Download Isosceles Triangles

10 Oct 2016 8:00 - 9:30 AM
Geometry Proposed Agenda
1) Bulletin clarkmagnet.net
2) Angles of a Triangle Homework corrections from Oct 4 - Doc
Camera
3) Homework from Oct 6 - discuss and decide next step
4) Review pages 17 and 18 information
5) Isosceles Triangles (includes a new construction - have
yellow booklet and compass and straightedge ready)
6) Homework
Homework: Find a Picture
Find a real world picture of congruent triangles
on the internet and print it out, or draw a picture
from life - or take your own picture and print
that out. Explain how you know the triangles
are congruent.
Postulates
Segment addition postulate: If B is
between A and C, then AB + BC = AC
A
B
C
Angle Addition Postulate: If P is in the
interior of angle RST,
then m∠RST = m ∠RSP +m∠PST
R
P
S
T
If angle RST is a straight angle and B is
any point not on the line is in the interior
of angle RST,
then m∠RSB + m ∠BST = 180°
B
R
S
T
Through any 2 points there is exactly one
line
What’s missing
on my diagram?
Through any 2 points there is exactly one
line
Letters for the the points and arrows
to show this is line, not a line
segment
Read the bold headings so you have an
idea what information you can find on
these 2 pages when you need it.
For today’s work - highlight Definition of
Isosceles triangle
Isosceles Triangles
A triangle with AT LEAST two congruent sides.
G-CO Congruent angles in isosceles
triangles
Below is an isosceles triangle ABC with
A
|AB| = |AC|
B
C
Three students propose different
arguments for why m∠(B) = m∠(C)
Could they all be correct?
A
B
C
Ravi: If I draw the bisector of ∠A then this is a
line of symmetry for triangle ABC and so
m∠B
= m∠C. Is he correct?
A
B
C
Reflection about the line will map A onto
itself; line segment AB onto AC and AC
onto AB - the angle at the vertex has
been cut in half and AB and AC are the
same length as each other.
A
Reflections are rigid transformations so
length is not changed.
Now we see we have also interchanged
the angles at B and C, so they must be
congruent which means they have equal
measures.
B
C
Brittney: If M is the midpoint of BC triangle ABM
is congruent to triangle ACB and so ∠B and
∠C are congruent. Is she correct?
A
B
M
C
M is the midpoint of line segment BC
which means BM = MC using the Midpoint
theorem (or definition of midpoint).
A
B
M
C
M is the midpoint of line segment BC
which means BM = MC using the Midpoint
theorem (or definition of midpoint).
We know by the reflexive property that AM
= AM.
A
B
M
C
By Side-Side-Side Congruence postulate
we can see that △ABM is congruent to
△ACB.
A
B
M
C
By Side-Side-Side Congruence postulate
we can see that △ABM is congruent to
△ACM.
A
Since ∠B and ∠C are corresponding
parts of congruent triangles they are
congruent.
B
M
C
Courtney: If P is a point on BC such that line AP
is perpendicular to BC then triangle ABP is
congruent to triangle ACP and so
∠B = ∠C. Is she correct?
A
B
P
C
Theorem: If two lines are perpendicular,
they form congruent adjacent angles.
We know AP = AP by reflexive property.
By Pythagorean theorem we know that
BP = PC
SSS Congruence Theorem, triangle ABP
is congruent to triangle ACP.
Since ∠B and ∠C are corresponding
parts of congruent triangles they are
congruent.
A
B
P
C
Are there any other ways to show this?
A
B
C
Isosceles Triangle
Definition?
Given AB = AC
Prove: ∠ABC = ∠ACB
Draw line segment AD to bisect angle BAC.
Statement
Reason
1. AB = AC
1. Given
2. ∠BAD = ∠CAD
Statement
Reason
1. AB = AC
1. Given
2. ∠BAD = ∠CAD 2. Definition of angle bisector
3. AD = AD
Statement
Reason
1. AB = AC
1. Given
2. ∠BAD = ∠CAD 2. Definition of angle bisector
3. AD = AD
4. △BAD =△CAD
3.Reflexive property
Statement
Reason
1. AB = AC
1. Given
2. ∠BAD = ∠CAD 2. Definition of angle bisector
3. AD = AD
3.Reflexive property
4. △BAD =△CAD
4. SAS congruence postulate
5. ∠ABC = ∠ACB
Statement
Reason
1. AB = AC
1. Given
2. ∠BAD = ∠CAD 2. Definition of angle bisector
3. AD = AD
3.Reflexive property
4. △BAD =△CAD
4. SAS congruence postulate
5. ∠ABC = ∠ACB
5. Corresponding parts of congruent
triangles are congruent.
Isosceles Triangle - bottom of page
Theorem.
If two sides of a triangle are congruent, then the
angles opposite those sides are congruent.
Draw it:
A perpendicular bisector
Is a ray, line, or line segment that bisects and
is perpendicular to a line segment.
DC _|_ AB
D
AC = CB
Both have to be true.
C
Construct A perpendicular bisector?
What would we do?
(Try in yellow
booklet)
1)
2)
Perpendicular bisector
Given AB = AC, ∠BAD = ∠CAD
Prove: BC = CD, BC _|_ AD
2 Column proof, complete, then start homework
page 23.
Statement
Reason
1. AB = AC, ∠BAD
=∠CAD
1. Given
2. AD = AD
2. Reflexive property
3. △ABD = △ACD
3. SAS congruence postulate
4 BD = CD
4. Corresponding parts of
congruent triangles are
congruent (CPCTC)
5. ∠ADB = ∠ADC
5. CPCTC
Statement
1.
AB = AC, ∠BAD =∠CAD
Reason
1.
Given
2. AD = AD
2. Reflexive property
3. △ABD = △ACD
3. SAS congruence postulate
4 BD = CD
4. Corresponding parts of congruent
triangles are congruent (CPCTC)
5. ∠ADB = ∠ADC
5. CPCTC
6. BC _|_ AD
6. If two lines form congruent
adjacent angles then the lines
are perpendicular.
Homework
Complete Properties of Isosceles Triangles
Homework problems 1 - 3 on page 23.
Show your thinking clearly and neatly.
Bring you congruent triangles in the real world
photographs or drawings on Wednesday for
use on a class poster, if you did not turn yours
in today in class.