Benchmark #3 Review
Warm-Up
1.
2. What is the measure of angle ABC.
3/30/15
Warm-Up
3/31/15
1.
2. Five exterior angles of a convex hexagon have measure 74°, 84°, 42°,
13°, 26°. What is the measure of the 6th exterior angle?
Warm-Up
4/1/15
1.
a. 5, 7, 9
b. 16, 30, 34
2. What is the area of the largest square in the figure below?
Right Triangles:
• In a right triangle, the side opposite the right angle is the longest
side, called the hypotenuse. The other two sides are the legs of a
right triangle.
Theorem 7.1 Pythagorean Theorem:
In a right triangle, the sum of the squares of the lengths of the legs
is equal to the square of the length of the hypotenuse.
a2 + b2 = c2
 The Converse of the Pythagorean Theorem is used to determine
if a triangle is a right triangle, acute triangle, or obtuse triangle.
If c2 = a2 + b2, then the triangle is a right triangle.
 If c2 > a2 + b2, then the triangle is an obtuse
triangle.
 If c2 < a2 + b2, then the triangle is an acute
triangle.

45°-45°-90° Special Right Triangle
 In a triangle 45°-45°-90° , the hypotenuse is
2 times as long as
a leg.
Example:
45°
45°
Leg
X
Hypotenuse
5 2 cm
5 cm
X 2
45°
Special Right Triangles
X
Leg
45°
5 cm
30°-60°-90° Special Right Triangle
 In a triangle 30°-60°-90° , the hypotenuse is twice as long as the
shorter leg, and the longer leg is
times as long as the shorter leg.
3
Hypotenuse
30°
Longer Leg
X 3
Example:
2X
30°
10 cm
5 3 cm
60°
X
Shorter Leg
Special Right Triangles
60°
5 cm
If the reference angle is
A
then
A
Hyp
Adj
B
9
C then
A
Opp
Opp
Hyp
C
B
Adj
C
AC  Hyp
AC  Hyp
BC  Opp
AB  Opp
AB  Adj
BC  Adj
Right Triangle Trigonometry
Definitions of Trig Ratios
Opp
Sin 
Hyp
Adj
Cos 
Hyp
Opp
Tan 
Adj
hypotenuse
hypotenuse

adjacent
opposite
SOH- CAH -TOA
Opp
Sin 
Hyp
Adj
Opp
Cos 
Tan 
Hyp
Adj
SOHCAHTOA
4
opp 8
sin  


hyp 10 5
adj
6 3
cos  

5
hyp 10
opp
8
4
tan  


adj
6
3
10
8

6
Solving Trigonometric Equations
There are only three possibilities for the placement of the variable ‘x”.

sin
A
Opp
=
x
sin  =
A
x
12 cm
25
B
sin 25
=
=
1
x =
12
sin 25
x = 28.4 cm
13
C
sin 25
25

sin 25 =
1
25 cm
12 cm
B
12
x
12
x
A
12 cm
x
sin 25
x
Hyp
Opp
Hyp
Sin X =
=
x
12
x
12
x = (12) (sin 25)
x = 5.04 cm
x
B
C
sin x
=
C
12
25
x = sin 1(12/25)
x = 28.7
Note you are looking for an
angle here!
CH 7.3
When you write a proportion
comparing the legs lengths
of ∆CBD and ∆ACD, you can
see that CD is the geometric
mean of BD and AD.
C
A
D
B
B
C
D
C
A
D
B
BD
CD
Shorter leg of ∆ACD
A
C
Longer leg of ∆CBD.
Shorter leg of ∆CBD.
=
CD
AD
Longer leg of ∆ACD.
Geometric Mean Theorems
C
 Theorem 7.6: In a right triangle, the altitude
from the right angle to the hypotenuse divides
the hypotenuse into two segments. The length
of the altitude is the geometric mean of the
lengths of the two segments
 Theorem 7.7: In a right triangle, the altitude
from the right angle to the hypotenuse divides
the hypotenuse into two segments. The length
of each leg of the right triangle is the
geometric mean of the lengths of the
hypotenuse and the segment of the
hypotenuse that is adjacent to the leg.
A
D
BD
CD
=
CD
AD
AB
CB
=
CB
DB
AB
AC
=
AC
AD
B
What does that mean?
2
x
6
6
x
y
5
3
√9 · √2 = x
5+2 =
y
7
=
y
14 = y2
3 √2 = x
√14 = y
=
18 = x2
√18 = x
x
3
y
2
y
2
If a convex polygon has n sides, then the
sum of the measure of the interior angles
is (n – 2)(180°)
If a regular convex polygon has n
sides, then the measure of one of
the interior angles is
(n  2)180
n
Parallelogram
Definition:
A quadrilateral whose opposite sides are parallel.
B
C
AB CD and BC AD
Symbol:
a smaller version
of a parallelogram
A
D
Naming:  A parallelogram is named using all four vertices.
 You can start from any one vertex, but you must
continue in a clockwise or counterclockwise direction.
 For example, the figure above can be either
ABCD or
ADCB.
A
Properties of Parallelogram
B
P
D
1.
C
Both pairs of opposite sides are congruent.
AB  CD and BC  AD
2.
Both pairs of opposite angles are congruent.
A  C
3.
and B  D
Consecutive angles are supplementary.
mA  mB  180 and mA  mD  180
mB  mC  180 and mC  mD  180
4.
Diagonals bisect each other but are not congruent
P is the midpoint of
AC and BD.
AP  PC
BP  PD
Theorems
Theorem 8.7: If both pairs of
opposite sides of a
quadrilateral are congruent,
then the quadrilateral is a
D
parallelogram.
ABCD is a parallelogram.
A
B
C
Theorems
ABCD is a parallelogram.
Theorem 8.8: If both pairs of
opposite angles of a
quadrilateral are congruent,
then the quadrilateral is a
D
parallelogram.
A
B
C
Theorems
ABCD is a parallelogram.
Theorem 8.9: If one
pair of opposite sides
of a quadrilateral are
congruent and
parallel, then the
quadrilateral is a
parallelogram.
C
B
D
A
Theorems
ABCD is a parallelogram.
A
Theorem 8.10: If the
diagonals of a
quadrilateral bisect each
other, then the
D
quadrilateral is a
parallelogram.
B
C
Properties of Special Parallelograms
 In this lesson, you will study three special types of
parallelograms: rhombuses, rectangles and squares.
A rectangle is a parallelogram with four
right angles.
A rhombus is a parallelogram
with four congruent sides
A square is a parallelogram with four
congruent sides and four right angles.
Take note:
 Corollaries about special quadrilaterals:
 Rhombus Corollary: A quadrilateral is a rhombus
if and only if it has four congruent sides.
 Rectangle Corollary: A quadrilateral is a rectangle
if and only if it has four right angles.
 Square Corollary: A quadrilateral is a square if and
only if it is a rhombus and a rectangle.
Venn Diagram shows relationships
 Each shape has the properties of every group that it belongs to.
For instance, a square is a rectangle, a rhombus and a
parallelogram; so it has all of the properties of those shapes.
parallelograms
rhombuses
rectangles
squares