Download Notes on Congruency Statements

Notes on Congruency Statements
Name_________________________
Tick marks are used to show the congruence of specific segments. Two segments with an
identical number of tick marks are congruent. If two segments have a different number of
number of ticks marks, then they are not congruent.
A
E
Using the information presented above and
the figures to the left, complete the
congruency statements below.
B
C
1.
AB  ______
F
D
2.
DF  ______
ED  ______
3.
4.
Is the following statement true or false: EF  AC ?
--------------------------------------------------------------------------------------------------------------------Similarly, angle arcs are used to show the congruence of specific angles. Two angles with an
identical number of angle arcs are congruent. If two angles have a different number of
number of angle arcs, then they are not congruent.
G
I
Using the information presented above and the figure to the right,
complete the congruency statements below.
5.
G  ______
6.
J  ______
7.
GHI  ______
8.
What term describes the two angles
with one angle arc in the figure?
H
J
K
9.
Are angle G and angle J alternate interior angles?
--------------------------------------------------------------------------------------------------------------------There are two ways in which one can write a triangle congruency statement (ex. LTN  OPY )
based on the figures of the triangles: using the tick marks or using the angle marks. It is
important to remember that the letters in the triangle congruency statement must be in the proper
order.
If the tick marks are to be used, then one must proceed in the same order around both figures.
For example, consider the two triangles at the top of the page. Suppose the triangles will be
identified by going from the one tick mark side to the two tick mark side. The triangle on the left
would be identified as CAB . The triangle on the right would be DEF . Therefore, an
example of a correct triangle congruency statement would be CAB  DEF .
10.
Using the same triangles, write a triangle congruency statement based on a naming of the
triangles that uses the three tick mark side, then the two tick mark side.
G
I
The second method is to use the angle arcs. Suppose the two triangles to the
left are congruent. The triangles can be identified in the following order:
one angle arc, two angle arcs, then three angle arcs.
H
J
11.
K
Therefore, the triangle on the top would be called HIG , and the triangle on
the bottom would be HJK . Hence, the triangle congruency statement
would HIG  HJK .
Write a triangle congruency statement for the triangles above by going in the following
order: two angle arcs, one angle arc, then three angle arcs.
--------------------------------------------------------------------------------------------------------------------Review of Special Quadrilaterals
Quadrilaterals
Three types of quadrilaterals are kites, parallelograms,
and trapezoids.
kites
parallelograms
all other
trapezoids quadrilaterals
--------------------------------------------------------------------------------------------------------------------A kite is a quadrilateral that has exactly two distinct pairs of adjacent congruent sides.
The diagonals of a kite are perpendicular.
--------------------------------------------------------------------------------------------------------------------The Five Properties of All Parallelograms
1.
The diagonals bisect each other.
2.
Consecutive angles are supplementary.
3.
Opposite angles are congruent.
4.
Opposite sides are parallel.
5.
Opposite sides are congruent.
--------------------------------------------------------------------------------------------------------------------Parallelograms
rhombuses squares
rectangles
all other
parallelograms
There are three types of parallelograms: rhombuses,
rectangles, and squares. Therefore, these three shapes have
all of the properties of parallelograms!
--------------------------------------------------------------------------------------------------------------------A rhombus is a type of parallelogram in which all four sides are congruent.
The diagonals of a rhombus are perpendicular.
Also, the diagonals bisect opposite angles. Essentially, this means that the diagonals cut each
interior angle into two congruent pieces.
Review of Special Quadrilaterals - Cont.
Name_________________________
A rectangle is a type of parallelogram in which all of the interior angles are congruent. This
means that each interior angle measures 90 .
The diagonals of a rectangle are congruent.
--------------------------------------------------------------------------------------------------------------------A square is a type of parallelogram in which all four sides are congruent AND all four interior
angles are congruent. A square is like a mixture of a rhombus and a rectangle. This means it has
all of the characteristics of both of them.
--------------------------------------------------------------------------------------------------------------------Finally, completely unrelated to parallelograms, a trapezoid is a quadrilateral in which exactly
one pair of opposite sides are parallel.
A special type of trapezoid is the isosceles trapezoid. An isosceles trapezoid is a trapezoid in
which the two non-parallel sides are congruent.
A special property of the isosceles trapezoid (not all trapezoids) - its diagonals are congruent.
--------------------------------------------------------------------------------------------------------------------Homework on Congruency Statements
V
S
For Questions 1-4, consider the triangles to the left, and
complete the congruency statements.
D
R
M
1. MR  ______
2. SV  ______
3. DR  ______
4. VST  ______
T
--------------------------------------------------------------------------------------------------------------------Z
For Questions 5-8, consider the congruent triangles to the right,
and complete the congruency statements.
N
5. Q  ______
B
6. E  ______
E
7. Z  ______
Q
8. NBQ  ______
F
--------------------------------------------------------------------------------------------------------------------The two triangles to the left are congruent. For Questions 9-12,
C
X
G
complete the congruency statements for the triangles.
Y
K
L
9. KG  ______
10. CY  ______
11. C  ______
12. CXY  ______
13.
Write three different segment congruency statements and three different angle
congruency statements based on the following triangle congruency statement:
RUN  IDK .
--------------------------------------------------------------------------------------------------------------------14.
Consider the following triangle congruency statement
for the figure to the right: TYH  OPH . Label the
figure with the appropriate tick marks and angle arcs
based on the given triangle congruency statement.
Y
H
T
O
P
--------------------------------------------------------------------------------------------------------------------N
A
15.
Suppose CAM  END in the diagram to the
left. Follow the instructions below.
A) Label AM with one tick mark.
C
M
B) Label EN with two tick marks.
C) Label C with one angle arc.
D) Label D with two angle arcs.
E) Completely label the remainder of the diagram,
making sure to place the proper number of tick
marks/angle arcs on the side/angle.
--------------------------------------------------------------------------------------------------------------------D
E
For Questions 16-19, consider parallelogram WXYU
to the right. Explain why the given statement is true.
Your answer should be based on one of the five
properties of all parallelograms (refer to notes if needed).
The example below is provided to guide you.
W
X
U
Z
Y
Example:
Question - WX is parallel to UY .
Answer - Opposite sides of a parallelogram are parallel.
16.
UW  XY
17.
UW is parallel to XY .
18.
WZ  ZY
19.
WXY  WUY
20.
Unrelated to the properties of a parallelogram, why is this statement true:
WZX  YZU ?