4-6 Objective: Use Congruent Triangles to Prove Corresponding Parts Congruent.

4-6 Objective: Use Congruent

Triangles to Prove Corresponding Parts

Congruent

Reminder about Congruent Figures

• If two triangles are congruent, then all the corresponding parts are congruent.

• If we can say two triangles are congruent, then we can say certain parts (sides or angles) are congruent, even if we didn’t use those parts.

GUIDED PRACTICE for Example 1

1. Explain how you can prove thatA C.

SOLUTION

GivenAB BCGivenAD DC

Reflexive propertyBD BD

ABD BCDThus the triangle by SSS

ANSWER

EXAMPLE 1 Use congruent triangles

Explain how you can use the given information to prove that the hanglider parts are congruent.

SOLUTION

GIVEN 1 2,∠RTQ RTS

PROVE QT ST

If you can show that QRT SRT, you will know that QT ST.First, copy the diagram and mark the giveninformation.

EXAMPLE 1 Use congruent triangles

Then add the information that you can deduce. In this case, RQT and RST are supplementary to congruent angles, so∠RQT RST. Also, RT RT .

Mark given information. Add deduced information.

Two angle pairs and a non-included side are congruent, so by the AAS Congruence Theorem, . Because corresponding parts of congruent triangles are congruent,

QRT SRT

QT ST.

EXAMPLE 2 Use congruent triangles for measurement

Surveying

Use the following method to find the distance across a river, from point N to point P.

• Place a stake at K on thenear side so that NK NP

• Find M, the midpoint of NK .

• Locate the point L so that NK KL and L, P, and Mare collinear.

EXAMPLE 2 Use congruent triangles for measurement

• Explain how this plan allows you to find the distance.

SOLUTION

Because NK NP and NK KL , N and K are congruent right angles.

Then, because corresponding parts of congruent triangles are congruent, KL NP . So, you can find the distance NP across the river by measuring KL .

MLK MPN by the ASA Congruence Postulate.

Because M is the midpoint of NK , NM KM . The vertical angles KML and NMP are congruent. So,

EXAMPLE 3 Plan a proof involving pairs of triangles

Use the given information to write a plan for proof.

SOLUTION

GIVEN 1 2, 3 4

PROVE BCE DCE

In BCE and DCE, you know 1 2 and CE CE . If you can show that CB CD , you can use the SAS Congruence Postulate.

EXAMPLE 3 Plan a proof involving pairs of triangles

CBA CDA. You are given 1 2 and 3 4. CA CA by the Reflexive Property. You can use the ASA Congruence Postulate to prove that CBA CDA.

To prove that CB CD , you can first prove that

Plan for ProofUse the ASA Congruence Postulate to prove that CBA CDA. Then state that CB CD . Use the SAS Congruence Postulate to prove that BCE DCE.

GUIDED PRACTICE for Examples 2 and 3

2. In Example 2, does it matter how far from point N you place a stake at point K ? Explain.

SOLUTIONNo, it does not matter how far from point N you place a stake at point K . Because M is the midpoint of NK

GivenNM MKDefinition of right triangle

MNP MKL areboth right triangles

Vertical angleKLM NMPASA congruence MKL MNP

GUIDED PRACTICE for Examples 2 and 3

No matter how far apart the strikes at K and M are placed the triangles will be congruent by ASA.

3. Using the information in the diagram at the right, write a plan to prove that PTU UQP.

GUIDED PRACTICE for Examples 2 and 3

Given TU PQ

Given PT QU

Reflexive property PU PU

This can be done by showing right triangles QSP and TRU are congruent by HL leading to right triangles USQ and PRT being congruent by HL which gives you PT UQ

STATEMENTS REASONS

SSSPTU UQP

PTU UQP By SSS

In Conclusion

• To show any two corresponding parts of a triangle are congruent show they are congruent using one of the postulates– Write a congruence statement– Make sure the parts are in the right places in

order to show they’re congruent

Examples

• Go to number 3 on page 259

Homework

• 1, 3-11, 16 – 20 evens, 23, 24, 29, 30, 33 – 35, 37

• Bonus: 27, 38, 40