
𝑁𝐶 bisects ∠INE, because bisector cuts the angle
in to two equal parts.
5. Compare the measures of ∠NRE and ∠CRE. What did you observe?
The angles are both equal to 90°.
6. What angle pair do ∠NRE and ∠CRE form? Why? The angles formed a linear pair
because the linear pair are formed by two right angles
7. How are the diagonals NC and IE related to each other?
̅̅̅̅
̅.
NC is perpendicular to ̅̅
IE̅̅ in symbols ̅̅̅̅
NC ⊥ IE
Activity 10: Theorems on Rhombus
G
O
C
D
What to
measure
Measurement
L
∠GDL
90°
̅̅̅̅
𝑮𝑳 and
̅̅̅̅̅
𝑶𝑫
2 inches
∠GCO and
∠OCL
90°
∠GDO and
∠ODL
45°
∠GOD and
∠LOD
45°
Questions and Answers:
1. What is the measure of ∠GDL? The measure of ∠GDL = 90°.
a. If ∠GDL is a right angle, can you consider square a rectangle? Yes, square GOLD is
a rectangle.
b. If yes, what theorem on rectangle justifies that a square is a rectangle? Because of
theorem No. 1, which state that, “If a parallelogram has one right angle, then it has four
right angles and the parallelogram is a rectangle”
2. What can you say about the lengths of GL and DO? They have the same length.
a. If GL and DO have the same measures, can you consider a square a rectangle? Yes,
square GOLD is a rectangle.
b. If yes, what theorem on rectangles justifies that a square is a rectangle? Because of
theorem No. 2, which state that, “The diagonals of a rectangle are congruent.”
3. What can you say about the measures of ∠GCO and ∠OCL? The measure of are
equal to 90°.
a. If GL and DO meet to form right angles, can you consider a square a rhombus? Yes,
square GOLD is a rhombus.
b. If yes, what theorem on rhombuses justifies that a square is a rhombus? Because of
theorem No. 3, which state that, “The diagonals of a rhombus are perpendicular.”
4. What can you say about the measures of ∠GDO and ∠ODL as a pair and ∠GOD and
∠LOD as another pair? The m∠GDO = m∠ODL and the m ∠GOD = ∠LOD.
a. If GL divides opposite angles equally, can you consider a square a rhombus?
Yes, Square GOLD is a rhombus.
b. If yes, what theorem on rhombuses justifies that a square is a rhombus? Because of
theorem No. 4, which state that, “Each diagonal of a rhombus bisects opposite angles.”
Activity 13: The Midsegment Theorem of
Trapezoid
T
R
O
G
A
P
Questions and Answers:
1. Does GO look parallel to the bases of the trapezoid? Yes
2. Measure GO. How long is it? It is 6.5 cm
3. What is the sum of the bases of TRAP? It is 13 cm.
4. Compare the sum of the bases and the length of GO. What did you find? The length
̅̅̅̅ is half of the sum of the bases of Trapezoid TRAP.
of 𝐺𝑂
5. Make a conjecture about the sum of the bases and the length of the segment joined
by the midpoints of the legs. Explain your answer. The length of ̅̅̅̅
𝐺𝑂 is half of the sum of
the bases of Trapezoid TRAP. Therefore the length of the segment joined by the
midpoints of the legs is one-half the sum of the bases.
Activity 14: Theorems on Isosceles Trapezoid
W
G
N
A
X
I
What to
measure
∠AGN
∠GAI
Measurement
100°
80°
∠AIN
80°
∠INW
̅̅
̅̅
𝑮𝑰
̅̅̅̅
𝑨𝑵
100°
7.5 cm
7.5 cm
Questions and Answers:
1. What two pairs of angles formed are base angles? The pairs of angle formed by the
bases angles are supplementary.
2. Compare the measures of the angles in each pair. What did you find? The angles in
each pair of base angles have the same measure. The sum of opposite angles in
an isosceles trapezoid is 180 degrees.
3. Make a conjecture about the measures of the base angles of an isosceles trapezoid.
Explain your answer. The angles in each pair of base angles have the same measure.
Base angles in an isosceles trapezoid are congruent and also the diagonals of the
trapezoid have equal length, therefore the diagonals of an isosceles trapezoid are
congruent.
Activity 15: Theorems on Kite
What to
measure
Measurement
∠CXU
∠UXT
90°
90°
∠EXT
90°
∠CXE
𝑪𝑿
̅̅̅̅
𝑿𝑻
90°
1.5 cm
1.5 cm
Questions and Answers:
1. What do you observe about the measures of the angles above? They have the same
measures which are all equal to 90°.
2. How are the diagonals related to each other? The diagonals are perpendicular.
3. Make a conjecture about the diagonals of a kite based on the angles formed. Explain
your answer. The angles have the same 90°measures, and they are all right angles.
Therefore the diagonals of a kite are perpendicular with each other.
4. Compare the lengths of the segments given above. What do you see?
5. What does UE do to CT at X? Why? These are the diagonals of the kite CUTE, the
diagonals bisect the other diagonal but they do not have the same length. As the figure
shown, the longer diagonal bisect the shorter one.
5. Make a conjecture about the diagonals of a kite based on the pair of congruent
segments formed. Explain your answer.
The length of the segments given in the table are the same, thus, a diagonal bisects the
other diagonal. Therefore the longer diagonal bisect the shorter one