Then choose any three adjacent angles, and notice that they form a straight angle.. B The opposite angles in a parallelogram must be equal, and any two “consecutive angles” as you move a
Trang 1Concept Review 1: Lines and Angles
Questions 1 and 2 refer to the diagram above
1 List all of the different pairs of angles that are congruent (equal)
2 List all of the different sets of angles that have a sum of 180°
Mark the figure to show the following information: AD ⏐⏐ HN, AI ⏐⏐ BM, and HD ⏐⏐ JL Then list the angles in the
figure that have the given characteristic:
H
N M
J
I
L
G F
6
7 8
9
10
11
12
13
14 15
a b c d e
r
5 Two angles supplementary to ∠9 6 One angle equal to ∠15
7 Three angles supplementary to ∠13
State whether each of the following pairs is supplementary (has a sum of 180 °), equal, or neither.
14 ∠6 and ∠7
l1and l2are lines and r is a ray.
Trang 2SAT Practice 1: Lines and Angles
Note: Figure not drawn to scale.
1. The figure above shows the intersection of
three lines x=
(A) 16
(B) 20
(C) 30
(D) 60
(E) 90
2. The figure above shows a parallelogram with
one side extended If z = 40, then y =
(D) 110
(E) 120
Note: Figure not drawn to scale
3. In the figure above, if 艎1⏐⏐艎2, then a + b =
(A) 130
(B) 270
(C) 280
(D) 290
(E) 310
130°
ᐉ1
ᐉ2
y° z°
z°
z°
4. In the figure above, if l1⏐⏐l2, then what is the
value of n in terms of m?
(A) 355 − 2m
(B) 185 − 2m
(C) 175 − 2m
(D) 95 − 2m
(E) 85 − 2m
5. In the figure above, if l1⏐⏐l2, then x=
(D) 101 (E) 111
6. In the figure above, if and FJ ––bisects
∠HFG, what is the measure of ∠FJH?
(A) 14 (B) 38 (C) 40 (D) 56 (E) 76
FG HJ
F
J H
G
76°
43°
36°
x°
ᐉ1
ᐉ2
m°
m + 5°
n°
ᐉ1
ᐉ2
x°
2x° 3x°
Trang 3Note: Figure not drawn to scale.
7. In the figure above, if 艎1⏐⏐艎2and 艎2⏐⏐艎3, then a =
(A) 50
(B) 55
(C) 60
(D) 65
(E) 70
130°
80°
a°
ᐉ1 ᐉ2 ᐉ3
Note: Figure not drawn to scale
8. In the diagram above, if 艎1⏐⏐艎2, then x= (A) 65
(B) 60 (C) 50 (D) 45 (E) 40
50°
40°
30°
ᐉ2
There’s a lot of detail to learn and understand to do well on the SAT For more tools and resources that will help, visit our Online Practice Plus at www.MHPracticePlus.com/SATmath.
Trang 4Concept Review 1
1 Only c and e are congruent.
2 a + b + c = 180°, c + d = 180°, d + e = 180°,
e + a + b = 180°
3 ∠5
4 ∠7 and ∠13
5 ∠12 and ∠14
6 ∠4
7 ∠11, ∠8, and ∠6
Answer Key 1: Lines and Angles
8 equal
9 supplementary
10 neither
11 supplementary
12 neither
13 supplementary
14 supplementary
4 C Notice that the m° angle has a “corresponding” angle below that has the same measure (Notice
that they form an F.) Then (m + 5) + n + m = 180.
Subtract (5 + 2m): n = 175 − 2m
5 C Draw an extra line through the vertex of the angle that is parallel to the other two Notice that
this forms two “Z” pairs Therefore, x= 36 + 43 = 79
43°
36°
ᐉ1
ᐉ2
43 ° 36°
m°
m+5°
n°
ᐉ1
ᐉ2
m°
SAT Practice 1
1 C Draw in the three angles that are “vertical,”
and therefore congruent, to the angles that are
shown Then choose any three adjacent angles,
and notice that they form a straight angle
There-fore, x + 2x + 3x = 180 So 6x = 180 and x = 30.
2 B The opposite angles in a parallelogram must be
equal, and any two “consecutive angles” as you
move around the figure must be supplementary
(Notice that consecutive angles form C’s or U’s If
you’re not sure why this theorem is true, sketch a
few sample parallelograms and work out the angles.)
The angle opposite the y ° must also measure y°, and
when this is added to the three z° angles, they form
a straight angle Therefore, y+ 40 + 40 + 40 = 180
and y= 60
3 E In the triangle, the angles must have a sum of
180° (See the next lesson for a simple proof.)
Therefore, the other two angles in the triangle
must have a sum of 50° Pick values for these two
angles that add up to 50°, and write them in It
doesn’t matter how you do it: 25° and 25°, 20° and
30°, 40° and 10°, as long as they add up to 50° You
can then find the values of a and b by noticing that
they form straight angles with the interior angles
So if the interior angles are 25° and 25°, then a and
b must both be 155.
Trang 56 B There are many relationships here to take
advantage of Notice that ∠HFG and the 76° angle
form a “Z,” so ∠HFG = 76° Remember that
“bi-sect” means to divide into two equal parts, so
∠HFJ = ∠ JFG = 38° Then notice that ∠JFG and
∠FJH form a “Z,” so ∠FJH = 38°.
7 A Consider the triangle with the a° angle The
other two angles in the triangle can be found from
the given angles Notice the “Z” that contains the two
80° angles and the “U” that contains the 130° and 50°
angles Therefore, a + 80 + 50 = 180, so a = 50.
130°
80°
a°
ᐉ1 ᐉ2 ᐉ3
50 °
80 °
F
J H
G
76°
38 °
38 °
38°
1 0 4 °
8 E Draw two more parallel lines and work with the Z’s Your figure should look like the one above
50 °
40 °
30°
ᐉ1
ᐉ2
30 °
10 °
10 °
40 °
40 °
Trang 6Angles in Polygons
Remembering what you learned about parallel lines
in the last lesson, consider this diagram:
We drew line 艎 so that it is parallel to the opposite side
of the triangle Do you see the two Z’s? The angles
marked a are equal, and so are the angles marked c.
We also know that angles that make up a straight line
have a sum of 180°, so a b c 180 The angles
in-side the triangle are also a, b, and c.
Therefore, the sum of angles in a triangle is
always 180°
Every polygon with n sides can be divided into n− 2
triangles that share their vertices (corners) with the
polygon:
Therefore, the sum of the angles in any polygon
with n sides is 180(n 2)°.
Angle-Side Relationships in Triangles
A triangle is like an alligator mouth with a stick in it:
The wider the mouth, the bigger the stick, right?
Therefore, the largest angle of a triangle is
al-ways across from the longest side, and vice
versa Likewise, the smallest angle is always
across from the shortest side
Example:
In the figure below, 72 70, so a b.
70° 72°
a b
a°
a°
b°
c°
c° ᐉ
An isosceles triangle is a triangle with two equal sides If two sides in a triangle are equal, then the angles across from those sides are equal, too, and vice versa
The Triangle Inequality
Look closely at the figure below The shortest path
from point A to point B is the line segment connect-ing them Therefore, unless point C is “on the way” from A to B, that is, unless it’s on AB ––, the distance
from A to B through C must be longer than the direct
route In other words:
The sum of any two sides of a triangle is always greater than the third side This means that the length of any side of a triangle must be be-tween the sum and the difference of the other two sides
The External Angle Theorem
The extended side of a triangle forms an exter-nal angle with the adjacent side The exterexter-nal angle of a triangle is equal to the sum of the two “remote interior” angles Notice that this follows from our angle theorems:
a + b + x = 180 and c + x = 180;
therefore, a + b = c
a°
b°
c°
x°
A
C
B
12
10
Lesson 2: Triangles
12 − 10 < AB < 12 + 10
2 < AB < 22
5 sides, 3 triangles
= 3(180°) = 540° 7 sides, 5 triangles= 5(180°) = 900°
Trang 71 The sum of the measures of the angles in a quadrilateral is .
2 The sum of the measures in an octagon is
3 In ΔABC, if the measure of ∠A is 65° and the measure of ∠B is 60°, then which side is longest? .
4 The angles in an equilateral triangle must have a measure of
5 Can an isosceles triangle include angles of 35° and 60°? Why or why not?
6 Draw a diagram to illustrate the external angle theorem
7 If a triangle has sides of lengths 20 and 15, then the third side must be less than but greater than
8 Is it possible for a triangle to have sides of lengths 5, 8, and 14? Why or why not?
9 If an isosceles triangle includes an angle of 25°, the other two angles could have measures of and or and
10 In the figure above, QS –– and RT –– are diameters of the circle and P is not the center Complete the statement
below with >, <, or =
PQ + PR + PS + PT QS + RT
(not shown)
R
T
S
Q P
R
T
S Q
Concept Review 2: Triangles
Trang 81. In the figure above, if AB = BD, then x =
(A) 25
(B) 30
(C) 35
(D) 50
(E) 65
2. In the figure above, a + b + c + d =
a°
b ° c°
d°
40°
50°
x°
A B C
D
3. The three sides of a triangle have lengths
of 9, 16, and k Which of the following could equal k?
I 6
II 16 III 25 (A) I only (B) II only (C) I and II only (D) II and III only (E) I, II, and III
Note: Figure not drawn to scale.
4. Which of the following statements about a and
b in the figure above must be true?
I a = b
II a + b = 90 III a < 60
(A) I only (B) II only (C) I and II only (D) II and III only (E) I, II, and III
5. In the figure above, if AD = DB = DC, then
x + y =
(D) 100 (E) 120
B
D
y°
x° 100°
b°
a°
SAT Practice 2: Triangles
1
2
3
4
5
7
8
9
6
1
0
2
3
4
5
7
8
9
6
1 0
2 3 4 5
7 8 9 6
1 0
2 3 4 5
7 8 9 6
Trang 96. In the figure above, which of the following
expresses a in terms of b and c?
(A) 180 − (b + c)
(B) 180 − (b − c)
(C) 90 − (b + c)
(D) 90 − (b − c)
(E) b + c
Note: Figure not drawn to scale
7. Which of the following represents the correct
ordering of the lengths of the five segments in
the figure above?
(A) AD > AB > DB > BC > DC
(B) AD > DB > AB > DC > BC
(C) AD > DB > AB > BC > DC
(D) AD > AB > DB > DC > BC
(E) AD > DB > DC > AB > BC
30°
35°
60°
65°
C D
a°
b°
c°
8. A triangle has two sides of lengths 4
centi-meters and 6 centicenti-meters Its area is n square centimeters, where n is a prime number What
is the greatest possible value of n?
(A) 11 (B) 12 (C) 19 (D) 23 (E) 24
Trang 10Concept Review 2
1 360°
2 1,080°
3 Draw a diagram If the measure of ∠A is 65° and
the measure of ∠B is 60°, then the measure of ∠C
must be 55°, because the angles must have a sum
of 180° Since ∠A is the largest angle, the side
op-posite it, BC, _ must be the longest side
4 60° Since all the sides are equal, all the angles
are, too
5 No, because an isosceles triangle must have two
equal angles, and the sum of all three must be
180° Since 35 + 35 + 60 ≠ 180, and 35 + 60 + 60 ≠
180, the triangle is impossible
6 Your diagram should look something like this:
a°
b°
c°
x°
a + b = c
7 If a triangle has sides of lengths 20 and 15, then the third side must be less than 35 (their sum) but greater than 5 (their difference)
8 No The sum of the two shorter sides of a triangle
is always greater than the third side, but 5 + 8 is not greater than 14 So the triangle is impossible
9 25° and 130° or 77.5° and 77.5°
10 Draw in the line segments PQ, PR, PS, and PT.
Notice that this forms two triangles, ΔPQS and ΔPRT Since any two sides of a triangle must have
a sum greater than the third side, PQ + PS > QS, and PR + PT > RT Therefore,
PQ + PR + PS + PT > QS + RT.
Answer Key 2: Triangles
SAT Practice 2
1 A If AB = BD, then, by the Isosceles
Trian-gle theorem, ∠BAD and ∠BDA must be
equal To find their measure, subtract
50° from 180° and divide by 2 This
gives 65° Mark up the diagram with
this information Since the angles in
the big triangle have a sum of 180°,
65 + 90 + x = 180, so x = 25.
2 500 Drawing two diagonals shows
that the figure can be divided into
three triangles (Remember that an
n-sided figure can be divided into n
− 2 triangles.) Therefore, the sum
of all the angles is 3 × 180° = 540°
Subtracting 40° leaves 500°
3 B The third side of any triangle must have a
length that is between the sum and the
differ-ence of the other two sides Since 16 − 9 = 7
and 16 + 9 = 25, the third side must be between
(but not including) 7 and 25
4 A Since the big triangle is a right
triangle, b + x must equal 90 The
two small triangles are also right
triangles, so a + x is also 90 Therefore,
a = b and statement I is true In one
“solution” of this triangle, a and b are 65 and x is 25 (Put the values into the diagram
and check that everything “fits.”) This solu-tion proves that statements II and III are not necessarily true
5 C If AD = DB, then, by
the Isosceles Triangle theorem, the angles opposite those sides must be equal You should mark the other angle with an
x also, as shown here Similarly, if DB = DC, then
the angles opposite those sides must be equal also,
and they should both be marked y Now consider
the big triangle Since its angles must have a sum
of 180, 2x + 2y = 180 Dividing both sides by 2 gives
x + y = 90 (Notice that the fact that ∠ADB
mea-sures 100° doesn’t make any difference!)
50°
x°
A B C
D
65 ° 65°
25 °
130 °
a°
b ° c°
d°
40°
b°
a°
x°
B
D
y°
x° 100°
x ° y°