Download Draw a Diagram One angle of an isosceles triangle measures 50

Problem-solving guidelines:
 Understanding the
problem
 Developing a plan and
strategy
 Carrying out the plan
 Checking the answer
EXAMPLE 1
Draw a Diagram
One angle of an isosceles triangle
measures 50. What are the measures of
the other two angles?
Problem-Solving Strategies:
 Guessing and checking
 Drawing a table/
diagram
 Writing an equation
 Simplifying the problem
 Looking for patterns
 Using logical reasoning
Solution:
1.
Understanding Problem: Read and understand the given information.
Summarize the problem.
1). Think about the information you are given and what you need to find.
a. What is the measure of one angle of the triangle?
b. What kind of triangle is involved in this problem?
c. What is special about this kind of triangle?
d. Summarize the goal of this problem.
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2.
Arranging a plan and strategy: Decide on a strategy to solve the problem
A good strategy to use here is to draw a diagram. Although you are given the
measure of one angle of the triangle, you do not know which angle. In particular,
you do not know if this angle is one of the two congruent angles in an isosceles
triangle or if this angle is not congruent to another angle.
3.
Carrying out the plan and strategy
2). Draw a diagram that shows the 50 angle as one of the two congruent
angles.
a. What is the measure of the angle that is congruent to the given
b.
angle?
What is the measure of the third angle?
3). Draw a diagram that shows the 50 angle in a different location. What
are the measures of the other angles?
4). Are there any other ways that make an isosceles triangle have a 50
angle? Why or why not?
5). Write a solution to the original question by summarizing what you
have found.
6). Describe how drawing a diagram can help you to solve the problem.
4.
Checking the Answer
Check that the sum of the three angles is 180.
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/ Student’s Book – Similarity, Congruency, and Tessellations
EXAMPLE 2
Write an Equation
Suppose you know that ABC  ZXY . What can you conclude from the angles and
sides of XYZ ? Why?
B
A
35
2 cm
C
Y
Z
X
Solution:
1). Understanding the problem
a. What is the unknown?
B, X, Z
AB, BC, XY, YZ, XZ
b. What are the data?
ABC  ZXY
A  35
AC = 2 cm
2). Arranging a plan and strategy
Since ABC  ZXY , we can use the properties of congruent triangles to find
the other sides and angles.
Mathematics for Junior High School Grade 9 / 45
3). Carrying out the plan
Since ABC  ZXY , the corresponding sides are of the same length,
namely:
AC = ZY, CB = YX and AB = ZX.
Furthermore, the corresponding angles are of the same measure,
namely:
A = Z, C= Y, and B = X.
4). Checking the answer
The answer refers to the original question and ABC  ZXY . Then the
answer can be accepted.
Solve the following problems using one or more strategies.
1.
The diagram on the right side shows parts of a
design made of toothpicks. The top level uses
three toothpicks. The second level uses six
toothpicks. The third level uses nine toothpicks,
and so on.
a.
b.
2.
If you decide to continue the design so that it has seven levels, how
many toothpicks do you need altogether?
Which level will use 24 toothpicks?
Find the missing angle measure.
40
45
x
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/ Student’s Book – Similarity, Congruency, and Tessellations
3.
Classify the triangle by its angles.
4.
Draw a horizontal line on your paper and draw A' B' C' so that it is
congruent to ABC below. Explain how you did it.
A
C
B
5.
Draw PQR using P, Q, and PQ below. Start by drawing a horizontal
ray with its initial point labeled by P. Next, place P at the end point P
with one of its legs coincides with (on top of) PQ . Then, by placing Q
appropriately, the shape of PQR is determined. Explain how.
Q
P
Q
P
6. Given PQR below. Draw ray AB on your paper. Find two rays AC and AD so
that BAC  PQR and BAD  PQR
P
Q
R
7.
a.
Draw a large scalene triangle and label the vertices D, E, and F. Construct
JK so that JK = DE .
b.
Construct an angle with vertex J and side JK that is equal to D .
c.
Construct an angle with vertex K and side KJ that is equal to E .
Mathematics for Junior High School Grade 9 / 47
d.
Extend the size of the angles in parts (b) and (c) so that they intersect at a
point, namely L. What can you say about DEF and JKL ?
8. In ABC and PQR , AB and PQ are of the same length. BC and QR are of the
same length. Must AC and PR be of the same length? Why or why not?
9. Is ABC congruent to DEF ? Why or why not?
B
43
A
34
E
43
24
C
D
34
24
F
Q
10. Darmawan carelessly drew a picture of
C
two triangles even though his textbook
says that ABC  PQR. Fortunately,
P
you know how to use the congruence
properties to figure out which pairs of
sides and angles are equal. Name all six
pairs of corresponding, congruent parts.
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/ Student’s Book – Similarity, Congruency, and Tessellations
A
B
R