Transcript Algebra I - Denise Kapler

Unit 3
Congruent Triangles
Agenda
I
Congruent Triangles
that if two triangles are congruent,
know then one shape can become
another through a
Rotate
?
Flip
Slide
Within a plane are there other possible
movements? Yes, the figure can be resized,
but congruency is not preserved.
Word Movement in the plane
October 2014
Congruent Triangles
Agenda
I know that if two triangles are congruent,
then the name of the triangles
indicate the corresponding parts.
∆𝐴𝐵𝐶 ≅ ∆𝐷𝐸𝐹
?
How are construction marks used to show
how a triangle has been moved?
Word Corresponding parts
October 2014
Agenda
I
Congruent Triangles
by SSS
In Euclidean geometry any 3 points,
know when non-collinear, determine a
unique triangle and a unique plane (i.e.
a 2-dimensional Euclidean space).
?
How many ways can a triangle be
proved congruent? (5)
Word Non-collinear
October 2014
Agenda
I know
?
Congruent Triangles
By SAS
That congruent triangles are of the
same size and shape. That they
have the same side lengths and
same angle measures .
Why do the angles of a
triangle sum to 180°?
Word Classification of 𝐓𝐫𝐢𝐚𝐧𝐠𝐥𝐞𝐬
November 2014
Agenda
Congruent Triangles
by ASA
I know That the exterior angle of a triangle equals
the sum of the remote interior angles
?
What is a linear pair?
Word Remote Angles
Exterior Angle
Solve for x.
November 2014
Agenda
Congruent Triangles
By AAS
I know
That two triangles can share a
side. Through the reflexive
property these two shared sides
can be shown to be congruent.
𝑅𝑁 ≅ 𝑅𝑁 (see proof)
?
If you know the measures of two angles of a
triangle, how do you calculate the third?
Word Third Angle
November 2014
Given: ∠NER ≅ ∠𝑁𝑉𝑅
𝑅𝑁 bisects ∠𝐸𝑅𝑉
Prove: ∆𝐸𝑁𝑅 ≅ ∆𝑉𝑁𝑅
Congruent Triangles
Agenda
by HL
I know
?
Given a right triangle that the hypotenuse
is the longest side length. It is opposite
the right angle. The right angle is formed
by the legs.
Does it matter which leg is
congruent when using HL?
Word Hypotenuse
Leg
November 2014
Congruent Triangles
Agenda
I know that if two triangles are congruent,
then all corresponding parts are
congruent. ∆𝐴𝐵𝐶 ≅ ∆𝐷𝐸𝐹
NOTE: First prove ∆′ 𝒔 are
≅ then use CPCTC to
prove corresponding
parts are ≅
?
How does the nomenclature of triangles
help you determine correspondence?
Word CPCTC
October 2014
Agenda
Congruent Triangles
I know
Equilateral triangles are equiangular. All side
lengths are congruent and all angle measures
are congruent.
?
If you know the measures of two angles of a
triangle, how do you calculate the third?
Word Equi – Latin root word
means same or equal
November 2014
Agenda
Congruent Triangles
I know
If two sides of a triangle are congruent, then
the angles opposite those sides are
congruent. Converse is true.
?
What is the definition of an
isosceles triangle?
Word Isos – Latin root word
means equal.
November 2014
Math Review October 29, 2014
1. Given the
angles below,
solve for ∠𝐶.
(34° and 120°)
1
2
A= bh A=s1+s2+s3
A=lw; P=2l+2w
A=πr2 C=2πr
2. What is the
relationship
between ∠3
and ∠4? If
∠3= 37⁰, and
l ‖m solve for ∠4.
3. Graph the
following
system. What
is the
relationship
between the
two lines?
𝑦 =𝑥+1
2𝑥 − 2𝑦 = −2
y= mx + b
defines linear
Math Review October 30, 2014
1. Given the 2
angles below,
solve for x.
(85° and 123°)
2. What is the
3. Graph the
relationship
following
system. What
between ∠5
is the
and ∠8? If
relationship
∠5= 29⁰ and
between
the
l ‖m solve for ∠8.
two lines?
𝑦 =𝑥+1
8𝑥 − 8𝑦 = −8
1 sum
Angle
2
Triangle
180°
Quadrilateral 360°
Pentagon (think)
y= mx + b
defines linear
Math Review October 31, 2014
1. Given the
angles below,
solve for x,y,z.
(x°, y°, z°, 73° and
75°)
2. What is the
3. Graph the
relationship
following
system. What
between ∠2
is the
and ∠7? If
relationship
∠2= 101⁰,
between
the
and l ‖m , solve
two lines?
for ∠8.
4𝑦 = 4𝑥 + 4
−2𝑥 + 2𝑦 = 2
Angle sum
Triangle 180°
Quadrilateral 360°
Pentagon (think )
y= mx + b
defines linear
Math Review November 3, 2014
1. Given the
angles below,
solve for y.
(42°, 85°, w°, x°
and 88°)
Angle sum
Triangle 180°
Quadrilateral 360°
Pentagon (think)
2. What is the
3. Graph the
relationship
following
system. What
between ∠5
is the
and ∠8? If
relationship
∠5= 29⁰, and
between
the
l ‖m , solve for ∠8.
two lines?
𝑦 =𝑥+1
−2𝑥 + 2𝑦 = 2
y= mx + b
defines linear
Math Review November 4, 2014
1. Given the
angles below,
solve for x.
(x°, 144°, 72°)
2. What is the
3. Graph the
relationship
following
system. What
between ∠2
is the
and ∠7? If
relationship
∠2= 101⁰,
between
the
and l ‖m solve
two lines?
for ∠8.
4𝑦 = 4𝑥 + 4
−2𝑥 + 2𝑦 = 2
Angle sum
Triangle 180°
Quadrilateral 360°
Pentagon (think)
y= mx + b
defines linear
Math Review November 5, 2014
1. Given the
angles below,
solve for x.
(61°, 73°, 72°)
Angle sum
Triangle 180°
Quadrilateral 360°
Pentagon (think )
2. What is the
3. Graph the
relationship
following
system. What
between ∠1
is the
and ∠5? If
relationship
∠1= 37⁰, and
between
the
l ‖m , solve for ∠8.
two lines?
1
𝑦= 𝑥
3
3𝑥 + 𝑦 = 2
y= mx + b
defines linear
Side-Side-Side Congruence Postulate
“If three sides of one triangle are congruent
to three sides of another triangle,
then the triangles are congruent”
6
4
8
4
6
8
Side-Angle-Side Postulate
“If two sides and the included angle of one
triangle are congruent to two sides and the
included angle of another triangle, then the
triangles are congruent”
5
5
7
7
Math Review November 6, 2014
1. Given the
angles below,
solve for
∠1 𝑎𝑛𝑑 ∠2.
(30°, 42°)
Angle sum
Triangle 180°
Quadrilateral 360°
Pentagon (think )
2. What is the
3. Graph the
relationship
following
system. What
between ∠1
is the
and ∠5? If
relationship
∠1= 37⁰, and
between
the
l ‖m , solve for ∠8.
two lines?
1
𝑦= 𝑥
3
3𝑥 + 𝑦 = 2
y= mx + b
defines linear
Math Review November 7, 2014
1. Given the
angles below, 2. What is the
solve for x.
relationship
between ∠4
[132°, 93°, (9x+3)°]
Angle sum
Triangle 180°
Quadrilateral 360°
Pentagon (think )
3. Graph the
following
system. What
is the
and ∠5? If
relationship
∠5= 63⁰ and
between
the
l ‖m, solve for ∠3.
two lines?
1
𝑦= 𝑥
3
3𝑥 + 𝑦 = 2
y= mx + b
defines linear
Math Review November 10, 2014
1. Given the
angles below, 2.
solve for
∠𝐷𝐴𝐶 & ∠DCA
[40°, 70°, 90°]
What is the
3. Graph the
relationship
following
system. What
between ∠6
is the
and ∠5? If
relationship
∠5= 21⁰, and
between
the
l ‖m,solve for ∠2.
Angle sum
Triangle 180°
Quadrilateral 360°
Pentagon (think )
two lines?
1
𝑦= 𝑥
3
3𝑥 + 𝑦 = 2
y= mx + b
defines linear
Math Review November 11, 2014
1. Given the
angles below in 2. What is the
relationship
the irregular
between
hexagon, solve
for ∠𝑥
∠1and ∠7? If
Angle sum
Triangle 180°
Quadrilateral 360°
Pentagon (think )
3. Graph the
following
system. What
is the
relationship
∠1= 3⁰, and
between
the
l ‖m , solve for ∠7.
two lines?
1
𝑦= 𝑥
4
4𝑥 + 𝑦 = 2
y= mx + b
defines linear
Math Review November 12, 2014
1. Given the
angles below in 2.
the irregular
pentagon, solve
for ∠𝑋
What is the
3. Graph the
relationship
following
system. What
between ∠4
is the
and ∠8? If
relationship
∠4= 63⁰, and
between
the
l ‖m solve for ∠7.
Angle sum
Triangle 180°
Quadrilateral 360°
Pentagon (think )
two lines?
1
𝑦= 𝑥
3
3𝑥 + 𝑦 = 2
y= mx + b
defines linear
Math Review September 11, 2014
1a. 400
If real: state if
rational or
irrational.
2. Find the
distance between
(-5, 24) and (2, 0)
3. Write the
equation of the
line defined by:
(3, -2) and (-4, 5)
1b. Plot
-10 < x <-1 on
a number line
Real numbers exist
on the number line
Write the line
equation in
slope intercept
form.
d
 y  y  x  x 
2
2
1
2
1
2
y= mx + b
defines linear
Math Review September 15, 2014
1a. Determine
area of square
with side
lengths of 2”
1b. Determine
perimeter of
square with
side lengths of
2”
Area = lw
Perimeter= 2l + 2w
3. Graph y= 1
2. Find the
x
midpoint between
(-6, 24) and (2, 0)
Name its function
family
y= mx + b
defines linear
Math Review September 25, 2014
1a. Determine
area of
equilateral
triangle s= 1cm
1b. Determine its
perimeter
2. 2. Solve for x.
Give algebraic
reasons for
each step.
6x + 7 = 22 - 3
3. Graph
f(x) = -2x
Name its function
family
Area = lw
Perimeter= 2l + 2w
A =  r2
C = 2 r
Given
Properties of Real Numbers
1. Equality Under Operations:
reflective, symmetric and
transitive
2. Equality: Substitution
3. Simplify (Arithmetic)
y= mx + b
defines linear