Download Right Triangle

Goal: I will be able to
and similarity.
Tool Bag
Formulas, equations,
Vocabulary, etc.
Vocabulary
determine triangle/shape correspondence
Here’s How…Notes & Examples
When studying triangles & shapes, it is essential to be able
to communicate about the parts of a triangle/shape without
any confusion. The following terms are used to identify
particular angles or sides:
• between
• adjacent to
• opposite to
• the included (side/angle)
Use figure △ABC to answer the following:
between
1. ∠A is ___________________
sides AB and AC.
adjacent
2. ∠B is ___________________
side AB and side BC.
B
opposite
3. Side AB is __________________
∠C
A
BC
4. Side _________
is the included side of ∠B and ∠C
∠B
5. ___________
is opposite to Side AC.
∠C
∠B
6. Side BC is between angles __________
and __________.
7. What is the included angle of AB and BC?
∠B
C
Triangle Names
Acute Triangle
has 3 acute angles
Acute Triangle
Right Triangle
has 1 right angle.
Obtuse Triangle has 1 obtuse angle.
Right Triangle
Obtuse Triangle
Equilateral
Triangle
has 3 congruent sides.
5”
5”
5”
Equilateral Triangle
Isosceles
Triangle
has at least 2 congruent sides
and 2 congruent angles.
8”
34°
8”
34°
Isosceles Triangle
Scalene
Triangle
has no congruent sides and
no congruent angles.
Scalene Triangle
Given two triangles, we need to determine a method to
establish alignment between the sides and angles.
Correspondence The vertices of triangles/shapes are paired up.
Y
B
A
C
X
We can chose to assign correspondence so that A
matches to X, B matches to Y, and C matches to Z.
There are 6 possible correspondences.
Note:
A
X
A
X
A
X
B
We use double
Y
B
Y
B
Y
C
Z
arrows to show
C
Z
C
Z
correspondence.
A
X
A
X
A
X
B
Y
B
Y
B
Y
C
Z
C
Z
C
Z
We write the correspondence as follows:
ABC
ABC
XZY
ABC
XYZ
ABC
YXZ
ABC
YZX
ABC
Z
ZXY
ZYX
Y
B
A
C
X
Z
Why do we bother setting up correspondence?
A correspondence provides a systematic way to compare
parts of two triangles. Without a correspondence, it would
be difficult to discuss the parts of a triangle because we
would have no way of referring to particular sides, angles, or
vertices.
Assume the correspondence
ABC
XYZ
What can be concluded about the vertices?
Vertex A corresponds to X, B corresponds to Y,
and C corresponds to Z
How is it possible for any two triangles to have
a total of six correspondences?
The first vertex can be matched with any of three vertices.
Then, the second vertex can be matched with any of the
remaining two vertices.
Example 1
Given the following triangles, use double arrows
to show correspondence between vertices,
angles, and sides.
T
B
A
S
C
R
Triangle
Correspondence
ABC
STR
Correspondence
of Vertices
A
B
C
S
T
R
Correspondence
of Angles
∠A
∠B
∠C
∠S
∠T
∠R
Correspondence
of Sides
AB
BC
CA
ST
TR
RS
Looking at the figure, it is impossible to tell the
triangles apart. They look exactly the same. One
triangle could be picked up and put on top of the
other.
A
C
B
Two triangles are identical if there is a triangle
correspondence so that corresponding sides and angles
of each triangle are equal in measurement.
For triangles, it is useful to have a way to indicate
those sides and angles that are equal. We mark sides
with tick marks and angles with arcs if we want to draw
attention to them. If two angles or two sides have the
same number of marks, it means they are equal.
F
C
A
In this figure…
B
D
E
AC = EF = DE
As you can see, tick marks are used to indicate
equal sides and angles
Example 2
Two identical triangles are shown. Give a triangle
correspondence that matches equal sides and angles.
T
C
B
A
Solution:
You Try
ABC
R
S
TSR
Draw two triangles that have correspondence. Have
your partner check your work.
Other Shapes
Quadrilateral
A four sided shape
square
trapezoid
rhombus
Pentagon
A five sided shape
Octagon
A eight sided shape
rectangle
Example 3
Given the following quadrilaterals, find x, y, and z:
PQSR  WTUV
R
x+5
70°
P
T
S
75°
132°
12 = z
W
18
75°
11z°
Q
24
Find x:
x + 5 = 12
PR
x + 5 - 5 = 12 - 5
x =7
Find z:
132 = 11z
132 = 11z
11
11
6y
R

V
U
 WV
V
12
Find y:
6y = 24
6y = 24
6
6
y =4
WT  PQ
Closing
1. Two shapes and their respective parts can be compared
once a correspondence has been assigned to the two shapes.
Once a correspondence is selected, corresponding sides and
corresponding angles can also be determined.
2. Corresponding vertices are notated by double arrows;
Shape correspondences can also be notated with double
arrows.
3. Shapes are identical if there is a correspondence so that
corresponding sides and angles are equal.
4. An equal number of tick marks on two different sides
indicates the sides are equal in measurement. An equal
number of arcs on two different angles indicates the angles
are equal in measurement.