Download 4-3 to 4-5 Student Notes (No HL)-DMW

Geometry C
Name:______________________
Date: ___________
Section 4.3/4.4/4.5: Proving Triangles Congruent
DO NOW
1. What are sides AC and BC called? Side AB?
2. Which side is in between A and C?
Objectives of Lesson:

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 using SSS, SAS, ASA, and AAS
Corresponding angles and corresponding sides are in the same position in
polygons with an equal number of sides.
Properties of Congruent Polygons
Diagram
Corresponding Angles
Corresponding Sides
Helpful Hint: To name a polygon, write the vertices in consecutive order.
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Side-Side-Side Congruence (SSS)
Postulate
Figure
If three sides of one
triangle are congruent
to three sides of
another triangle, then
the triangles are
congruent.
Example 1: Using SSS to Prove Triangle Congruence
Use SSS to explain why ∆𝐴𝐵𝐶 ≅ ∆𝐷𝐵𝐶.
Example 2
Use SSS to explain why ∆𝐴𝐵𝐶 ≅ ∆𝐶𝐷𝐴.
An included angle is an angle formed by two
adjacent sides of a polygon. B is the
̅̅̅̅ .
included angle between sides ̅̅̅̅
𝐴𝐵 and 𝐵𝐶
Conclusion
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Side-Angle-Side Congruence (SAS)
Postulate
Figure
Conclusion
If two sides and the
included angle of one
triangle are congruent
to two sides and the
included angle of
another triangle, then
the triangles are
congruent.
CAUTION: The letters SAS are written in that order because the
congruent angles MUST be between pairs of congruent corresponding sides.
Example 3: Engineering Application
The diagram shows part of the support structure for a tower. Use
SAS to explain why ∆𝑋𝑌𝑍 ≅ ∆𝑉𝑊𝑍.
Example 4
Use SAS to explain why ∆𝐴𝐵𝐶 ≅ ∆𝐷𝐵𝐶.
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Example 5: Verifying Triangle Congruence
Show that the triangles are congruent for the given value of the variable.
∆𝑀𝑁𝑂 ≅ ∆𝑃𝑄𝑅, when x = 5.
Example 6: Proving Triangles Congruent
Statements
Reasons
5
Example 7: Proving Triangles Congruent
Statements
Reasons
An included side is the common side to two consecutive angles in a polygon.
Angle-Side-Angle Congruence (ASA)
Postulate
If two angles and the included
side of one triangle are
congruent to two angles and
the included side of another
triangle, then the triangles are
congruent.
Figure
Conclusion
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Example 8: Applying ASA Congruence
Determine if you can use ASA to prove the triangles congruent. Explain.
Example 9
Determine if you can use ASA to prove NKL  LMN. Explain.
Angle-Angle-Side Congruence (AAS)
Postulate
Figure
If two angles and a nonincluded side of one triangle
are congruent to the
corresponding angles and nonincluded side of another
triangle, then the triangles are
congruent.
Example 10: Using AAS to Prove Triangles Congruent
Given: X  V, YZW  YWZ, XY  VY
Prove:  XYZ  VYW
Conclusion