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In baseball, home plate is a pentagon. Pentagon ABCDE is a diagram of a regulation home plate. The baseball rules are very specific about the exact dimensions of this pentagon so that every home plate is congruent to every other home plate. If pentagon PQRST is another home plate, identify each congruent corresponding part.
Mr. X is an inventive person. He takes pleasure in drawing a triangle and seeing if another person can recreate his drawing from piecemeal information. For each exercise, draw a diagram to support your answer. (Hint: Begin each exercise by drawing a triangle. Measure the parts of your triangle that Mr. X gives you and try to draw a different triangle with those parts. If the two triangles are congruent, you drew Mr. X’s triangle.)
1. If Mr. X gives you the measures of the sides of a triangle, could you be sure you would draw Mr. X’s triangle?
Yes; possible answer:
2. If Mr. X gives you the measures of the angles of a triangle, could you be sure you would draw Mr. X’s triangle?
No; possible answer:
3. If Mr. X gives you the measures of one angle and of both sides of that angle, could you be sure you would draw Mr. X’s triangle?
Yes; possible answer:
4. If Mr. X gives you the measures of one side and both angles that share that side, could you be sure you would draw Mr. X’s triangle?
Yes; possible answer:
5. If Mr. X gives you the measures of one angle, one adjacent side, and the side opposite the angle, could you be sure you would draw Mr. X’s triangle? (Hint: Start with an angle less than 45� and a long adjacent side.)
Triangles are congruent if they have the same size and shape. Their corresponding parts,the angles and sides that are in the same positions, are congruent.
Corresponding Parts
Congruent Angles Congruent Sides
�A � �J�B � �K�C � �L
_AB �
_JK_
BC �_KL _
CA �_LJ
To identify corresponding parts of congruent triangles, look at the order of the vertices in the congruence statement such as �ABC � �JKL.
Given: �XYZ � �NPQ. Identify the congruent corresponding parts.
When two geometric figures are congruent, each is the image of the other under a rigid transformation. The first diagram at right shows two triangles, each positioned on an identical 3-by-3 array of dots. Are the triangles congruent?
In the second diagram at right, the triangles appear on the same array, and each dot is named as a point. The first triangle is �BEG, and the second is �HEA. It is clear that �HEA is a reflection of �BEG across a line,
‹__›DF. So �HEA
is congruent to �BEG.
� � �
� � �
� � �
Each triangle is congruent to �BEG. Identify the transformation that relates the triangle to �BEG.
On each grid, sketch a triangle congruent to �BEG different from those given above. Use only the labeled points as vertices. Name the transformation that relates the new triangle to �BEG.
5. � � �
� � �
� � �
6. � � �
� � �
� � �
7. � � �
� � �
� � �
8. � � �
� � �
� � �
reflection reflection reflection rotation 180°
across‹__›CG across
‹__›AJ across
‹__›CG about point E
9. Refer to the 3-by-3 grids in Exercises 1–8. Using the labeled points as vertices, how many triangles congruent to �BEG are there in all? List them.
10. Refer to the 3-by-3 grids in Exercises 1–8. Using the labeled points as vertices, how many triangles can be formed on each grid? List them on a separate sheet of paper, dividing the list into groups of congruent triangles.
There are 76 triangles in all.
ChallengeCongruence and Transformations on an Array4-3
1. How can you tell which angle corresponds to �L?
�O does because they both have two arcs.
2. How can you tell which side corresponds to _KL?
It is side_NO because both sides have three tick marks.
Answer the following questions based on these two triangles.
�
�
��
�
3. What angle corresponds to �LMP? �OMN
4. What angle corresponds to �P? �N
5. What side corresponds to _PL ?
_NO
6. What side corresponds to _LM ?
_OM
These two triangles are congruent. This statement can be written
� �
�
� �
�
as follows: �ABC � �XYZ.Labeling triangles in this way is meaningful because it states that in these two triangles, �A � �X; �B � �Y; and �C � �Z. The order in which the letters are placed tells which angles are congruent.
Answer the following questions based on these two triangles.
� �
�
� �
�
7. Write a congruence statement for these two triangles. �MNP � �TRS
8. How did you determine the order of the letters in your congruence statement?
Corresponding angles of congruent triangles have the same measure, and
the order of the letters indicates which angles are congruent.
One arc shows these angles are corresponding.
Two tick marks show these sides are corresponding.