Download Packet 1 for Unit 3 M2Geo

M2 GEOMETRY PACKET 1 FOR UNIT 3 – SECTIONS 4-1 TO 4-3
ASSIGNMENTS FOR UNIT 3, PACKET 1
DUE
NUMBER
ASSIGNMENT
3A
p. 241-242
# 36, 37, 40-43 all
3B
p. 251
# 17-22 all
3C
p. 259 # 9
p. 261 # 29, 30
TOPICS
4-1:
Vocabulary: acute/right/obtuse triangles,
equilateral/isosceles/ scalene triangles
Classify triangles by lengths of sides and
measures of angles
Use algebra to solve problems involving sides
or angles of triangles
4-2:
Use algebra to solve problems involving
interior and exterior angles of a triangle
4-3:
Vocabulary: congruent triangles, corresponding
parts
Name pairs of congruent triangles
Use corresponding parts to solve problems
1
M2 GEOMETRY PACKET 1 FOR UNIT 3 – SECTIONS 4-1 TO 4-3
Classifying Triangles
Any triangle can be classified in 2 different ways:
…by the measure of its angles:
 Acute – all 3 angles are acute
 Right – one angle is right
 Obtuse – one angle is obtuse
…by the lengths of its sides:
 Equilateral – all 3 sides are congruent
 Isosceles – at least 2 sides are congruent
 Scalene – none of the sides are congruent
Classify each triangle as acute, obtuse, or right and also as equilateral, isosceles, or scalene.
1.
2.
12
3.
40 3
30.7
28.3
18
4.
5.
14.2
14.2
6.
8
8 2
27.7
24
8
12
7. Find x and the length of each side if RST is equilateral.
x = _____
RS = ______
ST = ______
TR = ______
8. Find y and the length of each side if ABC is isosceles with AB = BC.
y = ______
18
18
40
40
AB = ______
BC = ______
2
CA = ______
13
M2 GEOMETRY PACKET 1 FOR UNIT 3 – SECTIONS 4-1 TO 4-3
Angles of Triangles
1. Fill in the blank in the theorem below:
Triangle Angle
Sum Theorem
The sum of the measures of the angles of a triangle is ______.
In the figure at the right, mA  mB  mC  ______ .
An exterior angle is formed by extending a side of a polygon. For each exterior angle, the
remote interior angles are the interior angles that are not adjacent to that exterior angle.
2. Use RST shown at right to answer the questions below:
a. Find the missing measures:
mRTS  ______
m1  ______
b. Which angle in the diagram is an exterior angle, and which angles are remote interior
to this angle? (Give angle names, not measures.)
Exterior: __________
Remote interior: __________ and __________
c. How is the measure of the exterior angle related to the measures of its remote interior
angles in this diagram?
3. Use QRS shown at right to answer the questions below:
a. Find the missing measures:
mRQS  ______
mS  ______
b. Which angle in the diagram is an exterior angle, and which angles are remote interior
to this angle? (Give angle names, not measures.)
Exterior: __________
Remote interior: __________ and __________
c. How is the measure of the exterior angle related to the measures of its remote interior
angles in this diagram?
3
M2 GEOMETRY PACKET 1 FOR UNIT 3 – SECTIONS 4-1 TO 4-3
Exterior Angle
Theorem
The measure of an exterior angle of a triangle is equal to
the sum of the measures of the two remote interior angles.
In the diagram, m1  mA  mB .
4. Complete the proof of the Exterior Angle Theorem:
Given: 1 is an exterior angle of ABC
Prove: m1  mA  mB
Statements
Reasons
1. mA  mB  mBCA  ______
1.
2. m1  mBCA  180
2.
3. m1  mBCA 
3. Transitive Property
4.
4.
5. Use the Exterior Angle Theorem to find the measures of each numbered angle.
a.
b.
m1  ____  ____  ____
m1  ____  ____  ____
m2  ____
c.
m3  ____  ____  ____
m1  ____
____  m2  80  m2  ____
4
M2 GEOMETRY PACKET 1 FOR UNIT 3 – SECTIONS 4-1 TO 4-3
Congruent Triangles
Triangles that have the same size and same shape are
congruent triangles.
If 2 triangles are congruent, then all 3 pairs of corresponding angles and all 3 pairs of
corresponding sides are congruent.
Conversely, if all 3 pairs of corresponding angles and all 3 pairs of corresponding sides of 2
triangles are congruent, then the triangles are congruent.
In the figure above, ABC  RST . The order of the letters must indicate how the angles and
sides correspond. For example, ABC  RST indicates that A  B because A and B are
the 1st letters in each name. As another example, using the 1st and 3rd letters, AC  RT .
Third Angles
Theorem
If 2 angles of one triangle are congruent to 2 angles of a 2nd triangle, then
the 3rd angles of the triangles are congruent.
Abbreviation: if 2  s in 2  s are  , so are the 3rd  s
1. Show that the triangles are congruent by identifying all congruent corresponding parts.
Then write a congruence statement about the triangles.
a.
Angles:
Triangles:
Sides:
 ____   ____

 ____   ____

 ____   ____

 ______   ______
b.
Angles:
Triangles:
Sides:
 ____   ____

 ____   ____

 ____   ____

 ______   ______
5
M2 GEOMETRY PACKET 1 FOR UNIT 3 – SECTIONS 4-1 TO 4-3
1. (continued) Show that the triangles are congruent by identifying all congruent
corresponding parts. Then write a congruence statement about the triangles.
c.
Angles:
Triangles:
Sides:
 ____   ____

 ____   ____

 ____   ____

 ______   ______
2. Suppose ABC  DEF . Find the values of x and y.
6
M2 GEOMETRY PACKET 1 FOR UNIT 3 – SECTIONS 4-1 TO 4-3
PRACTICE PROBLEMS FOR 4-1 TO 4-3
In #1-6, classify each triangle as acute, obtuse, or right.
1.
2.
3.
4.
5.
6.
In #7-10, classify each triangle as equilateral, isosceles, or scalene.
7. ABE
8. EDB
9. EBC
10. DBC
11. Find x and the length of each side if ABC is isosceles with AB  BC .
x = ______
AB = ______
BC = ______
AC = ______
12. Find x and the length of each side if FGH is equilateral.
x = ______
FG = ______
GH = ______
FH = ______
7
M2 GEOMETRY PACKET 1 FOR UNIT 3 – SECTIONS 4-1 TO 4-3
13. Find each measure.
m1  ______
m2  ______
m3  ______
14. Find each measure.
m1  ______
m2  ______
m3  ______
m4  ______
m5  ______
15. Show that the triangles are congruent by identifying all congruent corresponding parts. Then write a
congruence statement about the triangles.
a.
Angles:
Triangles:
Sides:
 ____   ____

 ____   ____

 ____   ____

 ______   ______
b.
Angles:
Triangles:
Sides:
 ____   ____

 ____   ____

 ____   ____

 ______   ______
8
M2 GEOMETRY PACKET 1 FOR UNIT 3 – SECTIONS 4-1 TO 4-3
REVIEW PROBLEMS FOR 4-1 TO 4-3
Classify each triangle as acute, equiangular, obtuse, or right and as scalene, isosceles, or equilateral.
1.
2.
8
8
12.25
3.
15.8
10
17.9
Sketch and label each triangle. Then find x and the measure of each side.
4. FGH is equilateral, with FG = x +5, GH = 3x – 9, and FH = 2x –2.
Sketch:
x = ______
FG = ______
GH = ______
FH = ______
5. LMN is isosceles, with LM =LN, LM = 3x –2, LN =2x +1, and MN = 5x –2.
Sketch:
x = ______
LM = ______
LN = ______
MN = ______
9
6
6 3
12
M2 GEOMETRY PACKET 1 FOR UNIT 3 – SECTIONS 4-1 TO 4-3
6. If K  3, 2  , P  2,1 , and L  2, 3 , find the lengths of the sides of KPL , and classify the triangle by the
lengths of its sides.
In #7-8, find the measure of each numbered angle.
7.
8.
9. Show that the triangles are congruent by identifying all congruent corresponding parts. Then write a
congruence statement about the triangles.
Angles:
Triangles:
Sides:
 ____   ____

 ____   ____

 ____   ____

 ______   ______
10