Chapter 4 Congruent Triangles. 4.3 Proving Triangles Congruent: SSS and SAS.
Holt Geometry 4-3, 4-4, and 4-5 Congruent Triangles 4-3, 4-4, and 4-5 Congruent Triangles Holt...
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Transcript of Holt Geometry 4-3, 4-4, and 4-5 Congruent Triangles 4-3, 4-4, and 4-5 Congruent Triangles Holt...
![Page 1: Holt Geometry 4-3, 4-4, and 4-5 Congruent Triangles 4-3, 4-4, and 4-5 Congruent Triangles Holt Geometry Warm Up Warm Up Lesson Presentation Lesson Presentation.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649cee5503460f949bc5ef/html5/thumbnails/1.jpg)
Holt Geometry
4-3, 4-4, and 4-5Congruent Triangles4-3, 4-4, and 4-5
Congruent Triangles
Holt Geometry
Warm Up
Lesson Presentation
Lesson Quiz
![Page 2: Holt Geometry 4-3, 4-4, and 4-5 Congruent Triangles 4-3, 4-4, and 4-5 Congruent Triangles Holt Geometry Warm Up Warm Up Lesson Presentation Lesson Presentation.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649cee5503460f949bc5ef/html5/thumbnails/2.jpg)
Holt Geometry
4-3, 4-4, and 4-5Congruent Triangles
Let’s Get It Started . . .
1. Name all sides and angles of ∆FGH.
2. What is true about K and L? Why?
3. What does it mean for two segments to be congruent?
FG, GH, FH, F, G, H
;Third s Thm.
They have the same length.
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Holt Geometry
4-3, 4-4, and 4-5Congruent Triangles
• Use properties of congruent triangles. • Prove triangles congruent by using the
definition of congruence.• Apply SSS, SAS, ASA, and AAS to
construct triangles and solve problems. • Prove triangles congruent by using
SSS, SAS, ASA, and AAS.
Objectives
![Page 4: Holt Geometry 4-3, 4-4, and 4-5 Congruent Triangles 4-3, 4-4, and 4-5 Congruent Triangles Holt Geometry Warm Up Warm Up Lesson Presentation Lesson Presentation.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649cee5503460f949bc5ef/html5/thumbnails/4.jpg)
Holt Geometry
4-3, 4-4, and 4-5Congruent Triangles
corresponding anglescorresponding sidescongruent polygonstriangle rigidityincluded angleIncluded side
Vocabulary
![Page 5: Holt Geometry 4-3, 4-4, and 4-5 Congruent Triangles 4-3, 4-4, and 4-5 Congruent Triangles Holt Geometry Warm Up Warm Up Lesson Presentation Lesson Presentation.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649cee5503460f949bc5ef/html5/thumbnails/5.jpg)
Holt Geometry
4-3, 4-4, and 4-5Congruent Triangles
Geometric figures are congruent if they are the same size and shape. Corresponding angles and corresponding sides are in the same position in polygons with an equal number of sides.
Two polygons are congruent polygons if and only if their corresponding sides are congruent. Thus triangles that are the same size and shape are congruent.
![Page 6: Holt Geometry 4-3, 4-4, and 4-5 Congruent Triangles 4-3, 4-4, and 4-5 Congruent Triangles Holt Geometry Warm Up Warm Up Lesson Presentation Lesson Presentation.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649cee5503460f949bc5ef/html5/thumbnails/6.jpg)
Holt Geometry
4-3, 4-4, and 4-5Congruent Triangles
![Page 7: Holt Geometry 4-3, 4-4, and 4-5 Congruent Triangles 4-3, 4-4, and 4-5 Congruent Triangles Holt Geometry Warm Up Warm Up Lesson Presentation Lesson Presentation.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649cee5503460f949bc5ef/html5/thumbnails/7.jpg)
Holt Geometry
4-3, 4-4, and 4-5Congruent Triangles
To name a polygon, write the vertices in consecutive order.
Helpful Hint
![Page 8: Holt Geometry 4-3, 4-4, and 4-5 Congruent Triangles 4-3, 4-4, and 4-5 Congruent Triangles Holt Geometry Warm Up Warm Up Lesson Presentation Lesson Presentation.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649cee5503460f949bc5ef/html5/thumbnails/8.jpg)
Holt Geometry
4-3, 4-4, and 4-5Congruent Triangles
Naming Polygons Start at any vertex and list the
vertices consecutively in a clockwise or counterclockwise direction.
D
I
AN
EDIANE
IANED
ANEDI
NEDIA
EDIAN
DENAI
ENAID
NAIDE
AIDEN
IDENA
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Holt Geometry
4-3, 4-4, and 4-5Congruent Triangles
When you write a statement such as ABC DEF, you are also stating which parts are congruent.
Helpful Hint
![Page 10: Holt Geometry 4-3, 4-4, and 4-5 Congruent Triangles 4-3, 4-4, and 4-5 Congruent Triangles Holt Geometry Warm Up Warm Up Lesson Presentation Lesson Presentation.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649cee5503460f949bc5ef/html5/thumbnails/10.jpg)
Holt Geometry
4-3, 4-4, and 4-5Congruent Triangles
Say What?
Given: ∆PQR ∆STW
Identify all pairs of corresponding congruent parts.
Angles: P S, Q T, R W
Sides: PQ ST, QR TW, PR SW
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Holt Geometry
4-3, 4-4, and 4-5Congruent Triangles4-4 Triangle Congruence: SSS and SAS
Holt Geometry
Warm Up
Lesson Presentation
Lesson Quiz
![Page 12: Holt Geometry 4-3, 4-4, and 4-5 Congruent Triangles 4-3, 4-4, and 4-5 Congruent Triangles Holt Geometry Warm Up Warm Up Lesson Presentation Lesson Presentation.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649cee5503460f949bc5ef/html5/thumbnails/12.jpg)
Holt Geometry
4-3, 4-4, and 4-5Congruent Triangles
Let’s Get It Started
1. Name the angle formed by AB and AC.
2. Name the three sides of ABC.
3. ∆QRS ∆LMN. Name all pairs of congruent corresponding parts.
Possible answer: A
QR LM, RS MN, QS LN, Q L, R M, S N
AB, AC, BC
![Page 13: Holt Geometry 4-3, 4-4, and 4-5 Congruent Triangles 4-3, 4-4, and 4-5 Congruent Triangles Holt Geometry Warm Up Warm Up Lesson Presentation Lesson Presentation.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649cee5503460f949bc5ef/html5/thumbnails/13.jpg)
Holt Geometry
4-3, 4-4, and 4-5Congruent Triangles
Apply SSS and SAS to construct triangles and solve problems.
Prove triangles congruent by using SSS and SAS.
Objectives
![Page 14: Holt Geometry 4-3, 4-4, and 4-5 Congruent Triangles 4-3, 4-4, and 4-5 Congruent Triangles Holt Geometry Warm Up Warm Up Lesson Presentation Lesson Presentation.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649cee5503460f949bc5ef/html5/thumbnails/14.jpg)
Holt Geometry
4-3, 4-4, and 4-5Congruent Triangles
triangle rigidityincluded angle
Vocabulary
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Holt Geometry
4-3, 4-4, and 4-5Congruent Triangles
In Lesson 4-3, you proved triangles congruent by showing that all six pairs of corresponding parts were congruent.
The property of triangle rigidity gives you a shortcut for proving two triangles congruent. It states that if the side lengths of a triangle are given, the triangle can have only one shape.
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Holt Geometry
4-3, 4-4, and 4-5Congruent Triangles
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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Holt Geometry
4-3, 4-4, and 4-5Congruent Triangles
Adjacent triangles share a side, so you can apply the Reflexive Property to get a pair of congruent parts.
Remember!
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Holt Geometry
4-3, 4-4, and 4-5Congruent Triangles
Example 1: Using SSS to Prove Triangle Congruence
Use SSS to explain why ∆ABC ∆DBC.
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Holt Geometry
4-3, 4-4, and 4-5Congruent Triangles
Example 2
Use SSS to explain why ∆ABC ∆CDA.
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Holt Geometry
4-3, 4-4, and 4-5Congruent Triangles
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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Holt Geometry
4-3, 4-4, and 4-5Congruent Triangles
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Holt Geometry
4-3, 4-4, and 4-5Congruent Triangles
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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Holt Geometry
4-3, 4-4, and 4-5Congruent Triangles
Example 3: Engineering Application
The diagram shows part of the support structure for a tower. Use SAS to explain why ∆XYZ ∆VWZ.
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Holt Geometry
4-3, 4-4, and 4-5Congruent Triangles
Example 4
Use SAS to explain why ∆ABC ∆DBC.
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Holt Geometry
4-3, 4-4, and 4-5Congruent Triangles
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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Holt Geometry
4-3, 4-4, and 4-5Congruent Triangles
Example 5: Verifying Triangle Congruence
Show that the triangles are congruent for the given value of the variable.
∆MNO ∆PQR, when x = 5.
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Holt Geometry
4-3, 4-4, and 4-5Congruent Triangles
Example 6: Proving Triangles Congruent
Given: BC ║ AD, BC AD
Prove: ∆ABD ∆CDB
ReasonsStatements
5. SAS Steps 3, 2, 45. ∆ABD ∆ CDB
4. Reflex. Prop. of
2. Given
3. Alt. Int. s Thm.3. CBD ADB
1. Given1. BC || AD
2. BC AD
4. BD BD
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Holt Geometry
4-3, 4-4, and 4-5Congruent Triangles
Example 7
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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Holt Geometry
4-3, 4-4, and 4-5Congruent Triangles
An included side is the common side of two consecutive angles in a polygon. The following postulate uses the idea of an included side.
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Holt Geometry
4-3, 4-4, and 4-5Congruent Triangles
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Holt Geometry
4-3, 4-4, and 4-5Congruent Triangles
Example 1: Problem Solving Application
A mailman has to collect mail from mailboxes at A and B and drop it off at the post office at C. Does the table give enough information to determine the location of the mailboxes and the post office?
![Page 32: Holt Geometry 4-3, 4-4, and 4-5 Congruent Triangles 4-3, 4-4, and 4-5 Congruent Triangles Holt Geometry Warm Up Warm Up Lesson Presentation Lesson Presentation.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649cee5503460f949bc5ef/html5/thumbnails/32.jpg)
Holt Geometry
4-3, 4-4, and 4-5Congruent Triangles
The answer is whether the information in the table can be used to find the position of points A, B, and C.
List the important information: The bearing from A to B is N 65° E. From B to C is N 24° W, and from C to A is S 20° W. The distance from A to B is 8 mi.
1 Understand the Problem
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Holt Geometry
4-3, 4-4, and 4-5Congruent Triangles
Draw the mailman’s route using vertical lines to show north-south directions. Then use these parallel lines and the alternate interior angles to help find angle measures of ABC.
2 Make a Plan
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Holt Geometry
4-3, 4-4, and 4-5Congruent Triangles
mCAB = 65° – 20° = 45°
mCAB = 180° – (24° + 65°) = 91°
You know the measures of mCAB and mCBA and the length of the included side AB. Therefore by ASA, a unique triangle ABC is determined.
Solve3
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Holt Geometry
4-3, 4-4, and 4-5Congruent Triangles
One and only one triangle can be made using the information in the table, so the table does give enough information to determine the location of the mailboxes and the post office.
Look Back4
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Holt Geometry
4-3, 4-4, and 4-5Congruent Triangles
Example 8: Applying ASA Congruence
Determine if you can use ASA to prove the triangles congruent. Explain.
Two congruent angle pairs are give, but the included sides are not given as congruent. Therefore ASA cannot be used to prove the triangles congruent.
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Holt Geometry
4-3, 4-4, and 4-5Congruent Triangles
Example 9
Determine if you can use ASA to prove NKL LMN. Explain.
By the Alternate Interior Angles Theorem. KLN MNL. NL LN by the Reflexive Property. No other congruence relationships can be determined, so ASA cannot be applied.
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Holt Geometry
4-3, 4-4, and 4-5Congruent Triangles
You can use the Third Angles Theorem to prove another congruence relationship based on ASA. This theorem is Angle-Angle-Side (AAS).
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Holt Geometry
4-3, 4-4, and 4-5Congruent Triangles
Example 10: Using AAS to Prove Triangles Congruent
Use AAS to prove the triangles congruent.
Given: X V, YZW YWZ, XY VY
Prove: XYZ VYW