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GEOMETRY
8-1 Proving Triangles Congruent SSS & SAS
Warm UpWarm Up
Lesson PresentationLesson Presentation
Lesson QuizLesson Quiz
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GEOMETRY
8-1 Proving Triangles Congruent SSS & SAS
Warm UpClassify each angle as acute, obtuse, or right.
1. 2.
3.
4. If the perimeter is 47, find x and the lengths of the
three sides.
rightacute
x = 5; 8; 16; 23
obtuse
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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!
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GEOMETRY
8-1 Proving Triangles Congruent SSS & SAS
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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
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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.
Interior
Exterior
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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.
Interior
Exterior
4 is an exterior angle.
3 is an interior angle.
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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.
Interior
Exterior
3 is an interior angle.
4 is an exterior angle.
The remote interior angles of 4 are 1 and 2.
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GEOMETRY
8-1 Proving Triangles Congruent SSS & SAS
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GEOMETRY
8-1 Proving Triangles Congruent SSS & SAS
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GEOMETRY
8-1 Proving Triangles Congruent SSS & SAS
Apply SSS and SAS to construct triangles and solve problems.
Prove triangles congruent by using SSS and SAS.
Objectives
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GEOMETRY
8-1 Proving Triangles Congruent SSS & SAS
triangle rigidityincluded angle
Vocabulary
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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.
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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.
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GEOMETRY
8-1 Proving Triangles Congruent SSS & SAS
Adjacent triangles share a side, so you can apply the Reflexive Property to get a pair of congruent parts.
Remember!
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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.
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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.
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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.
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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.
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GEOMETRY
8-1 Proving Triangles Congruent SSS & SAS
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GEOMETRY
8-1 Proving Triangles Congruent SSS & SAS
The letters SAS are written in that order because the congruent angles must be between pairs of congruent corresponding sides.
Caution
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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.
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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.
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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.
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GEOMETRY
8-1 Proving Triangles Congruent SSS & SAS
Proving Triangles Congruent
Given: BC ║ AD, BC ADProve: ∆ABD ∆CDB
ReasonsStatements
5. SAS Steps 3, 2, 45. ∆ABD ∆ CDB
4. Reflex. Prop. of
3. Given
2. Alt. Int. s Thm.2. CBD ADB
1. Given1. BC || AD
3. BC AD
4. BD BD
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GEOMETRY
8-1 Proving Triangles Congruent SSS & SAS
TEACH! Proving Triangles Congruent
Given: QP bisects RQS. QR QS
Prove: ∆RQP ∆SQP
ReasonsStatements
5. SAS Steps 1, 3, 45. ∆RQP ∆SQP
4. Reflex. Prop. of
1. Given
3. Def. of bisector3. RQP SQP
2. Given2. QP bisects RQS
1. QR QS
4. QP QP
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GEOMETRY
8-1 Proving Triangles Congruent SSS & SAS
Lesson Quiz: Part I
1. Show that ∆ABC ∆DBC, when x = 6.
ABC DBCBC BCAB DBSo ∆ABC ∆DBC by SAS
Which postulate, if any, can be used to prove the triangles congruent?
2. 3.none SSS
26°
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GEOMETRY
8-1 Proving Triangles Congruent SSS & SAS
Lesson Quiz: Part II
4. Given: PN bisects MO, PN MO
Prove: ∆MNP ∆ONP
1. Given2. Def. of bisect3. Reflex. Prop. of 4. Given5. Def. of 6. Rt. Thm.7. SAS Steps 2, 6, 3
1. PN bisects MO2. MN ON3. PN PN4. PN MO 5. PNM and PNO are rt. s6. PNM PNO
7. ∆MNP ∆ONP
Reasons Statements