Download Investigating Geometry Activity: Congruence Statements

Name———————————————————————— lesson
4.2
Date —————————————
Investigating Geometry Activity:
Congruence Statements
For use before the lesson “Apply Congruence and Triangles”
Materials: Question
tracing paper
How do you write congruence statements?
Two geometric figures that are congruent have exactly the same size and shape.
Imagine cutting out one of the congruent figures. You could then position the cutout
so that it fits perfectly over the other figure.
In two congruent figures, all the parts of one figure are congruent to the corresponding
parts of the other figure. In congruent polygons, this means that the corresponding
sides and corresponding angles are congruent.
explore
Explore congruent triangles
Study the congruent triangles below and their congruence statements. Trace one of
the triangles in each pair on tracing paper. Place the tracing paper on top of the
second triangle. Determine the corresponding sides and angles for each pair of
congruent triangles.
B
Y
C
F
X
Lesson 4.2
E
draw
conclusions
D
L
N
nCAB > nDEF
M
Z
nXYZ > nMLN
Use your observations to complete these exercises.
1. List the corresponding sides and corresponding angles for nCAB and nDEF
and for nXYZ and nMLN.
2. What is the relationship between the corresponding angles and the way the
congruence statement is written?
3. Explain how to write a congruence statement. Provide an example to support
your explanation.
4. Write congruence statements for each pair of triangles.
a.
C
K
S
L
D
Geometry
Chapter Resource Book
M
O
E
4-20
b.
N
J
R
T
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
A
By the Triangle Sum Theorem, you know that
m∠ 2 1 m∠ 4 1 m∠ 6 5 1808. The exterior angles
of the triangle are ∠ 1, ∠ 3, and ∠ 5. To find the
sum of these three angles, subtract 1808 from the
sum of all six angles, 5408, to obtain 3608.
Lesson 4.2 Apply Congruence
and Triangles
Teaching Guide
Check drawings; 5 different colors must be
used. 2. Check drawings. There are 4 large right
triangles (in the corners), 16 small right triangles,
8 squares, 8 large parallelograms, and 16 small
parallelograms.
1.
Investigating Geometry Activity
} } } } } }
1. ​CA​ > DE​
​  ; AB​
​  > EF​
​  ; BC​
​  > FD​
​  ; ∠ C > ∠ D;
} } } }
​  ; YZ​
​  > LN​
​  ;
∠ A > ∠ E; ∠ B > ∠ F; XY​
​  > ML​
} }
​  ; ∠ X > ∠ M; ∠ Y > ∠ L; ∠ Z > ∠ N.
​ZX​ > NM​
2. The congruence statement is written so that
corresponding vertices are written in the same
order. 3. Sample answer: To write a congruence
statement for two triangles, always list the
corresponding vertices in the same order. For
} } } } } }
​  , BC​
​  > YZ​
​ ,  CA​
​  > ZX​
​  ,
example, if AB​
​  > XY​
∠ A > ∠ X, ∠ B > ∠ Y, and ∠ C > ∠ Z, then
n ABC > nXYZ. 4. a. nCED > nJLK
b. nNOM > nSTR.
Practice Level A
}
} }
}
1. Check student diagram; CP​
​  ù BI​
​  ; PN​
​  ù IY​
​  ; ​
}
}
NC​ ù YB​
​ ;  ∠ C ù ∠ B; ∠ P ù ∠ I; ∠ N ù ∠ Y
}
2. ​OP​  3. ∠ F 4. ∠ G 5. 1108 6. 7 km
7. n QPO 8. n EIG ù n QOM; all
corresponding sides and angles are congruent.
9. none 10. n KLN ù n MNL; all
corresponding sides and angles are congruent.
11. n DEG ù n HEF; all corresponding sides and
angles are congruent. 12. 25 13. 70 14. 11
15. 19
y E(2, 4)
16. Sample answer:
}
} }
}
1. Check student diagram; AM​
​  ù CD​
​  ; AT​
​  ù ​CN​ ; ​
}
}
MT​ ù DN​
​  ; ∠ A ù ∠ C; ∠ M ù ∠ D; ∠ T ù ∠ N
}
2. ∠ T 3. ​HS​  4. 488 5. 738 6. 5 cm
7. n JTM 8. n DEG ù n FGE; all
­corresponding sides and angles are congruent.
9. none 10. n XWY ù n ZWY; all
corresponding sides and angles are congruent.
11. 25 12. 25 13. x 5 17; y 5 17
14. a 5 9; b 5 8
y
15. Sample answer:
(5, 9)
(1, 7) C
2
A
(7, 7)
B
2
x
16. Reflexive Property of Congruence;
∠ A ù ∠ C; Definition of congruent triangles
17. Transitive Property of Congruent Triangles
Practice Level C
}
1. IW​
​   2. ∠ L 3. ∠ G 4. 338 5. 4 cm
6. n IGW 7. n ZTX ù n WVY; all
corresponding sides and angles are congruent.
8. CDEF ù HIJK; all corresponding sides and
angles are congruent. 9. x 5 60; y 5 66
10. x 5 2; y 5 24 11. The corresponding sides
need to be shown as congruent. 12. 808
13. Reflexive Property of Congruence; Third
Angles Theorem; Definition of Congruent
Triangles 14. Definition of Midpoint; Alternate
Interior Angles Theorem; Vertical Angles
Theorem; Definition of Congruent Triangles
Study Guide
1. 3 2. 5 3. 7 4. 11 5. 65 6. 2
Math and History Application
1. about 186 meters 2. 21,390 m2
3. 20,736 m2 4. Sample answer: The area of the
square in Exercise 3 and the triangle in Exercise 2
appear to be close enough to support Herodotus’
claim.
Challenge Practice
1. Sample answer:
F(6, 2)
1
Practice Level B
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
answers
Lesson 4.1 Apply Triangle Sum
Properties, continued
(4, 0)
D(2, 0)
7
x
(8, 22)
(4, 24)
A46
Geometry
Chapter Resource Book
CS10_CC_G_MECR710761_C4AK.indd 46
4/28/11 6:14:06 PM