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Bell Ringer
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Using Congruent Triangles
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Example 1 Use Corresponding Parts
In the diagram, AB and CD bisect each other at M. Prove that A B.
Because A and B are corresponding angles in ∆ADM and ∆BCM, show that ∆ADM ∆BCM to prove that A B.
2.
SOLUTION
First sketch the diagram and label any congruent segments and congruent angles.
1.
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Example 1 Use Corresponding Parts
Statements Reasons
1. AB and CD bisect each other at M.
Given1.
Definition of segment bisector2.2. MA MB
AMD BMC3. Vertical Angles Theorem3.
∆ADM ∆BCM5. SAS Congruence Postulate5.
A B6. Corresponding parts of congruent triangles are congruent.
6.
Definition of segment bisector4.4. MD MC
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Example 2 Visualize Overlapping Triangles
SOLUTION
Sketch the triangles separately and mark any given information. Think of ∆JGH moving to the left and ∆KHG moving to the right.
1.
Sketch the overlapping triangles separately. Mark all congruent angles and sides. Then tell what theorem or postulate you can use to show ∆JGH ∆KHG.
Mark GJH HKG and JHG KGH.
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Example 2 Visualize Overlapping Triangles
Look at the original diagram for shared sides, shared angles, or any other information you can conclude.
2.
Add congruence marks to GH in each triangle.
In the original diagram, GH and HG are the same side, so GH HG.
You can use the AAS Congruence Theorem to show that ∆JGH ∆KHG.3.
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Example 3 Use Overlapping Triangles
In the original diagram, C is the same in both triangles (BCA ECD).
SOLUTION
Sketch the triangles separately. Then label the given information and any other information you can conclude from the diagram.
1.
Write a proof that shows AB DE.
ABC DEC
AB DE
CB CE
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Show ∆ABC ∆DEC to prove that AB DE.
Statements Reasons
1. ABC DEC Given1.
Example 3 Use Overlapping Triangles
Given2.2. CB CE
C C3. Reflexive Prop. of Congruence3.
ASA Congruence Postulate4.∆ABC ∆DEC4.
Corresponding parts of congruent triangles are congruent.
5.5. AB DE
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Now You Try Use Overlapping Triangles
ANSWER SAS.
1. Tell which triangle congruence theorem or postulate you would use to show that AB CD.
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Checkpoint Use Overlapping Triangles
ANSWER
Statements Reasons
1. Given1.
Given2.2. J L
ASA Congruence Postulate4.∆KJN ∆KLM4.
Corresponding parts of triangles are .5.5. NJ ML
KJ KL
Reflexive Prop. of Congruence3.3. K K
2. Given KJ KL and J L, show NJ ML.
Redraw the triangles separately and label all congruences. Explain how to show that the triangles or corresponding parts are congruent.
Now You Try
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Checkpoint Use Overlapping Triangles
Given SPR QRP and Q S, show ∆PQR ∆RSP.3.
ANSWER
Statements Reasons
1. Given1.
Given2.2. Q S
AAS Congruence Theorem4.∆PQR ∆RSP4.
SPR QRP
Reflexive Prop. of Congruence3.3. PR RP
Now You Try
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CheckpointNow You Try
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CheckpointNow You Try
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CheckpointNow You Try
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CheckpointNow You Try
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Page 268
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Complete Pages 268-270 #s 6-20 Even Only