1. 4.64.6 Use Congruent Triangles
Bell Thinger
Suppose that ∆XYZ ≅ ∆RST. Complete each statement.
ANSWER T
ANSWER RS
1. XY ≅ ?
2. Z ≅ ?
ANSWER Y
3. m S = m ?
ANSWER 50
4. If A ≅ B, m A = (2x + 40)º, and m B = (3x – 10)º,
find x.
2. 4.6Example 1
Explain how you can use
the given information to
prove that the hang glider
parts are congruent.
SOLUTION
RTQ ≅ RTS
GIVEN: 1 ≅ 2,
PROVE: QT ≅ ST
If you can show that QRT ≅ SRT, you will know that
QT ≅ ST.
3. 4.6Example 1
First, copy the diagram and mark the given information.
Then add the information 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, QRT ≅ SRT.
Because corresponding parts of congruent
triangles are congruent, QT ≅ ST.
4. 4.6Guided Practice
1. Explain how you can prove that
A ≅ C.
Since BD ≅ BD by the Reflexive Property, the
triangles are congruent by SSS. So, ∠A ≅ ∠C
because they are corresponding parts of congruent
triangles.
ANSWER
5. 4.6Example 2
Surveying
Use the following method to
find the distance across a
river, from point N to point P.
• Place a stake at K on the
near side so that NK NP
• Find M, the midpoint of NK .
• Locate the point L so that NK KL and L, P, and M
are collinear.
• Explain how this plan allows you to find the distance.
6. 4.6Example 2
SOLUTION
The vertical angles ∠KML and ∠NMP are congruent.
So, MLK ≅ MPN by the ASA Congruence Postulate.
Then, because corresponding parts of congruent
triangles are congruent, KL ≅ NP. So, you can find the
distance NP across the river by measuring KL.
Because NK NP and NK KL,
∠N and ∠K are congruent
right angles. Because M is the
midpoint of NK , NM ≅ KM .
7. 4.6Example 3
Use the given information to write a
plan for proof.
SOLUTION
GIVEN: 1 ≅ 2, 3 ≅ 4
PROVE: BCD ≅ 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.
8. 4.6Example 3
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.
9. 4.6Guided Practice
3. Using the information in the
diagram at the right, write a plan to
prove that ∆PTU ∆≅ UQP.
ANSWER
Since you already know that TU ≅ QP and UP ≅ PU, you
need only show PT ≅ UQ to prove the triangles are
congruent by SSS. 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.
10. 4.6Example 4
Write a proof to verify that the construction for copying
an angle is valid.
SOLUTION
Add BC and EF to the diagram. In the
construction, AB , DE , AC , and DF are
all determined by the same compass
setting, as are BC and EF . So, you can
assume the following as given
statements.
GIVEN: AB ≅ DE, AC ≅ DF, BC ≅ EF
PROVE: ∠D ≅ ∠A
11. 4.6Example 4
STATEMENTS REASONS
Plan in Action
Plan For Proof
Show that CAB ≅ FDE, so you can conclude that the
corresponding parts ∠A and ∠D are congruent.
1. Given1. AB ≅ DE,
AC ≅ DF, BC ≅ EF
2. SSS Congruence
Postulate
2. FDE ≅ CAB
3. ∠ D ≅ ∠ A 3. Corresp. parts of ≅
are ≅.
12. 4.6Exit Slip
Tell which triangles you can show are congruent
in order to prove AE = DE. What postulate or
theorem would you use?
1.
ANSWER AEC ≅ DEB by the AAS Cong.
Thm. or by the ASA Cong. Post.
