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8-1
Proving Triangles Congruent SSS & SAS
Warm Up
Lesson Presentation
Lesson Quiz
GEOMETRY
8-1
Proving Triangles Congruent SSS & SAS
Warm Up
Classify each angle as acute, obtuse, or right.
1.
right
2.
acute
3.
obtuse
4. If the perimeter is 47, find x and the lengths of the
three sides.
x = 5; 8; 16; 23
GEOMETRY
8-1
Proving Triangles Congruent SSS & SAS
Homework: Page 410-411
You should be done with 1-16
For Tonight work on Problems#: 16*, 22, 23
Include a graph for each problem, use a ruler & compass!
GEOMETRY
8-1
Proving Triangles Congruent SSS & SAS
GEOMETRY
8-1
Proving Triangles Congruent SSS & SAS
An auxiliary line is a line that is added to a figure to
aid in a proof.
An auxiliary line
used in the
Triangle Sum
Theorem
GEOMETRY
8-1
Proving Triangles Congruent SSS & SAS
The interior is the set of all points inside the figure.
The exterior is the set of all points outside the figure.
Exterior
Interior
GEOMETRY
8-1
Proving Triangles Congruent SSS & SAS
An interior angle is formed by two sides of a triangle.
An exterior angle is formed by one side of the
triangle and extension of an adjacent side.
4 is an exterior angle.
Exterior
Interior
3 is an interior angle.
GEOMETRY
8-1
Proving Triangles Congruent SSS & SAS
Each exterior angle has two remote interior angles. A
remote interior angle is an interior angle that is not
adjacent to the exterior angle.
The remote interior angles
of 4 are 1 and 2.
Exterior
Interior
4 is an exterior angle.
3 is an interior angle.
GEOMETRY
8-1
Proving Triangles Congruent SSS & SAS
GEOMETRY
8-1
Proving Triangles Congruent SSS & SAS
GEOMETRY
8-1
Proving Triangles Congruent SSS & SAS
Objectives
Apply SSS and SAS to construct triangles and solve problems.
Prove triangles congruent by using SSS and SAS.
GEOMETRY
8-1
Proving Triangles Congruent SSS & SAS
Vocabulary
triangle rigidity
included angle
GEOMETRY
8-1
Proving Triangles Congruent SSS & SAS
The property of triangle rigidity states
that if the side lengths of a triangle are
given, the triangle can have only one
shape.
GEOMETRY
8-1
Proving Triangles Congruent SSS & SAS
For example, you only need to know that
two triangles have three pairs of congruent
corresponding sides.
This can be expressed as the following
postulate.
GEOMETRY
8-1
Proving Triangles Congruent SSS & SAS
Remember!
Adjacent triangles share a side, so you can apply the Reflexive
Property to get a pair of congruent parts.
GEOMETRY
8-1
Proving Triangles Congruent SSS & SAS
Example 1: Using SSS to Prove Triangle Congruence
Use SSS to explain why ∆ABC  ∆DBC.
It is given that AC  DC and that AB  DB. By the
Reflexive Property of Congruence, BC  BC.
Therefore ∆ABC  ∆DBC by SSS.
GEOMETRY
8-1
Proving Triangles Congruent SSS & SAS
TEACH! Example 1
Use SSS to explain why
∆ABC  ∆CDA.
It is given that AB  CD and BC  DA.
By the Reflexive Property of Congruence, AC  CA.
So ∆ABC  ∆CDA by SSS.
GEOMETRY
8-1
Proving Triangles Congruent SSS & SAS
An included angle is an angle formed
by two adjacent sides of a polygon.
B is the included angle between sides
AB and BC.
GEOMETRY
8-1
Proving Triangles Congruent SSS & SAS
It can also be shown that only two pairs of
congruent corresponding sides are needed to
prove the congruence of two triangles if the
included angles are also congruent.
GEOMETRY
8-1
Proving Triangles Congruent SSS & SAS
GEOMETRY
8-1
Proving Triangles Congruent SSS & SAS
Caution
The letters SAS are written in that order
because the congruent angles must be
between pairs of congruent corresponding
sides.
GEOMETRY
8-1
Proving Triangles Congruent SSS & SAS
Example 2: Engineering Application
The diagram shows part of
the support structure for a
tower. Use SAS to explain
why ∆XYZ  ∆VWZ.
It is given that XZ  VZ and that YZ  WZ.
By the Vertical s Theorem. XZY  VZW.
Therefore ∆XYZ  ∆VWZ by SAS.
GEOMETRY
8-1
Proving Triangles Congruent SSS & SAS
TEACH! Example 2
Use SAS to explain why
∆ABC  ∆DBC.
It is given that BA  BD and ABC  DBC.
By the Reflexive Property of , BC  BC.
So ∆ABC  ∆DBC by SAS.
GEOMETRY
8-1
Proving Triangles Congruent SSS & SAS
The SAS Postulate guarantees that if you are
given the lengths of two sides and the
measure of the included angles, you can
construct one and only one triangle.
GEOMETRY
8-1
Proving Triangles Congruent SSS & SAS
Proving Triangles Congruent
Given: BC ║ AD, BC  AD
Prove: ∆ABD  ∆CDB
Statements
Reasons
1. BC || AD
1. Given
2. CBD  ADB
2. Alt. Int. s Thm.
3. BC  AD
3. Given
4. BD  BD
4. Reflex. Prop. of 
5. ∆ABD  ∆ CDB
5. SAS Steps 3, 2, 4
GEOMETRY
8-1
Proving Triangles Congruent SSS & SAS
TEACH! Proving Triangles Congruent
Given: QP bisects RQS. QR  QS
Prove: ∆RQP  ∆SQP
Statements
Reasons
1. QR  QS
1. Given
2. QP bisects RQS
2. Given
3. RQP  SQP
3. Def. of bisector
4. QP  QP
4. Reflex. Prop. of 
5. ∆RQP  ∆SQP
5. SAS Steps 1, 3, 4
GEOMETRY
8-1
Proving Triangles Congruent SSS & SAS
Lesson Quiz: Part I
1. Show that ∆ABC  ∆DBC, when x = 6.
26°
ABC  DBC
BC  BC
AB  DB
So ∆ABC  ∆DBC by SAS
Which postulate, if any, can be used to prove the
triangles congruent?
2.
3.
none
SSS
GEOMETRY
8-1
Proving Triangles Congruent SSS & SAS
Lesson Quiz: Part II
4. Given: PN bisects MO, PN  MO
Prove: ∆MNP  ∆ONP
Statements
Reasons
1.
2.
3.
4.
5.
6.
1.
2.
3.
4.
5.
6.
7.
PN bisects MO
MN  ON
PN  PN
PN  MO
PNM and PNO are rt. s
PNM  PNO
7. ∆MNP  ∆ONP
Given
Def. of bisect
Reflex. Prop. of 
Given
Def. of 
Rt.   Thm.
SAS Steps 2, 6, 3
GEOMETRY