4-6 Triangle Congruence: ASA, AAS, and HLEXAMPLESEXERCISES1 Given: B is the midpoint of AE.ZA = ZE,16. Given: Cis the midpointof AG.HA || GBHZABC = ZEBDProve: AABC=AEBDProve: AHAC= ABGCB17. Given: WX 1 XZ,YZ 1 ZX,WZ = YXProof:StatementsReasonsProve: AWZX=AYXZZ,1. Given2. Given3. Given4. Def. of mdpt.5. ASA Steps 1, 4, 21. ZA = ZE2. ZABC E ZEBDW3. B is the mdpt. of AE.4. AB = EB5. ΔΑΒC= ΔΕBD18. Given: ZS and ZVRare right angles.RT = UW.mZT= mZWProve: ARST :AUVW4-7 Triangle Congruence: CPCTCEXAMPLESEXERCISES1 Given: JL and HK bisect each other.19. Given: M is the midpointof BD.BC = DCProve: ZJHG= ZLKGKProve: 21 = 2BMD20. Given: PQ= RQ,PS = RSProve: QS bisects ZPQR.Proof:StatementsReasons1. JI and HK bisect1. Giveneach other.2. JG = LG, andHG = KG.2. Def. of bisectR3. ZJGH = LLGK3. Vert. A Thm.21. Given: H is the midpoint of GJ.L is the midpoint of MK.GM = KJ, GJ = KM,G4. AJHG = ALKG5. ZJHG = ZLKG4. SAS Steps 2, 35. СРСТСZG= ZKProve: ZGMH= ZKJLM LK298Chapter 4 Triangle Congruence 4-8 Introduction to Coordinate ProofEXAMPLESEXERCISES1 Given: ZB is a right angle in isosceles rightAABC. E is the midpoint of AB.D is the midpoint of CB. AB = CBPosition each figure in the coordinate plane and givethe coordinates of each vertex.22. a right triangle with leg lengths r and sProve: CE = AD23. a rectangle with length 2p and width pProof: Use the coordinates A(0, 2a) , B(0,0),and C(2a, 0). Draw AD and CE.24. a square with side length 8mFor exercises 25 and 26 assign coordinates to eachvertex and write a coordinate proof.25. Given: In rectangle ABCD, E is the midpoint ofEAB, Fis the midpoint of BC, Gis themidpoint of CD, and His the midpointof AD.BDProve: EF= GHBy the Midpoint Formula,(0 +0 2a+026. Given: APQR has a right ZQ.Mis the midpoint of PR.E == (0, a) and2(0 + 2a 0+ 0D=|= (a, 0)Prove: MP = MQ= MR227. Show that a triangle with vertices at (3, 5), (3, 2),and (2, 5) is a right triangle.By the Distance Formula,CE = V(2a – 0)² + (0 – a)²=V4a? + a² = a/5AD=V(a – 0)² + (0 – 2a)²a² + 4a² = a/5Thus CE = AD by the definition of congruence.4-9 Isosceles and Equilateral TrianglesEXAMPLEEXERCISES1 Find the value of x.Find each value.EmZD+ mZE+ mZF= 180°28. хMK(45 – 3x)°by the Triangle SumTheorem. mZE = mZFby the IsoscelesTriangle Theorem.D42°mZD+2mZE= 180°Substitution29. RSR42 + 2(3x) = 180Substitute the givenvalues.2у — 4.51.5y6x = 138Simplify.x = 23Divide both sides by 6.30. Given: AACD is isosceles with ZD as the vertexangle. Bis the midpoint of AC.АВ — х+ 5, ВС3 2х — 3, and CD 3D 2x + 6.Find the perimeterAACD.Study Guide: Review299

As per our company guidelines we are supposed to answer only one question kindly post other question…

Answered: EXERCISES 16. Given: Cis the midpoint…

Math

Geometry

EXERCISES 16. Given: Cis the midpoint of AG. HA || GB A Prove: AHAC= ABGC

Elementary Geometry for College Students

6th Edition

ISBN:9781285195698

Author:Daniel C. Alexander, Geralyn M. Koeberlein

Publisher:Daniel C. Alexander, Geralyn M. Koeberlein

Chapter10: Analytic Geometry

Section10.CT: Test

Problem 13CT: Using as few variables as possible, state the coordinates of each point if DEF is isosceles with ...

See similar textbooks

Related questions

Q: 26. Given: B is the midpoint of DC. ABL DC 1. B is the midpt of DC, AB 1 DC 2. 3. 4. Prove: AABD =…

A: Consider the triangle

Q: Theorem-e Gaey-Hemiltion)let Aexz motrix

Q: R. 4. P. ove that PR bisects ORN. ve that RÔS = PÂT.

A: Since solve the first question for you. If you want any specific question to be solved then please…

Q: Given: AB // DC, Eis the midpoint of BC Prove: AABE=ADCE

A: Given that AB // DC E is the midpoint of BC .

Q: Find the angle between v and w if v · w = 1/2 ||v|| ||w||.

Q: Find z such that: IZI = 2, arg Z : 6.

Q: 13. Let A be the set A = {a, b, c, d}. Consider the relation R on A given by R = {(a, a), (b, b),…

Q: 2. If B is the midpoint of AC, which of the, following must be true? I. АВ + ВС 3D АС II. 2ВС - АС…

Q: Find all angles between 0 and 2T satisfying the condition -1 Cs x = Separate your answers with…

Q: AB DC AB = DC C is the midpoint of BE AC | DE Given: Prove: A D E

A: Given, AB∥DCAB≅DC C is the midpoint of BE

Q: Order the following statements and reasons to prove ABDE ACDE. Start with the given statement. #…

A: We have to show congruence of triangle

Q: ABC is a right-angled triangle at A and AB > AC. Bisect BC in D draw DE perpendicular to the…

Q: Find the cosecant (if it exists) of the angle √2 √√2 2 0-√2 O 0-220 √2 0 = 5л 4

Q: Does AD lie in plane M?

Q: 8 units away from

A: Given that: Coordinates of A(0,0) and B(12,5)C is 8 units away from AB.

Q: Q3. Let ABC be a triangle in the E.G.such that for some mR(ZBAC)=a,MR(ABC) = B positive real number…

A: Note: We’ll answer the first question since the exact one wasn’t specified. Please submit a new…

Q: Verify . that the. functrun. u= 1+y. t.2? sais fies the thredimensianal.aplace eg. Ux. +Uyy +Uzz=o

A: Given u=1x2+y2+z2

Q: Laplace stanajoarm obtain sthe foslousing also. Oue sthe Specify hoc:- ci'as= t (et coset-se-2t)ult)…

A: i) Let the given equation for Laplace transform be x(t)=(e-t cos 2t - se-2t) u(t) + e2t u(-t) / 2…

Q: ABC is a triangle. BF, CG are any two lines drawn from the extremities of the base BC to meet AC and…

Q: Compute the area of the triangle. T = = ||OP x oQ|| = |(36, -36, 36) || = II ト

Q: Compute Cixj)xk and un h Find patro of he pMane gog ingi (-i, novmetit tj 4K

A: The equation of the plane passing through the given point and perpendicular(normal) to the given…

Q: 8 In AABC, Pis the midpoint of AC. RS // BP and AR AB - 3:5. Determine the following: A R B

A: Given problem:- To find:- ar△TPC/ar△RSC

Q: If Tis the midpoint of SU, what are ST, TU, and SU? 6x 3x +27

Q: 30 Evaluate .P S: с Tw ds (w=J²) J along a) ONP (b) OBP

A: We have to find the value of integral along ONP and OBP

Q: Measuring Segments Find the midpoint, M, of AB. TOO 0 M [?] B 26 Enter

Q: let foas= b"tz0?4acty fnd the Prst three denivahes.

Q: Consider AGRT and AARE in the figure to the right. Given: R is the midpoint of AG and ET. Prove:…

A: Given: A figure with two triangles EAR and GTR. As R is the midpoint of AG, therefore, GR=AR As,…

Q: O 3. How does the converse of the Hinge Theorem differ from the Hinge Theorem in terms of the sides…

Question

4-6 Triangle Congruence: ASA, AAS, and HL
EXAMPLES
EXERCISES
1 Given: B is the midpoint of AE.
ZA = ZE,
16. Given: Cis the midpoint
of AG.
HA || GB
H
ZABC = ZEBD
Prove: AABC=AEBD
Prove: AHAC= ABGC
B
17. Given: WX 1 XZ,
YZ 1 ZX,
WZ = YX
Proof:
Statements
Reasons
Prove: AWZX=AYXZ
Z,
1. Given
2. Given
3. Given
4. Def. of mdpt.
5. ASA Steps 1, 4, 2
1. ZA = ZE
2. ZABC E ZEBD
W
3. B is the mdpt. of AE.
4. AB = EB
5. ΔΑΒC= ΔΕBD
18. Given: ZS and ZV
R
are right angles.
RT = UW.
mZT= mZW
Prove: ARST :AUVW
4-7 Triangle Congruence: CPCTC
EXAMPLES
EXERCISES
1 Given: JL and HK bisect each other.
19. Given: M is the midpoint
of BD.
BC = DC
Prove: ZJHG= ZLKG
K
Prove: 21 = 2
B
M
D
20. Given: PQ= RQ,
PS = RS
Prove: QS bisects ZPQR.
Proof:
Statements
Reasons
1. JI and HK bisect
1. Given
each other.
2. JG = LG, and
HG = KG.
2. Def. of bisect
R
3. ZJGH = LLGK
3. Vert. A Thm.
21. Given: H is the midpoint of GJ.
L is the midpoint of MK.
GM = KJ, GJ = KM,
G
4. AJHG = ALKG
5. ZJHG = ZLKG
4. SAS Steps 2, 3
5. СРСТС
ZG= ZK
Prove: ZGMH= ZKJL
M L
K
298
Chapter 4 Triangle Congruence

4-8 Introduction to Coordinate Proof
EXAMPLES
EXERCISES
1 Given: ZB is a right angle in isosceles right
AABC. E is the midpoint of AB.
D is the midpoint of CB. AB = CB
Position each figure in the coordinate plane and give
the coordinates of each vertex.
22. a right triangle with leg lengths r and s
Prove: CE = AD
23. a rectangle with length 2p and width p
Proof: Use the coordinates A(0, 2a) , B(0,0),
and C(2a, 0). Draw AD and CE.
24. a square with side length 8m
For exercises 25 and 26 assign coordinates to each
vertex and write a coordinate proof.
25. Given: In rectangle ABCD, E is the midpoint of
E
AB, Fis the midpoint of BC, Gis the
midpoint of CD, and His the midpoint
of AD.
B
D
Prove: EF= GH
By the Midpoint Formula,
(0 +0 2a+0
26. Given: APQR has a right ZQ.
Mis the midpoint of PR.
E =
= (0, a) and
2
(0 + 2a 0+ 0
D=
|= (a, 0)
Prove: MP = MQ= MR
2
27. Show that a triangle with vertices at (3, 5), (3, 2),
and (2, 5) is a right triangle.
By the Distance Formula,
CE = V(2a – 0)² + (0 – a)²
=V4a? + a² = a/5
AD=V(a – 0)² + (0 – 2a)²
a² + 4a² = a/5
Thus CE = AD by the definition of congruence.
4-9 Isosceles and Equilateral Triangles
EXAMPLE
EXERCISES
1 Find the value of x.
Find each value.
E
mZD+ mZE+ mZF= 180°
28. х
M
K
(45 – 3x)°
by the Triangle Sum
Theorem. mZE = mZF
by the Isosceles
Triangle Theorem.
D
42°
mZD+2mZE= 180°
Substitution
29. RS
R
42 + 2(3x) = 180
Substitute the given
values.
2у — 4.5
1.5y
6x = 138
Simplify.
x = 23
Divide both sides by 6.
30. Given: AACD is isosceles with ZD as the vertex
angle. Bis the midpoint of AC.
АВ — х+ 5, ВС3 2х — 3, and CD 3D 2x + 6.
Find the perimeter
AACD.
Study Guide: Review
299

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Step 3

VIEW

Trending now

This is a popular solution!

Step by step

Solved in 3 steps with 2 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Factorization$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, geometry and related others by exploring similar questions and additional content below.