Download Investigate right triangle similarity

Name: _________________________________
Investigating Right Triangle Similarity
1. SAN and justify: If two triangles both have a 38 degree angle, then those triangles are similar.
2. Use the diagram below to determine if this equation:
a x c
= = is sometimes true, always true, or never true.
b y d
Justify.
d
b
y
38°
a
38°
38°
c
x
Some vocabulary:
A side is opposite an angle in a right triangle iff neither of its endpoints are the vertex of the angle.
A side is the hypotenuse of an angle in a right triangle iff it is opposite of the right angle.
A side is adjacent to an angle in a right triangle iff one of its endpoints is the vertex of the angle and it is NOT the
C
hypotenuse.
To the right, you see an angle labeled θ . We call this a “theta” and
opposite
variable typically used in mathematics to represent angles. We
called it x, or y, or any other variable.
The labeled sides are opposite and adjacent to the angle θ
A
hypotenuse it is a
could have
θ
adjacent
B
Critical Understanding!!!
As you should have determined in your SAN questions, if I have two right triangles that have a congruent acute angle,
the triangles will be similar by AA. This means that the ratio of their sides must be equal. Therefore, if a right triangle
has an angle of 50 degrees, the ratio of
opposite side
opposite side
will be equal to the ratio of
in any other right
hypotenuse
hypotenuse
triangle with a 50 degree angle! The same would be true of the ratios between any pairs of sides!
Let’s investigate with the activity that follows:
Open up the GSP document on moodle labeled Right Triangle Investigation. Fill out the following tables and plot your
points on the axis. Round your ratios to the nearest thousandth!
Angle Measure
opposite side
hypotenuse
5
y
1.4
1.3
1.2
1.1
22
1
30
0.9
0.8
38
0.7
45
0.6
53
0.5
0.4
60
0.3
68
0.2
0.1
75
x
87
Angle Measure
5
22
adjacent side
hypotenuse
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
y
1.4
1.3
1.2
1.1
1
30
38
0.9
0.8
0.7
45
53
60
0.6
0.5
0.4
0.3
68
75
87
0.2
0.1
x
Angle Measure
y
opposite side
adjacent side
20
18
5
16
22
14
30
12
38
10
45
8
53
6
60
4
68
2
75
x
-5
5
10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95
87
Answer the following:
3. If you know a right triangle has a 53 degree angle and the smallest leg is 5 cm, can you determine the lengths of
the other two sides?
4. Draw a 45-45-90 triangle and label its sides in terms of x. How does this diagram verify what you found for the
opposite side
ratio for the 45 degree angle?
adjacent side
5. Draw a 30-60-90 triangle and label its sides in terms of x. How does this diagram verify what you found for the
opposite side
ratio for the 30 degree angle?
hypotenuse