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46 Congruence in RightTriangles
Mathematics Florida StandardsMAFS.912.G-SRT.2.5 Use congruence... aiteria tosolve problems and prove relationships in geometricfigures.
MP1,MP3
Objective To prove right triangles congruent using the Hypotenuse-Leg Theorem
Getting Ready!
What does the
large triangle tellyou about angles inthe figure?
MATHEMATICAL
PRACTICES
In the diagram below, two sides and a nonincluded angle of one triangle are congruent
to two sides and the nonincluded angle of another triangle.
Q
Lesson
Vocabularyhypotenuselegs of a righttriangle
J
Notice that the triangles are not congruent. So, you can conclude that Side-Side-Angle
is not a valid method for proving two triangles congruent. This method, however, works
in the special case of right triangles, where the right angles are the nonincluded angles.
In a right triangle, the side opposite the
right angle is called the hypotenuse. It is
the longest side in the triangle. The other
two sides are called legs.
Tlie right anglealways "points" tothe hypotenuse. J
Hypotenuse
Essential Understanding You can prove that two triangles are congruentwithout having to show that all corresponding parts are congruent. In this lesson,
you will prove right triangles congruent by using one pair of right angles, a pair of
hypotenuses, and a pair of legs.
258 Chapter 4 Congruent Triangles
Theorem 4-6 Hypotenuse-Leg (HI) Theorem
Theorem
If the hypotenuse
and a leg of one right
triangle are congruent
to the hypotenuse
and a leg of another
right triangle, then
the triangles are
congruent.
If...
APQR and AXYZ are right A,
^ = )a,and^ = W
Then ...
APQR = AXYZ
To prove the HL Theorem you will need to draw auxiliary lines to make a third triangle.
Pr^f Proof of Theorem 4-6: Hypotenuse-Leg Theorem
Given: APQR and AXYZ are right triangles, with right angles
Q and K ̂ ̂ and PQ = )^,
Prove: APQR = AXYZ
Proof: On AXYZ, draw ZY.
Mark point S so that FS = QR. Then, APQR = AXYS by SAS.
Since corresponding parts of congruent triangles are congruent,
PR = XS. It is given that PR = XZ, so XS = XZ by the Transitive
Property of Congruence. By the Isosceles Triangle Theorem,AS = AZ, so AXYS = AXYZ by AAS. Therefore, APQR = AXYZby the Transitive Property of Congruence.
Key Concept Conditions for HL Theorem
To use the HL Theorem, the triangles must meet three conditions.
Conditions
• There are two right triangles.
• The triangles have congruent hypotenuses.
• There is one pair of congruent legs.
Lesson 4-6 Congruence in Right Triangles 259
Proof Problem 1 Using the HL Theorem
On the basketball backboard brackets shown below, Z.ADC and /LBDC are
right angles and AC = BC. Are AADC and ABDC congruent? Explain.
How can you
visualize the two
right triangles?Imagine cutting AABCalong On eitherside of the cut, you gettriangles with the sameleg DC.
\
• You are given that AADC and ABDC are right angles.So, AADC and ABDC are right triangles.
• The hypotenuses of the two right triangles are AC and BC.
You are given that AC = BC.
• DC is a common leg of both AADC and ABDC. DC = DC by the Reflexive Propertyof Congruence.
Yes, AADC = ABDC by the HL Theorem.
Got It? 1. a. Given: APRS and ARPQ are
right angles, BP = QR
Prove: APRS = ARPQ S
b. Reasoning Your friend says, "Suppose you have two right triangles withcongruent hypotenuses and one pair of congruent legs. It does not matter
which leg in the first triangle is congruent to which leg in the second
triangle. The triangles will be congruent." Is your friend correct? Explain.
260 Chapter 4 Congruent Triangles
Proof
How can you get
started?
Identify the hypotenuseof each right triangle.Prove that the
hypotenuses arecongruent.
Problem 2 Writing a Proof Using the HL Theorem
Given: bisects AD at C,
AB 1 EC, AB = W
Prove: AABC = ADEC
BE bisects AD.
Given
AS
DE 1 EC
Given
Z./AeC and
/.DEC are
right A.
Def. of 1 lines
AC = DC
Def. of bisector
A/teCand ADEC
are right A.
Def. of right triangle
AB = DE
Given
^ jjj Got It? 2. Given: CD = £-4, ̂ D is the perpendicularbisector of C£
Prove: ACBD s ABBA
AABC = ADEC
HL Theorem
C
BD
Lesson Check
Do you know HOW?
Are the two triangles congruent? If so, write the
congruence statement.
Do you UNDERSTAND?
5. Vocabulary A right triangle has side lengths of
5 cm, 12 cm, and 13 cm. What is the length of the
hypotenuse? How do you know?
6. Compare and Contrast How do the HL Theorem and
the SAS Postulate compare? How are they different?
Explain.
7. Error Analysis Your classmate says that there is not
enough information to determine whether the two
triangles below are congruent. Is your classmate
correct? Explain.
M
K
c PowerGeometry.com I Lesson 4-6 Congruence in Right Triangles 261
Practice and Problem-Solving Exercises
Practice 8. Developing Proof Complete the flow proof.
MATHEMATICAL
PRACTICES
Apply
Given: PS = Pf, /LPRS = APRT
Prove: APRS = APRT
jLPRS and ̂ PRTare =.
Given
^PffSand ̂ PRT
are supplementary.
A that form a linear
pair are supplementary.
LPRS and /.PRT
are right A.
ps^Wr
c.j_
' ̂= PR
d. ?
j ̂ See Problem 1.
APRS and APRT
are right A.
b. ?
APRS = APRT
e. ?
9. Developing Proof Complete the paragraph proof.
Given: AA and AD are right angles, AB = DE
Prove: AABE = ADEB
Proof: It is given that AA and AD are right angles. So, a. J_ by thedefinition of right triangles, b. because of the Reflexive Property of
Congruence. It is also given that C. ? .So, AABE = ADEB by d. ? .
L
n
ia Given: HV l GT, GH = TV,Proof / jg midpoint of HV
Prove: AIGH = AITV
11. Given: PM = Rf, ^ See Problem 2.Proof Pf ITf.mr ITf,
Mis the midpoint of TJ
Prove: APTM = ARMJ
H
V
Algebra For what values ofa; andy are the triangles congruent by HL?
12. 13. 3y + X
y+1y-x
x + 5
y+5
14. Study Exercise 8. Can you prove that APRS = APRT without using the HL
Theorem? Explain.
262 Chapter 4 Congruent Triangles
15. Think About a Plan AABC and APQR are
right triangular sections of a fire escape,
as shown. Is each story of the building the
same height? Explain.
• What can you tell from the diagram?
• How can you use congruent triangles here?
16. Writing "A HA!" exclaims your classmate.
"There must be an HA Theorem, sort of
like the HL Theorem!" Is your classmate
correct? Explain.
17. Given: RS = TU, RS 1 ST, TU 1 UV,Proof j js midpoint of RV
Prove: ARST^ ATUV
R
18. Given: ALNP is isosceles with base NP,Proof MN 1PL,ML = '^
Prove: AMNL = AQPL
L
-] T
UV
Constructions Copy the triangle and construct a triangle congruentto it using the given method.
20. HL19. SAS
21. ASA
23. Given: AGKE is isosceles with base G£,Proof angles, and
K is the midpoint of LD.
Prove: LG = DE
22. SSS
Given: LO bisects AMLN,Proof ^ W 1 In
Prove: ALMO = ALNO
M
25. Reasoning Are the triangles congruent? Explain.
C PowerGeo metiy.com
D
Lesson 4-6 Congruence in Right Triangles 263
\
26. a. Coordinate Geometry Graph the points y4(-5, 6), B(l, 3), -D(-8,0), and E{~2, -3).Draw AB, AE, ED, and DE. Label point C, the intersection of AE and BD.
b. Find the slopes of AE and BD. How would you describe Z-ACB and Z.ECD1
c. Algebra Write equations for AE and BD. What are the coordinates of C?
d. Use the Distance Formula to find AB, BC, DC, and DE.
e. Write a paragraph to prove that AABC = AEDC.
Challenge Geometry in 3 Dimensions For Exercises 27 and 28, use the hgure at the right.
27j Given: BE _L EA, BE L EC, AABC is equilateral
Prove: AAEB = ACEB
28. Given: AAEB = ACEB, M LEA,M EEC
Can you prove that AABC is equilateral? Explain.
SAT/Aa
Standardized Test Prep
29. You often walk your dog around the neighborhood.
Based on the diagram at the right, which of the
following statements about distances is true?
CK>SH=LH C^SH>LH
cj:>ph=ch c5:> bh<ch
School (5)
Park (P)
Cafe (C)
Short
Response
Library (L)
30. What is the midpoint of LM with endpoints L(2,7) and M{5, -1)?
CE> (3.5,3) CH^ (2,4.5)
CI5(3.5,4) CD (7, 6)
31. In equilateral AXYZ, name four pairs of congruent righttriangles. Explain why they are congruent.
Home (H)
Mixed Review"N
For Exercises 32 and 33, what type of triangle must ASTU be? Explain.
32. ASTU s AUTS 33. ASTU = AUST
4lb See Lesson 4-5.
Get Ready! To prepare for Lesson 4-7, do Exercises 34-36.
Can you conclude that the triangles are congruent? Explain.
34. AABC and ALMN 35. ALMN and AHJK
RN
^ See Lessons 4-3 and 4-6.
36. ARST and AABC
L J
K
MH
264 Chapter 4 Congruent Triangles