Buksis-9
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9.1 90
Types of Triangles
What are you going to learn? To determine the types of triangle To find the sum of the angles of a triangle To describe the relation between exterior and interior angles To find the formula for the perimeter and area of a triangle
Key Terms: • • • • • • • • •
congruent sides equilateral triangle isosceles triangle scalene triangle right triangle obtuse triangle acute triangle exterior angle interior angle
Look at the picture of a sailing boat above. In the picture, the sails are in the shape of quadrilaterals and triangles. What is the name of the quadrilateral?
Look at the triangular sails. What types of triangles are they?
1. TYPES OF TRIANGLES BASED ON THE LENGTHS OF THE SIDES F
C
P R
A
B (a)
E
D (b)
Q (c)
Figure 9.1
Look at the Figure 9.1 a. Use a ruler to measure the lengths of the sides of ΔABC. Mathematics for Junior High School Year 7 / 339
b. Are there any equal sides? If so, how many are they? c. By looking at the lengths of the sides, what kind of triangle is ΔABC? Explain it. d. Repeat questions (a) to (c) for ΔDEF. e. Repeat questions (a) to (c) for ΔPQR. • Equilateral triangle is a triangle having three congruent sides. • Isosceles triangle is a triangle having two congruent sides. • Scalene triangle is a triangle having no congruent sides.
Relation to the Real World
(a)
(b)
(c)
Figure 9.2
• Look at Figure 9.2 (a), what shape is the high building? • In Figure 9.2 (b), what shape is the sail of the motorboat? • Look at Figure 9.2 (c), what shape is the sail of the fisherman’s boat? Think and Discuss • Look at the figure on the right. • How many equilateral triangles are there?
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Student’s Book – Triangles
K
Look at isosceles triangle KLM on the left. ∠K is called the vertex angle. L
∠L and ∠M are called the base angles.
M
2. TYPES OF TRIANGLES BASED ON THE MEASURES OF THE ANGLES You have learnt about a rectangle and its properties. To remind you, discuss the following. GROUP WORK 1. 2. 3. 4. 5. 6. 7.
Draw rectangle ABCD 8 cm in length and 6 cm in width. Draw the diagonals. Cut the rectangle along the side a. Cut it along one of the diagonals. What is the shape of the fragments? Do both fragments have the same area? Look at both fragments. Is there one right angle in each fragment? If so, how do you measure the angle? Show the position of the angle and mention its name.
If rectangle ABCD is cut into two parts along one of the diagonals, we will get two congruent triangles. Because one of the angles in each triangle is a right angle (∠C or ∠B), the triangle is called a right triangle. A
B
K
L
P
C (a)
R
(b)
M
Figure 9.3
(c)
Q
Look at the above figure. Use a protractor to answer the following questions. a. Find the measures of the angles of ΔABC. b. Is there any 90° angle? c. What are the measures of the other two angles?
Mathematics for Junior High School Year 7 / 341
d. By observing the measures of the angles, what kind of triangle is ΔABC? Explain it. e. Repeat questions (a) to (d) for ΔKLM. f. Repeat questions (a) to (d) for ΔPQR. • A right triangle is a triangle that has one 90° angle. • An obtuse triangle is a triangle that has one obtuse angle. • An acute triangle is a triangle that has three acute angles
Related to the Real World
(a)
(b) • In figure (a), what shape is the sail of the boats?
• In figure (b) , what shape is the wooden terrace of the house? Look at the following figures. B
A ⎡
⎤
⎣
⎦
C
A
⎤ ⎣
D
B
D
C
Figure 9.4
Arranging Two Isosceles Triangles • Arrange two equal isosceles triangles by attaching the equal sides. • Draw each geometrical shape you get. • How many kinds of geometrical shape can you get? • What are the names of the shapes?
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Student’s Book – Triangles
3. TYPES OF TRIANGLES BASED ON THE ANGLES AND SIDES B
L R
A
(a)
C
K
(b)
M
Q
P (c)
Figure 9.5
Look at the above figure. Use a protractor and a ruler to answer the following questions. a. Find the measures of the angles of ΔABC. b. Measure the lengths of the sides of ΔABC. c. Are there any equal sides in ΔABC? d. Is there any 90° angle in ΔABC? e. What are the measures of the other two angles? f. By observing the angles, what shape is ΔABC? g. By observing the lengths of the sides, what shape is ΔABC? h. By observing the angles and the sides, what shape is ΔABC? Explain it. i. Repeat questions (a) to (h) for ΔKLM. j. Repeat questions (a) to (h) for ΔPQR. k. Can you find other types of triangle in this grouping? Explain it. A right isosceles triangle is a triangle that has one 90° angle and two equal sides. An obtuse isosceles triangle is a triangle that has one obtuse angle and two equal sides. An acute isosceles triangle is a triangle that has one acute angle and two equal sides.
Mathematics for Junior High School Year 7 / 343
Think and Discuss Look at the picture of Guyana’s flag on the left. What kinds of triangle are there on the flag?
The Sum of the Angles of a Triangle What is the sum of all angles of a triangle? To answer the question, do the following activity.
Mini - Lab WORK IN GROUPS Materials: Paper, a pencil, a protractor, a ruler, and scissors.
1. Draw three triangles as shown in the figure on the right. 2. Cut each triangle along the sides. 3. Share them with your friends, so that each person gets different triangle. 4. Draw a straight line g as you like. 1 3 5. Mark a number on each angle of the triangle you have. 2 6. Cut the edges of the triangles as shown in the figure on the right. 7. Choose point P on line g. Place 3 edges of the triangle 2 3 from those pieces of paper on P. Arrange the endpoints as g 1 in the figure on the right. 8. Compare your result with your friends’ results for different triangles. 9. What can you and your friends conclude? 10. Recheck to justify your conclusion by measuring each angle of the triangles using a protractor and calculate the sum. Do it carefully.
On the basis of the activities you have done, you have found the sum of the angles in a triangle. If all angles in a triangle fit together side by side, do they form a straight angle? The sum of all the angles of a triangle is 180°. Knowing that the sum of the angles in a triangle is 180°, you can determine the size of one of the angles if the sizes of the other two angles are given.
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Student’s Book – Triangles
CRITICAL THINKING Is it possible for a triangle to have two right angles? Explain. Is it possible for a triangle to have two obtuse angles? Explain.
57°
Discuss Music. Have you ever seen a piano like in the picture on the left? A supporting stick supports the top of the piano. The stick forms a 57° angle with the base of the piano, whereas the top forms a 90° angle with the stick. How many degrees is the other angle?
R 80°
PROBLEM 1 2 cm
2 cm
?
?
Given ΔPQR as in the figure on the right. a. What kind of triangle is ΔPQR? Explain it.
Q
P
b. What is the measure of ∠P? c. What is the measure of ∠Q? d. How can you determine the measures of ∠P and ∠Q? e. Is ∠P = ∠Q? Why?
PROBLEM 2 Calculate the measure of each angle of ΔABC.
(8x − 1)°
C
B ⎤
What is the sum of ∠A and ∠C? Explain. (4x + 7)° A Mathematics for Junior High School Year 7 / 345
PROBLEM 3 Look at the figure of ΔFGH on the right. a. Find the measure of each angle indicated by x, y, z. b. Based on the measures of the angles, what kind of triangle is F
ΔFGH?
G 39°
65°
21° x° y° J
z° H
c. Based on the measures of the angles, what kind of triangle is ΔGHJ? d. Based on the measures of the angles, what kind of triangle is ΔFGJ?
The Exterior and Interior Angles of a Triangle An exterior angle of a triangle is an angle formed by the side of a triangle with the extension of the other side of the triangle. Think about the definition of the interior angle of a triangle. Look at ΔXYZ on the left. The side XY is lengthened into WY.
Z c°
∠ Y, ∠Z, and ∠YXZ are the interior angles of ΔXYZ
Exterior Angle W
and ∠WXZ is the exterior angle of ΔYXZ. a. What conclusion can you draw about the relation a°
X
b°
Y
between ∠WXZ and ∠YXZ? b. What is the measure of ∠WXZ?
c. What conclusion can you draw about the relation between the size of the exterior angle (∠WXZ) and two interior angles (∠XYZ and ∠YZX)? d. How many exterior angles are there in a triangle?
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Student’s Book – Triangles
PROBLEM 4
D
Look at the figure on the right.
F
a. Name the exterior angle of ΔDEF. b. What is the measure of the exterior angle of ΔDEF? c. What is the measure of ∠DFE.
G
75° 35°
E
d. Measure the measure of ∠EDF. Based on the above explanation, you can draw the conclusion as follows: The measure of an exterior angle of a triangle is equal to the sum of two interior angles which are not supplementary to the exterior angle.
The Perimeter and Area of a Triangle To find the perimeter of a triangle, you should know first the lengths of the sides of the triangle because the perimeter of a triangle is the sum of the lengths of the sides forming the triangle.
PROBLEM 5 C
Look at the figure on the left. a. How can you calculate the perimeter of ΔABC shown A
B
in the figure on the left? Explain. b. What conclusion can you draw? c. Can you formulate the perimeter of ΔABC?
Mathematics for Junior High School Year 7 / 347
If P is the perimeter of a triangle having the sides a, b dan c, then the perimeter of the triangle can be stated as P=a+b+c
Think and Discuss 14 cm 10 m
8m
Garden Problem Pak Budi has a garden as shown in the figure on the left. He wants to build a fence around the garden.
6m
a. How can you calculate the perimeter of Pak Budi’s garden? b. How long is the fence that Pak Budi needs? c. What is the relation between the perimeter of the garden and the cost? Explain. d. If the cost of setting the fence is Rp 25.000,00 per meter, how much money will Pak Budi spend? Check Your Understanding
R
5 cm
1. Calculate the perimeter of the isosceles triangle PQR on the right.
7 cm
2. Explain how you calculate the perimeter of ΔPQR. P
Mini - Lab WORK IN GROUPS Materials: grid paper, a ruler, and scissors 1. Draw rectangle ABCD on the grid paper with the length of 12 squares and the width of 9 squares. 2. Cut the rectangle ABCD along the sides. 3. What is the area of rectangle ABCD? 4. Draw one of the diagonals of rectangle ABCD. 5. Cut the rectangle ABCD along the diagonals (step 4) into two parts. 6. What is the shape you get? Are the two shapes obtained equal? 7. Do the two shapes have the same area? 8. What is the area of each shape you get (step 7)? 9. What is the formula of each of the shapes you get?
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Student’s Book – Triangles
Q
Critical Thinking
C
Look at ΔABC in the figure on the right. y
a. Calculate the area of ΔABC. b. Is there any other way to calculate the area of ΔABC? c. What conclusion can you draw?
t
x ⎦
A
B
a
The results of the Mini Lab above show that the area of a triangle can be obtained from the area of a rectangle; the area of a right triangle is half of that of a rectangle. Therefore, it can be concluded that:
h
If A is the area of a triangle whose base is b and height is h, then the area of the triangle can be stated as: A=
b a
1 (b × h) 2
Think and Discuss Woodwork
5m
A carpenter is going to make a wooden wall for the back of a warehouse. If the price of 6m
wood is Rp 5,000.00/m2, how much is the cost to make the wall of the warehouse?
8m
M 13 cm
Checking Understanding Given ΔKLM as shown in the figure on the right. Calculate the area of ΔKLM.
N
12 cm
L
14 cm 15 cm
K
Mathematics for Junior High School Year 7 / 349
Critical Thinking Given the area of ΔPQR is 16 cm2 and the height is 4 cm. How can you calculate the length of the base of ΔPQR?
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Student’s Book – Triangles
1. Name the types of the following triangles. Use a ruler and a protractor if necessary. F
B
G
E
A C M L
H
Q
I S U
D O
P
V R
W
T
K
a. What kind of triangle is ΔABC? Explain it. b. What kind of triangle is ΔDEF? Explain it. c. What kind of triangle is ΔGHI? Explain it. d. What kind of triangle is ΔKLM? Explain it. e. What kind of triangle is ΔPQR? Explain it. f.
What kind of triangle is ΔSTO? Explain it.
g. What kind of triangle is ΔUVW? Explain it. 2. Draw the following triangles (if possible) on dotted paper: a. a triangle with three acute angles. b. a triangle with one right angle. c.
a triangle with one obtuse angle.
d. a triangle with one right angle and one obtuse angle. e. a triangle with three different lengths of its sides. f.
a triangle with two equal lengths of its sides.
g. a triangle with three equal lengths of its sides.
Mathematics for Junior High School Year 7 / 351
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3. Look at the flag of Jamaica on the right. a. Name the kinds of triangle in the picture of the flag of Jamaica. b. How many triangles are alike on the flag? Show them. c. Which triangles have the same size? 4. Look at the two triangle rulers on the right figure. a. Do they have similarity? Explain it. b. What are the differences between them? Explain. 5. Look at ΔABC on the right figure.
C
a. What kinds of triangles form ΔABC? b. How many congruent triangles are there in ΔABC? c. How many right-angled triangles are there? d. How many congruent isosceles triangles A are there in ΔABC? e. How many isosceles triangles are there in ΔABC? f. Is there any equilateral triangle in Δ ABC?
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Student’s Book – Triangles
B
6. Draw the following triangles. If the triangle cannot be drawn, then write “impossible” and state your reasons. a. a right triangle b. a right isosceles triangle c. an obtuse isosceles triangle d. an obtuse equilateral triangle 7. Critical thinking. a. Can b. Can c. Can d. Can e. Can f. Can
an isosceles triangle be an acute triangle? Explain. a scalene triangle be an acute triangle? Explain. a right triangle be an acute triangle? Explain. an equilateral triangle be an acute triangle? Explain. an acute triangle be an equilateral triangle? Explain. a scalene triangle be an obtuse triangle? Explain.
8. Look at the figure on the right. a. How many congruent triangles are G there? b. What kinds of triangles form square ACEG? H c. How many degrees is each of the base angles of each triangle? d. How many degrees is the top angle A of each triangle?
F
O
B
E
D
C
e. What angle has the equal size to ∠OAB? f. How many degrees is ∠OAB? 9. What kind of triangle has the following angles? a. 90°, 40°, 50°
b. 115°, 30°, 35°
b. 38°, 72°, 70°
10. Given a triangle with two of its angles are 35° and 50°. What is the measure of the third angle? What kind of triangle is it?
Mathematics for Junior High School Year 7 / 353
11. Look at the following figures. D
A 45°
105°
?
50°
G C
60° L
60°
80°
E
45°
B
M
I
?
?
? H
40°
K
F a. Measure the unknown angle in each of the triangles above. b. Based on the sizes of the angles, what kind of triangle is each triangle above?
12. Find the sizes of the unknown angles in the isosceles triangles below. C
D
52°
F
47°
?
2
3
3
?
?
R ?
2 5
? E
A
5 ?
Q
63°
P
B
13. Look at the figures below. a. Find the sizes of the unknown angles. b. What kind of triangle is each triangle on the right? c. What is the sum of the acute angles of each triangle on the right? d. What is the relation between both acute angles in each of the above triangle? 35°
45°
30°
(i)
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Student’s Book – Triangles
(ii)
(iii)
14. Find out the value of a, b, and c of each of the following triangles. 2b °
3a ° 2a °
c°
2b °
3c °
2b°
35°
(i)
c°
(ii)
(iii)
15. Critical Thinking. Given a triangle whose angles are 50°, 60° and 70°. a. What kind of triangle is it? Why? b. Can you classify the triangle based on the lengths of the sides? Explain it.
16. Open Question. Given triangle ABC with one of the angles being 18°. What kind of triangle is it? Explain it. 17. Look at the triangles in the figure below. a. Determine the sizes of the unknown angles. b. Sort ascending by the sizes of the angles in each triangle. c. Sort descending by the lengths of the sides in each triangle. d. Make a prediction about the relation between the results of (b) and (c). V O
H 45°
(1)
75°
135°
P
U
30°
S
(4)
T
(2) M
28°
(3)
N
60° Q
110°
F
G
Mathematics for Junior High School Year 7 / 355
18. What are the measures of the angles of a triangle if the lengths of the sides are as follows: a. AB = 8, BC = 5, and AC = 7. b. DE = 15, EF = 18, and DF = 5. c. XY = 2, YZ = 4, and XZ = 3. 19. What are the lengths of the sides of a triangle if the measures of the angles are as follows: a. ∠S = 90°, ∠R = 40°, ∠T = 50° b. ∠A = 20°, ∠B = 120°, ∠C = 40° c. ∠X = 70°, ∠Y = 30° , ∠Z = 80° d. ∠D = 80°, ∠E = 50°, ∠F = 50° 20. Observation . Is it possible to form a triangle from the following midribs of palm leaves if the lengths are as the following? State the reasons. a. 11 cm, 12 cm and 15 cm. c. 6 cm, 10 cm, and 13 cm. b. 2 cm, 3 cm and 6 cm. d. 5 cm, 10 cm and 15 cm. 21. The perimeter of a quadrilateral PQRS in the figure on the right is 22 m. a. Determine the lengths of PQ, SR, PS and RQ.
S
P
R
Q
b. How can you calculate the area of PQRS? c. What is the area of PQRS? 22. Look at the figure below. Which shape has the largest area? Explain.
(a)
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Student’s Book – Triangles
(b)
(c)
23. The perimeter of ΔKLM is 40 cm.
x
K
M
a. What shape is ΔKLM? b. Determine the lengths of the sides of ΔKLM.
2x − 5
24. Given the shapes below. L
(a)
(c)
(b)
a. Determine the area of each shape. b. Which shape has the largest area? 25 cm
25. Reni has a piece of square patterned cardboard forming a rectangle with the length of 25 cm. She is going to make a toy the shape of which is shown in the figure on the right. What is the area of the unused cardboard?
⎡
26. Critical Thinking. Look at the areas of triangles I and II. Compare the area I with that of II. Explain.
I x
II x
Mathematics for Junior High School Year 7 / 357
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Student’s Book – Triangles
Types of Triangles
What are you going to learn? To determine the types of triangle To find the sum of the angles of a triangle To describe the relation between exterior and interior angles To find the formula for the perimeter and area of a triangle
Key Terms: • • • • • • • • •
congruent sides equilateral triangle isosceles triangle scalene triangle right triangle obtuse triangle acute triangle exterior angle interior angle
Look at the picture of a sailing boat above. In the picture, the sails are in the shape of quadrilaterals and triangles. What is the name of the quadrilateral?
Look at the triangular sails. What types of triangles are they?
1. TYPES OF TRIANGLES BASED ON THE LENGTHS OF THE SIDES F
C
P R
A
B (a)
E
D (b)
Q (c)
Figure 9.1
Look at the Figure 9.1 a. Use a ruler to measure the lengths of the sides of ΔABC. Mathematics for Junior High School Year 7 / 339
b. Are there any equal sides? If so, how many are they? c. By looking at the lengths of the sides, what kind of triangle is ΔABC? Explain it. d. Repeat questions (a) to (c) for ΔDEF. e. Repeat questions (a) to (c) for ΔPQR. • Equilateral triangle is a triangle having three congruent sides. • Isosceles triangle is a triangle having two congruent sides. • Scalene triangle is a triangle having no congruent sides.
Relation to the Real World
(a)
(b)
(c)
Figure 9.2
• Look at Figure 9.2 (a), what shape is the high building? • In Figure 9.2 (b), what shape is the sail of the motorboat? • Look at Figure 9.2 (c), what shape is the sail of the fisherman’s boat? Think and Discuss • Look at the figure on the right. • How many equilateral triangles are there?
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Student’s Book – Triangles
K
Look at isosceles triangle KLM on the left. ∠K is called the vertex angle. L
∠L and ∠M are called the base angles.
M
2. TYPES OF TRIANGLES BASED ON THE MEASURES OF THE ANGLES You have learnt about a rectangle and its properties. To remind you, discuss the following. GROUP WORK 1. 2. 3. 4. 5. 6. 7.
Draw rectangle ABCD 8 cm in length and 6 cm in width. Draw the diagonals. Cut the rectangle along the side a. Cut it along one of the diagonals. What is the shape of the fragments? Do both fragments have the same area? Look at both fragments. Is there one right angle in each fragment? If so, how do you measure the angle? Show the position of the angle and mention its name.
If rectangle ABCD is cut into two parts along one of the diagonals, we will get two congruent triangles. Because one of the angles in each triangle is a right angle (∠C or ∠B), the triangle is called a right triangle. A
B
K
L
P
C (a)
R
(b)
M
Figure 9.3
(c)
Q
Look at the above figure. Use a protractor to answer the following questions. a. Find the measures of the angles of ΔABC. b. Is there any 90° angle? c. What are the measures of the other two angles?
Mathematics for Junior High School Year 7 / 341
d. By observing the measures of the angles, what kind of triangle is ΔABC? Explain it. e. Repeat questions (a) to (d) for ΔKLM. f. Repeat questions (a) to (d) for ΔPQR. • A right triangle is a triangle that has one 90° angle. • An obtuse triangle is a triangle that has one obtuse angle. • An acute triangle is a triangle that has three acute angles
Related to the Real World
(a)
(b) • In figure (a), what shape is the sail of the boats?
• In figure (b) , what shape is the wooden terrace of the house? Look at the following figures. B
A ⎡
⎤
⎣
⎦
C
A
⎤ ⎣
D
B
D
C
Figure 9.4
Arranging Two Isosceles Triangles • Arrange two equal isosceles triangles by attaching the equal sides. • Draw each geometrical shape you get. • How many kinds of geometrical shape can you get? • What are the names of the shapes?
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Student’s Book – Triangles
3. TYPES OF TRIANGLES BASED ON THE ANGLES AND SIDES B
L R
A
(a)
C
K
(b)
M
Q
P (c)
Figure 9.5
Look at the above figure. Use a protractor and a ruler to answer the following questions. a. Find the measures of the angles of ΔABC. b. Measure the lengths of the sides of ΔABC. c. Are there any equal sides in ΔABC? d. Is there any 90° angle in ΔABC? e. What are the measures of the other two angles? f. By observing the angles, what shape is ΔABC? g. By observing the lengths of the sides, what shape is ΔABC? h. By observing the angles and the sides, what shape is ΔABC? Explain it. i. Repeat questions (a) to (h) for ΔKLM. j. Repeat questions (a) to (h) for ΔPQR. k. Can you find other types of triangle in this grouping? Explain it. A right isosceles triangle is a triangle that has one 90° angle and two equal sides. An obtuse isosceles triangle is a triangle that has one obtuse angle and two equal sides. An acute isosceles triangle is a triangle that has one acute angle and two equal sides.
Mathematics for Junior High School Year 7 / 343
Think and Discuss Look at the picture of Guyana’s flag on the left. What kinds of triangle are there on the flag?
The Sum of the Angles of a Triangle What is the sum of all angles of a triangle? To answer the question, do the following activity.
Mini - Lab WORK IN GROUPS Materials: Paper, a pencil, a protractor, a ruler, and scissors.
1. Draw three triangles as shown in the figure on the right. 2. Cut each triangle along the sides. 3. Share them with your friends, so that each person gets different triangle. 4. Draw a straight line g as you like. 1 3 5. Mark a number on each angle of the triangle you have. 2 6. Cut the edges of the triangles as shown in the figure on the right. 7. Choose point P on line g. Place 3 edges of the triangle 2 3 from those pieces of paper on P. Arrange the endpoints as g 1 in the figure on the right. 8. Compare your result with your friends’ results for different triangles. 9. What can you and your friends conclude? 10. Recheck to justify your conclusion by measuring each angle of the triangles using a protractor and calculate the sum. Do it carefully.
On the basis of the activities you have done, you have found the sum of the angles in a triangle. If all angles in a triangle fit together side by side, do they form a straight angle? The sum of all the angles of a triangle is 180°. Knowing that the sum of the angles in a triangle is 180°, you can determine the size of one of the angles if the sizes of the other two angles are given.
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CRITICAL THINKING Is it possible for a triangle to have two right angles? Explain. Is it possible for a triangle to have two obtuse angles? Explain.
57°
Discuss Music. Have you ever seen a piano like in the picture on the left? A supporting stick supports the top of the piano. The stick forms a 57° angle with the base of the piano, whereas the top forms a 90° angle with the stick. How many degrees is the other angle?
R 80°
PROBLEM 1 2 cm
2 cm
?
?
Given ΔPQR as in the figure on the right. a. What kind of triangle is ΔPQR? Explain it.
Q
P
b. What is the measure of ∠P? c. What is the measure of ∠Q? d. How can you determine the measures of ∠P and ∠Q? e. Is ∠P = ∠Q? Why?
PROBLEM 2 Calculate the measure of each angle of ΔABC.
(8x − 1)°
C
B ⎤
What is the sum of ∠A and ∠C? Explain. (4x + 7)° A Mathematics for Junior High School Year 7 / 345
PROBLEM 3 Look at the figure of ΔFGH on the right. a. Find the measure of each angle indicated by x, y, z. b. Based on the measures of the angles, what kind of triangle is F
ΔFGH?
G 39°
65°
21° x° y° J
z° H
c. Based on the measures of the angles, what kind of triangle is ΔGHJ? d. Based on the measures of the angles, what kind of triangle is ΔFGJ?
The Exterior and Interior Angles of a Triangle An exterior angle of a triangle is an angle formed by the side of a triangle with the extension of the other side of the triangle. Think about the definition of the interior angle of a triangle. Look at ΔXYZ on the left. The side XY is lengthened into WY.
Z c°
∠ Y, ∠Z, and ∠YXZ are the interior angles of ΔXYZ
Exterior Angle W
and ∠WXZ is the exterior angle of ΔYXZ. a. What conclusion can you draw about the relation a°
X
b°
Y
between ∠WXZ and ∠YXZ? b. What is the measure of ∠WXZ?
c. What conclusion can you draw about the relation between the size of the exterior angle (∠WXZ) and two interior angles (∠XYZ and ∠YZX)? d. How many exterior angles are there in a triangle?
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PROBLEM 4
D
Look at the figure on the right.
F
a. Name the exterior angle of ΔDEF. b. What is the measure of the exterior angle of ΔDEF? c. What is the measure of ∠DFE.
G
75° 35°
E
d. Measure the measure of ∠EDF. Based on the above explanation, you can draw the conclusion as follows: The measure of an exterior angle of a triangle is equal to the sum of two interior angles which are not supplementary to the exterior angle.
The Perimeter and Area of a Triangle To find the perimeter of a triangle, you should know first the lengths of the sides of the triangle because the perimeter of a triangle is the sum of the lengths of the sides forming the triangle.
PROBLEM 5 C
Look at the figure on the left. a. How can you calculate the perimeter of ΔABC shown A
B
in the figure on the left? Explain. b. What conclusion can you draw? c. Can you formulate the perimeter of ΔABC?
Mathematics for Junior High School Year 7 / 347
If P is the perimeter of a triangle having the sides a, b dan c, then the perimeter of the triangle can be stated as P=a+b+c
Think and Discuss 14 cm 10 m
8m
Garden Problem Pak Budi has a garden as shown in the figure on the left. He wants to build a fence around the garden.
6m
a. How can you calculate the perimeter of Pak Budi’s garden? b. How long is the fence that Pak Budi needs? c. What is the relation between the perimeter of the garden and the cost? Explain. d. If the cost of setting the fence is Rp 25.000,00 per meter, how much money will Pak Budi spend? Check Your Understanding
R
5 cm
1. Calculate the perimeter of the isosceles triangle PQR on the right.
7 cm
2. Explain how you calculate the perimeter of ΔPQR. P
Mini - Lab WORK IN GROUPS Materials: grid paper, a ruler, and scissors 1. Draw rectangle ABCD on the grid paper with the length of 12 squares and the width of 9 squares. 2. Cut the rectangle ABCD along the sides. 3. What is the area of rectangle ABCD? 4. Draw one of the diagonals of rectangle ABCD. 5. Cut the rectangle ABCD along the diagonals (step 4) into two parts. 6. What is the shape you get? Are the two shapes obtained equal? 7. Do the two shapes have the same area? 8. What is the area of each shape you get (step 7)? 9. What is the formula of each of the shapes you get?
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Student’s Book – Triangles
Q
Critical Thinking
C
Look at ΔABC in the figure on the right. y
a. Calculate the area of ΔABC. b. Is there any other way to calculate the area of ΔABC? c. What conclusion can you draw?
t
x ⎦
A
B
a
The results of the Mini Lab above show that the area of a triangle can be obtained from the area of a rectangle; the area of a right triangle is half of that of a rectangle. Therefore, it can be concluded that:
h
If A is the area of a triangle whose base is b and height is h, then the area of the triangle can be stated as: A=
b a
1 (b × h) 2
Think and Discuss Woodwork
5m
A carpenter is going to make a wooden wall for the back of a warehouse. If the price of 6m
wood is Rp 5,000.00/m2, how much is the cost to make the wall of the warehouse?
8m
M 13 cm
Checking Understanding Given ΔKLM as shown in the figure on the right. Calculate the area of ΔKLM.
N
12 cm
L
14 cm 15 cm
K
Mathematics for Junior High School Year 7 / 349
Critical Thinking Given the area of ΔPQR is 16 cm2 and the height is 4 cm. How can you calculate the length of the base of ΔPQR?
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1. Name the types of the following triangles. Use a ruler and a protractor if necessary. F
B
G
E
A C M L
H
Q
I S U
D O
P
V R
W
T
K
a. What kind of triangle is ΔABC? Explain it. b. What kind of triangle is ΔDEF? Explain it. c. What kind of triangle is ΔGHI? Explain it. d. What kind of triangle is ΔKLM? Explain it. e. What kind of triangle is ΔPQR? Explain it. f.
What kind of triangle is ΔSTO? Explain it.
g. What kind of triangle is ΔUVW? Explain it. 2. Draw the following triangles (if possible) on dotted paper: a. a triangle with three acute angles. b. a triangle with one right angle. c.
a triangle with one obtuse angle.
d. a triangle with one right angle and one obtuse angle. e. a triangle with three different lengths of its sides. f.
a triangle with two equal lengths of its sides.
g. a triangle with three equal lengths of its sides.
Mathematics for Junior High School Year 7 / 351
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3. Look at the flag of Jamaica on the right. a. Name the kinds of triangle in the picture of the flag of Jamaica. b. How many triangles are alike on the flag? Show them. c. Which triangles have the same size? 4. Look at the two triangle rulers on the right figure. a. Do they have similarity? Explain it. b. What are the differences between them? Explain. 5. Look at ΔABC on the right figure.
C
a. What kinds of triangles form ΔABC? b. How many congruent triangles are there in ΔABC? c. How many right-angled triangles are there? d. How many congruent isosceles triangles A are there in ΔABC? e. How many isosceles triangles are there in ΔABC? f. Is there any equilateral triangle in Δ ABC?
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Student’s Book – Triangles
B
6. Draw the following triangles. If the triangle cannot be drawn, then write “impossible” and state your reasons. a. a right triangle b. a right isosceles triangle c. an obtuse isosceles triangle d. an obtuse equilateral triangle 7. Critical thinking. a. Can b. Can c. Can d. Can e. Can f. Can
an isosceles triangle be an acute triangle? Explain. a scalene triangle be an acute triangle? Explain. a right triangle be an acute triangle? Explain. an equilateral triangle be an acute triangle? Explain. an acute triangle be an equilateral triangle? Explain. a scalene triangle be an obtuse triangle? Explain.
8. Look at the figure on the right. a. How many congruent triangles are G there? b. What kinds of triangles form square ACEG? H c. How many degrees is each of the base angles of each triangle? d. How many degrees is the top angle A of each triangle?
F
O
B
E
D
C
e. What angle has the equal size to ∠OAB? f. How many degrees is ∠OAB? 9. What kind of triangle has the following angles? a. 90°, 40°, 50°
b. 115°, 30°, 35°
b. 38°, 72°, 70°
10. Given a triangle with two of its angles are 35° and 50°. What is the measure of the third angle? What kind of triangle is it?
Mathematics for Junior High School Year 7 / 353
11. Look at the following figures. D
A 45°
105°
?
50°
G C
60° L
60°
80°
E
45°
B
M
I
?
?
? H
40°
K
F a. Measure the unknown angle in each of the triangles above. b. Based on the sizes of the angles, what kind of triangle is each triangle above?
12. Find the sizes of the unknown angles in the isosceles triangles below. C
D
52°
F
47°
?
2
3
3
?
?
R ?
2 5
? E
A
5 ?
Q
63°
P
B
13. Look at the figures below. a. Find the sizes of the unknown angles. b. What kind of triangle is each triangle on the right? c. What is the sum of the acute angles of each triangle on the right? d. What is the relation between both acute angles in each of the above triangle? 35°
45°
30°
(i)
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Student’s Book – Triangles
(ii)
(iii)
14. Find out the value of a, b, and c of each of the following triangles. 2b °
3a ° 2a °
c°
2b °
3c °
2b°
35°
(i)
c°
(ii)
(iii)
15. Critical Thinking. Given a triangle whose angles are 50°, 60° and 70°. a. What kind of triangle is it? Why? b. Can you classify the triangle based on the lengths of the sides? Explain it.
16. Open Question. Given triangle ABC with one of the angles being 18°. What kind of triangle is it? Explain it. 17. Look at the triangles in the figure below. a. Determine the sizes of the unknown angles. b. Sort ascending by the sizes of the angles in each triangle. c. Sort descending by the lengths of the sides in each triangle. d. Make a prediction about the relation between the results of (b) and (c). V O
H 45°
(1)
75°
135°
P
U
30°
S
(4)
T
(2) M
28°
(3)
N
60° Q
110°
F
G
Mathematics for Junior High School Year 7 / 355
18. What are the measures of the angles of a triangle if the lengths of the sides are as follows: a. AB = 8, BC = 5, and AC = 7. b. DE = 15, EF = 18, and DF = 5. c. XY = 2, YZ = 4, and XZ = 3. 19. What are the lengths of the sides of a triangle if the measures of the angles are as follows: a. ∠S = 90°, ∠R = 40°, ∠T = 50° b. ∠A = 20°, ∠B = 120°, ∠C = 40° c. ∠X = 70°, ∠Y = 30° , ∠Z = 80° d. ∠D = 80°, ∠E = 50°, ∠F = 50° 20. Observation . Is it possible to form a triangle from the following midribs of palm leaves if the lengths are as the following? State the reasons. a. 11 cm, 12 cm and 15 cm. c. 6 cm, 10 cm, and 13 cm. b. 2 cm, 3 cm and 6 cm. d. 5 cm, 10 cm and 15 cm. 21. The perimeter of a quadrilateral PQRS in the figure on the right is 22 m. a. Determine the lengths of PQ, SR, PS and RQ.
S
P
R
Q
b. How can you calculate the area of PQRS? c. What is the area of PQRS? 22. Look at the figure below. Which shape has the largest area? Explain.
(a)
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Student’s Book – Triangles
(b)
(c)
23. The perimeter of ΔKLM is 40 cm.
x
K
M
a. What shape is ΔKLM? b. Determine the lengths of the sides of ΔKLM.
2x − 5
24. Given the shapes below. L
(a)
(c)
(b)
a. Determine the area of each shape. b. Which shape has the largest area? 25 cm
25. Reni has a piece of square patterned cardboard forming a rectangle with the length of 25 cm. She is going to make a toy the shape of which is shown in the figure on the right. What is the area of the unused cardboard?
⎡
26. Critical Thinking. Look at the areas of triangles I and II. Compare the area I with that of II. Explain.
I x
II x
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