Download Geometry 1 Final Exam Review

Geometry 1
Final Exam Review
1)
The larger of two supplementary angles is three times as big as the smaller angle. Find
the measure of the larger angle.
2)
If a ⊥ b and ∠ 2 = 54°,
then ∠ 1 = ______°.
1
a
2
b
3)
If M is the midpoint of XY , name two line segments that must be equal. NOTE – You
need to draw your own diagram to answer this question.
4)
GIVEN: G is the midpoint of JH ; GH ≅ HI
I
H
PROVE: JG ≅ HI
G
J
STATEMENTS
REASONS
1) G is the midpoint of JH
1)
2) JG ≅ GH
2)
3) GH ≅ HI
3)
4) JG ≅ HI
4)
5)
If x − 5 = 11, then x = 16 illustrates what algebra property?
6)
AB ≅ AB illustrates what algebra property?
7)
If m || n and ∠ 1 = 110°, then ∠ 3 = _______°.
m
8)
If m || n and ∠ 6 = (4x)° and ∠ 7 = (2x + 6)°,
then x = ______.
6
2
n
9)
5
1
If ∠ 1 ≅ ∠ 8, write the postulate or theorem that
justifies why m || n.
3
4
7
8
10)
Name a pair of alternate exterior angles.
11)
If ∠ 6 = 100°, then ∠ 8 must equal ______° for line m to be parallel to line n.
12)
Find the slope of a line that passes through the points (5, –2) and (–5, 10).
13)
Line 1 has a slope of –5 / 3. Line 2 is parallel to Line 1 and Line 3 is perpendicular to
Line 1. Find the slope of Line 2 and Line 3.
14)
A triangle with an obtuse angle and no equal sides would be classified as
a(n) _______ ______ triangle. NOTE – this problem requires two (2) answers.
15)
If ∠ A = 90° and ∠ BCD = 140°,
then ∠ B = ______°.
16)
If ∠ A = 90°, ∠ B = 50°, and
∠ BCA = (2x)°, then x = ______.
17)
Find the sum of the interior angles of an octagon.
18)
Find the sum of the exterior angles of a pentagon.
19)
Find the measure of each interior angle of a regular octagon.
20)
Find the measure of each exterior angle of a regular 18-gon.
B
A
C
D
21)
Find the number of sides of a regular polygon in which each interior angle has a
measure of 165°.
22)
Find the number of sides of a regular polygon in which each exterior angle has a
measure of 10°.
23)
If PR ≅ TR and QR ≅ SR, which
method would be used to verify
that Δ PQR ≅ ΔTSR?
P
Q
R
24)
If ∠ Q and ∠ S are right angles and
QR ≅ SR, which pair of angles or segments
must be congruent to verify that
Δ PQR ≅ ΔTSR using the HL method?
S
T
DIAGRAM FOR
QUESTIONS #23 – #26.
25)
If PQ ≅ TS and QR ≅ SR, what else must be
congruent if one wants to use the SSS method
to make Δ PQR ≅ ΔTSR?
26)
If Δ PQR ≅ ΔTSR, which segment would be
equal to PR ? Explain why.
27)
In Δ BCD, if BC ≅ CD, then which two angles must be equal? NOTE – You need to
draw your own diagram to answer this question.
28)
In Δ BCD, BC = 3, CD = 4, and DB = 5. Which angle in Δ BCD would be the largest?
NOTE – You need to draw your own diagram to answer this question.
29)
In Δ BCD, ∠ B = 60° and ∠ C = 100°. Which side of Δ BCD is the shortest?
NOTE – You need to draw your own diagram to answer this question.
30)
Can a triangle be formed by segments measuring 4 cm, 6 cm, and 9 cm?
31)
If TR ≅ ZX , TS ≅ ZY , ∠ T = 40°, and
∠ Z = 20°, what conclusion can you reach
about the diagrams to the right?
32)
If TR ≅ ZX , TS ≅ ZY , YX = 5 cm, and
S
R
SR = 8 cm, what conclusion can you reach
about the diagrams to the right?
T
Y
X
Z
33)
The coordinates of point A are (–1, –2) and the coordinates of point B are (3, 6). Find
the coordinates of the midpoint of AB using the Midpoint Formula.
34)
The coordinates of point A are (–1, –2) and the coordinates of point B are (3, 6). Find
the length of AB using the Distance Formula.
35)
Solve the equation 5(x – 6) = 20, and provide a reason for each step.
36)
In Δ XYZ, ∠ X = (3x – 5)º, ∠ Y = (2x + 10) º, and ∠ Z = (7x + 7) º. What is the measure
of ∠ X?
37)
Find the value of x in the diagram to the right.
(x + 23)º
145º
xº
38)
RS is the midsegment of Δ QTU.
Find the value of x.
S
Q
U
x-1
24
R
T
39)
A triangle has one side of length of 15 cm and another of length 10 cm. Write an
inequality that best represents the possible lengths of the third side.
40)
Find the perimeter of Δ VXW.
V
Which method can be used to prove
Δ ADB ≅ Δ CDB?
A
2x - 1
x-2
X
41)
W
D
B
x+5
C
42)
The endpoints of JK are J (2, 4) and K (3, − 4). Find the endpoints of J ' K '
the translation ( x, y ) → ( x − 12, y + 8).
after
43)
The endpoints of JK are J (2, 4) and K (3, − 4). Find the endpoints of J ' K '
the segment is reflected about the y-axis.
after
44)
The endpoints of JK are J (2, 4) and K (3, − 4). Find the endpoints of J ' K '
the segment after it is rotated 180º around the origin.
after
45)
Name a line segment that appears to be parallel to AF .
E
D
46)
Name a line segment that appears to be perpendicular
to EF .
F
A
47)
Name a line that is skew to BD.
48)
Draw a rectangle. How many lines of symmetry does it have?
C
B
*********************ANSWERS*********************
1)
∠ 1 – larger, ∠ 2 – smaller → ∠ 1 = 3x, ∠ 2 = x →
Supplementary ∠s add up to 180° → ∠ 1 + ∠ 2 = 180° → 3x + x = 180 → 4x = 180 →
x = 45 → ∠ 1 = 3 ⋅ 45 = 135° → Larger angle = 135°
2)
Perpendicular lines meet to form a right angle → If a ⊥ b, then ∠ 1 + ∠ 2 = 90° →
If ∠ 1 = 54°, then ∠ 2= 90 – 54 = 36°
3)
The midpoint of a segment is a point that divides
a segment into two ≅ segments → XM ≅ MY
X
Midpoint
M
4)
G is the midpoint of JH
→
Given
JG ≅ GH
→
Definition of a Midpoint
Y
The midpoint of a segment divides a segment into two ≅ segments.
GH ≅ HI
→
Given
JG ≅ HI
→
Transitive Property
JG and HI are both ≅ to GH , so they must be ≅ to each other.
5)
Addition Property – If a = b, then a + c = b + c →
x − 5 = 11 → x − 5 + 5 = 11 + 5 → x = 16
6)
Reflexive Property – a = a → AB ≅ AB
7)
∠ 1 and ∠ 3 are corresponding ∠s → If lines are parallel, then corresponding angles
are congruent → If ∠ 1 = 110°, then ∠ 3 = 110°
8)
∠ 6 and ∠ 7 are consecutive interior ∠s → If lines are parallel, then consecutive
interior angles are supplementary → ∠ 6 + ∠ 7 = 180° → ( 4 x ) + ( 2 x + 6 ) = 180 →
6 x + 6 = 180 → 6 x = 174 → x = 29
9)
∠ 1 and ∠ 8 are alternate exterior ∠s → If alternate exterior angles are congruent, then
the lines are parallel.
10)
Alternate exterior angles → both exterior, opposite sides of transversal, not adjacent.
Possible answers are ∠ 4 & ∠ 5 and ∠ 8 & ∠ 1
11)
∠ 6 and ∠ 8 are corresponding ∠s → If corresponding angles are congruent, then the
lines are parallel → If ∠ 6 = 100°, then ∠ 8 must equal 100° for m to be parallel to n.
12)
m=
13)
If Line 1 and Line 2 are parallel, then their slopes must be equal →
Slope of Line 2 = – 5 / 3; If Line 1 and Line 3 are perpendicular, then their slopes must
be opposite reciprocals → Slope of Line 3 = 3 / 5
14)
Obtuse Triangle → a triangle with one obtuse angle; Scalene Triangle → a triangle
with no congruent sides → Obtuse Scalene
15)
If ∠ BCD = 140, then ∠ BCA = 40° because they form a linear pair →
∠ B + ∠ A + ∠ BCA = 180° because the sum of the angles of a triangle is 180° →
∠ B = 180 – 90 – 40 = 50°
16)
∠ B + ∠ A + ∠ BCA = 180° because the sum of the angles of a triangle is 180° →
50 + 90 + ( 2 x ) = 180 → 2 x + 140 = 180 → 2 x = 40 → x = 20
10 − ( −2 ) 12
y2 − y1
6
6
→m=
=
=
=−
x2 − x1
−5 − 5
−10 −5
5
17)
Octagon = 8 sides → ( 8 − 2 ) ⋅180 = 6 ⋅180 = 1080°
18)
The sum of the exterior angles of a convex polygon is 360º, no matter how many sides
the polygon has.
19)
Octagon = 8 sides → ( 8 − 2 ) ⋅180 = 6 ⋅180 = 1080° → A regular polygon has all equal
sides and all equal angles → Measure of one interior angle = 1080° ÷ 8 = 135º
20)
The sum of exterior angles of a convex polygon is 360°, no matter how many sides the
polygon has → Measure of one exterior angle = 360° ÷ 18 = 20º
21)
If the interior angle = 165°, then the exterior angle adjacent to it = 15° → The sum of
the exterior angles is 360° → 360° ÷ 15° = 24 exterior angles → 24 exterior angles
means 24 sides
22)
The sum of the exterior angles is 360° → 360° ÷ 10° = 36 exterior angles → 36 exterior
angles means 36 sides
23)
∠ PRQ ≅ ∠ SRT because
vertical angles are congruent →
Side – Angle – Side or SAS
P
24)
PR ≅ TR – The needed segment is
marked in red.
P
Q
Q
R
R
S
25)
T
PR ≅ TR – The needed segment is
marked in red.
P
S
26)
Q
T
If Δ PQR ≅ ΔTSR, then all the
corresponding sides and corresponding
angles must be congruent (CPCTC).
The corresponding side to PR is TR
P
Q
R
R
S
T
S
27)
If two sides of a triangle are ≅,
28)
T
DB is the longest side →
then the angles opposite those sides
∠ C is opposite DB so
are ≅ → ∠ D is opposite BC ;
∠ C is the largest angle
∠ B is opposite CD → ∠ D ≅ ∠ B
B
B
C
D
3
C
5
4
D
29)
∠ D = 180 – 100 – 60 = 20° → ∠ D is the smallest angle →
BC is opposite ∠ D so BC is the shortest side
B
60°
100°
20°
C
D
30)
4 + 6 is greater than 9 → Yes, a triangle can be formed
31)
SR is opposite the larger ( 40° ) angle,
YX is opposite the smaller ( 20° ) →
32)
( )
∠ T is opposite the longer side ( SR ) ,
∠ Z is opposite the shorter side YX ,
∠ Z < ∠ T or ∠ T > ∠ Z
SR > YX or YX < SR
Y
Y
S
S
5 cm
20°
X
Z
X
40°
R
Z
8 cm
R
T
T
33)
y + y2 ⎞
⎛x +x
⎛ −1 + 3 −2 + 6 ⎞ ⎛ 2 4 ⎞
Midpoint = ⎜ 1 2 , 1
,
⎟ = ⎜ , ⎟ = (1, 2 )
⎟ → Midpoint = ⎜
2 ⎠
2 ⎠ ⎝2 2⎠
⎝ 2
⎝ 2
34)
Distance =
( x2 − x1 ) + ( y2 − y1 )
2
2
→ Distance =
( 3 − ( −1) ) + ( 6 − ( −2 ) )
2
Distance = 42 + 82 = 16 + 64 = 80 ≈ 8.9 → AB = 80 ≈ 8.9
35)
5( x − 6) = 20 − − Given
5 ⋅ x − 5 ⋅ 6 = 20 → 5 x − 30 = 20 − − Distributive Property
5 x − 30 + 30 = 20 + 30 → 5 x = 50 − − Addition Property
5 x / 5 = 50 / 5 → x = 10 − − Division Property
2
→
36)
∠ X + ∠ Y + ∠ Z = 180° because the sum of the angles of a triangle is 180° →
( 3x − 5 ) + ( 2 x + 10 ) + ( 7 x + 7 ) = 180 → 12 x + 12 = 180 → 12 x = 168 → x = 14 →
∠ X = ( 3 x − 5 ) ° = 3 ⋅14 − 5 = 37°
37)
38)
The 145° angle is an exterior angle → The exterior angle of a triangle is equal to the
sum of the two nonadjacent interior angles → 145 = x + ( x + 23) → 145 = 2 x + 23 →
122 = 2 x → 61 = x
1
1
RS = TU → x − 1 = ⋅ 24 → x − 1 = 12 → x = 13
2
2
39)
If the 3rd side is the shortest side, 3rd side + 10 > 15 → 3rd side > 5
If the 3rd side is the longest side, then 10 + 15 > 3rd side → 25 > 3rd side
5 < 3rd side < 25
40)
If ∠ V = ∠ X , then XW = VW → x + 5 = 2 x − 1 → 6 = x
XW = x + 5 = 6 + 5 = 11; VW = 2 x − 1 = 2 ⋅ 6 − 1 = 12 − 1 = 11; VX = x − 2 = 6 − 2 = 4
Perimeter of Δ VXW = XW + VW + VX = 11 + 11 + 4 = 26
41)
∠ ADB ≅ ∠ CDB → All right ∠s are ≅
DB ≅ DB → Reflexive Property
∠ A ≅ ∠ C → Given
A
Angle – Angle – Side or AAS
D
D
C
42)
( x, y ) → ( x − 12, y + 8) → J ( 2, 4 ) → J ' ( 2 − 12, 4 + 8 ) = J ' (10, 12 ) ;
K ( 3, − 4 ) → K ' ( 3 − 12, − 4 + 8 ) = K ' ( −9, 4 )
43)
Reflection about the y-axis: ( x, y ) → ( − x, y )
J (2, 4) → J ' ( − ( 2 ) , 4 ) = J ' ( −2, 4 ) ;
K ( 3, − 4 ) → K ' ( − ( 3) , − 4 ) = K ' ( −3, − 4 )
B
B
44)
180° rotation about the origin: ( x, y ) → ( − x, − y )
J (2, 4) → J ' ( − ( 2 ) , − ( 4 ) ) = J ' ( −2, − 4 ) ;
K (3, − 4) → K ' ( − ( 3) , − ( −4 ) ) = K ' ( −3, 4 )
45)
AF is part of the bottom face and the left face. You need lines that are also part of the
bottom face or left face that do not intersect AF . Possible answers are BC and DE.
46)
Perpendicular means “meets to form a right angle.” Possible answers are
ED, AF , and FC.
47)
BD is part of the top face. You need lines that are not part of the top face that do not
intersect BD. Possible answers are AF , EF , and FC.
48)
Two lines of symmetry