Download Chapter 6 Congruent Triangles and Quadrilaterals.

Warm-up
3/19
1) Sketch ΔABC and ΔRDH so that they are
congruent triangles.
2) What things must we know to say that two
polygons are congruent?
Announcements
1) Test needs to be made up by Friday.
2) Homework 6.1 due Monday.
3) New Bathroom passes.
4) Ways to raise your grade.
5) Participation
3/19
Chapter 6 Congruent Triangles
and Quadrilaterals.
6.1 Congruence Postulates and
Theorems
Recall:
Congruent polygons are polygons that have three
properties:
1) Same number of sides.
2) Corresponding sides are congruent.
3) Corresponding angles are congruent.
So to determine if two triangles are congruent you
would have to show that all three corresponding
sides are congruent and all three corresponding
angles are congruent.
Objective: Today we will learn that only three of
these six congruence's are needed to determine if
triangles are congruent.
Ex #1
Given: △SPQ and △RPQ;
SP  RP ; SQ  RP ; QPR  QPS; PSQ  PRQ
Prove: △SPQ ≅△RPQ
Think about what things you must show in order
to prove that the two triangles are congruent.
Pair with a partner and determine what you must
show.
Share
Ex #1
Now lets try the proof.
Statements
Reasons
1)
1) SP  RP ; SQ  RP ; QPR  QPS; PSQ  PRQ
△SPQ
△SPQ and
and △RPQ
△RPQ
2) QPR  QPS
2) QPR  QPS
3) PQ  QP
3) PQ  QP
4) △SPQ ≅△RPQ
4) △SPQ ≅△RPQ
1)
1) Given
Given
NOTE: The reflexive property
states that a side or angle is
Congruent to itself.
2) Theorem 5.11
2) Theorem 5.11
3) Reflexive Property
3) Reflexive Property
4) Definition of congruent
4)
Definition of congruent
polygons
polygons
We will now try to discover what three
congruence's we need to be able to tell if two
triangles are congruent.
First let us draw two angles that are congruent.
Second make one side of each angle congruent
line segments.
Third make the second side
of each angle congruent.
Now how many possible triangles can be formed?
Only one!
Thus if we know two sides and the angle that is
formed by these sides are congruent then we
know that the two triangles are congruent.
This leads us to a Postulate. That means we
cannot prove it but we can “see” that this
statement must be true.
Postulate 6.1 Side-Angle-Side Congruence
Postulate. If two sides and an included angle of
one triangle are congruent to the corresponding
two sides and included angle of another triangle,
then the two triangles are congruent.
Lets try another
First let us draw two segments that are congruent.
Second, using this segment as one side of an angle draw two
congruent angles.
Third, using the original segment draw two more congruent
angles from the other vertex.
Now how many possible tri-angles can be formed?
Only one!
Thus if we know two angles and the side that is
between these angles are congruent then we
know that the two triangles are congruent.
This leads us to a Postulate. That means we
cannot prove it but we can “see” that this
statement must be true.
Postulate 6.2 Angle-Side-Angle Congruence
Postulate. If two angles and an included side of
one triangle are congruent to the corresponding
two angles and included side of another triangle,
then the two triangles are congruent.
Warm-up
3/20
1) Use Postulates 6.1 and 6.2 to state whether the
triangles are congruent. State which postulate
you would use.
Announcements
1) Test needs to be made up by Friday.
2) Homework 6.1 due Monday.
3) New Bathroom passes.
4) Participation
3/19
Ex #2
Given: AD ∥BC; AB ∥ DC
Prove: △ABD ≅△CDB
Think about what things you must show in order
to prove that the two triangles are congruent.
Pair with a partner and determine what you must
show.
Share
Theorem 6.1 Side-Angle-Angle congruence
theorem. If two angles of a triangle and a side
opposite one of the two angles are congruent to
the corresponding angles and side of another
triangle, then the two triangles are congruent.
Theorem 6.2 Isosceles Triangle Theorem. In an
isosceles triangle the base angles are congruent.
Theorem 6.3 If two angles of a triangle are
congruent, then the sides opposite those angles
are congruent, and the triangle is an isosceles
triangle.
Theorem 6.4 A triangle is equilateral if and only if
it is equiangular.
Warm-up
1) Finish the proof
3/23
Statements
Reasons
1) LN Bisects MLO and MNO
1)
2)
MLN  NLO; LNM  LNO
2) Definition of angle bisector
3) LN  LN
3)
4) MLN  OLN
4)
5)
M  O
5)
L
M
N
O
Announcements
3/23
1) Homework 6.1 due Today.
2) Homework 6.1 Part 2 due Wednesday.
Cont of 6.1 Congruence Postulates
and Theorems
Review:
So what Triangle congruence Postulates and
theorems do we have so far?
-SAS
-ASA
-SAA
That leaves us with 3 more possibilities:
-AAA
-SSS
-ASS
Objective: Today we will find one more triangle
congruence theorem and will understand why the
other options do not give us congruence theorems.
SSS
First determine the 3 side lengths you will have to
make your two triangles.
Now how many possible tri-angles can be formed?
Only one!
Theorem 6.5 side-Side-Side Congruence Theorem. If
each side of one triangle is congruent to the
corresponding side of a second triangle then the two
triangles are congruent.
AAA
First determine the 3 angle measures you will
have to make your two triangles.
Now how many possible tri-angles can be
formed?
Infinitely many!
So if you know that two triangles have 3 angles
that are congruent you CANNOT say that the
triangles are congruent.
ASS
First draw an acute angle.
Second choose two lengths for your sides and
make one ray of your angle one of these lengths.
Now how many possible tri-angles can be
formed?
2!
So if you know that two triangles have an angle, a
side included in the angle, and a side opposite the
angle that are congruent(ASS) you CANNOT say
that the triangles are congruent.
Warm-up
3/24
Announcements
3/24
1) Homework 6.1 Part 2 due Wednesday.
2) CM #29.1 (Triangle congruence's) will be on
Wednesday.
3) Unit Quiz next Friday the 24th
4) Period 3, Bring socks.
5) Homework Questions????