Bell Ringer

Using Congruent Triangles

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.

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

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.

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.

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

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

Now You Try Use Overlapping Triangles

ANSWER SAS.

1. Tell which triangle congruence theorem or postulate you would use to show that AB CD.

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

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

CheckpointNow You Try

CheckpointNow You Try

CheckpointNow You Try

CheckpointNow You Try

Page 268

Complete Pages 268-270 #s 6-20 Even Only