Download Assignments Triangles

Geometry Assignments: Triangles
Day
Topics
Homework
1
Parallel lines and transversals
HW T - 1
2
Proofs with parallel lines and transversals
HW T - 2
3
Sum of the angles in a triangle
HW T - 3
4
Exterior angle theorem ***QUIZ***
HW T - 4
5
Triangle inequalities
HW T - 5
6
Centroid of a triangle
HW T - 6
7
Medians and midsegments ***QUIZ***
HW T - 7
8
Prooflets
HW T - 8
9
Review
HW T - Review
10
***TEST***
HW
Grade
Quiz
Grade
Geometry HW: Triangles - 1
1. The Alternate Interior Angle Theorem can also be proved using a rigid
motion. In the diagram at right, name a single transformation after which the
image of 3 will be 5 and the image of 4 will be 6. Be specific.
(We’re not going to do the actual proof because some of the details get very
confusing.)
t
P
l1
3 4
6 5
l2
Q
2. We wish to prove the following theorem: If parallel lines are cut by a
transversal, same side interior angles are supplementary.
Given: l1 || l2, transversal t
Prove: 3 and 2 are supplementary
t
1
On your paper, fill in the missing statements and reasons labeled (a) – (g).
3
Statement
1. l1 || l2, transversal t
2. 1 and 2 are supplementary
3. 1 + 2 = 180
4. 1  3
5. (d)
6. (f)
l1
2
Reason
1. Given
2. (a)
3. (b)
4. (c)
5. (e)
6. (g)
l2
a
115
125
3. In the diagram at right, which lines are parallel?
(1) a and b, only
(2) a and c, only
(3) b and c only
(4) All three of them
(5) None of them
b
)
65
c
4. Each diagram shows two parallel lines cut by a transversal. Find the value of x in each diagram.
a.
b.
(x2
c.
 x  30 


 2 
 30)
(5x + 6)
 3 x  130 


 8 
7

 x  24 
3

5. A problem showed the diagram at right and said “Write an appropriate
equation and find the value of x.”
Rufus wrote 4.5x + 40 = 190 – 3x and got x = 20.
Doofus wrote 6x + 10 = 190 – 3x and got x = 20.
Goofus wrote 4.5x + 40 = 6x + 10 and got x = 20.
The teacher only gave one student full credit. Who got full credit and what
made his right and the other two wrong?
6. In the diagram at right, l1 || l2 and each algebraic expression
represents an angle.
a. Determine the value of x.
b. Determine if l3 || l4 and give a reason.
(4.5x + 40)
(6x + 10)
(190 – 3x)
l1
l2
(4x + 10) (8x + 5)
(5x – 4)
(This assignment continued on the next page.)
(3x – 12)
l3
l4
7. In the diagram at right, if l1 || l2, find the values of x and y.
l1
(2x)
(x + 3y) (3x – 2y)
l2
Geometry HW: Triangles - 2
For 1 – 3, write complete statement-reason proofs.
l3
l4
2
A
R
3
A
1
L
R
l2
P
1. Given: l1 || l2 , l3 || l4
E

T
S
3. Given: AM  MS ,
APE  ALR,
ALR  LAR
Prove: AL || PE

N
B
2. Given: PAR , RLE ,
Prove: 1  3
 



M
L
l1
MS  SB ,
AMN BSR
Prove: LMN || RST
A
C
4. In the diagram at right, BDG , mABD = 110°,
mDGF = 130°, AB || CD ,and ED || FG . Find
mCDE. (Hint: First find mCDB and mEDG.)
E
110
F
130
B
D
G
B
A
50
G 5. In the diagram at left, AB || CED , CF || EG ,
mABC = 50, and mGED = 20. Find mBCF.
F
x
20
C
D
E
6. Using the diagram at right, determine if l1 || l2,
and give a reason for your answer.
(2x + 1)
(6x + 10) (3x – 19)
l1
l2
(y – 25)
(x + 10) (2y – 20)
(x + y)
l3
7. In the diagram at left, l1 || l2.
a. Find the values of x and y.
b. Determine if l3 || l4.
l4
(This assignment is continued on the next page.)
l1
l2
8. In the diagram at right, if l1 || l2, then ma
(1) is less than 50.
(2) is exactly 50
(3) is greater than 50
(4) could be any of (1) – (3).
l1
50
a
l2
9. In the diagram at right, which of the following must be true?
(1) 1  2.
(2) 1 is supplementary to 2.
(3) 1 is complementary to 2.
(4) None of these.
2
1
10. In the diagram, if l1 || l2, then which of the following must be true?
(1) 1  5
(2) 2 is supplementary to 3
(3) 3  5
(4) 2  4
(5) All of these.
l1
2
3
l2
l3
1
4
5
l4
11. Which of the following is always true?
(1) Alternate interior angles are congruent.
(2) Corresponding angles are congruent.
(3) Same side interior angles are supplementary.
(4) Vertical angles are congruent.
(5) All of them.
(6) None of them.
12. Memorize the following terms (you should already know most of them)
Polygon: A simple closed figure formed by line segments (see right).
Triangle: A polygon with exactly three sides.
Scalene triangle: A triangle with no sides congruent.
Isosceles triangle: A triangle with (at least) two sides congruent.
Equilateral triangle: A triangle with all three sides congruent.
Acute triangle: A triangle with all acute angles.
Right triangle: A triangle with one right angle.
Obtuse triangle: A triangle with one obtuse angle.
Polygon
Not
“simple”
Not “closed”
Sides not
all segments
Geometry HW: Triangles - 3
1. In an equilateral triangle, all three angles are congruent. What is the measure of each angle?
2. In an isosceles right triangle, the two acute angles are congruent. What is their measure?
3. The measures of the angles of a triangle are in the ratio 2:4:9. Find the measures of all three angles.
4. a. What is a right triangle?
b. What must be true about the acute angles of a right triangle?
5. Explain why each of the following triangles is impossible:
a. Equilateral right triangle
b. Obtuse right triangle
6. In ABC, the measure of B is 15 more than the measure of A and the measure of C is three times the
measure of B. Find the measures of all three angles of the triangle.
7. A triangle has perimeter 35. The length of the longest side is equal to the square of the length of the
shortest side. The length of the middle side is one less than four times the length of the shortest side. Find
the lengths of all three sides algebraically.
A
B
8. In the diagram at right, AB || CD , BEC , and
ED  BEC . If mABC = 50, find mCDE.
9. Triangle BUG has vertices B(5, –2), U(0, 3) and G(7, 4).
a. Find the perimeter of BUG in simplest radical form.
b. Find the coordinates of the point M on BG so that UM is a median of BUG.
c. Show using coordinate geometry that UM is also an altitude of BUG.
10. In triangles ABC and XYZ, A  X and B  Y. Outline a proof that C  Z.
mA = a, mB = b, etc.
E
C
D
For convenience, let
Geometry HW: Triangles - 4
1. In each diagram, find the number of degrees in the value of x.
a.
b.
20
40
x
c.
(140 – x)
d.
x
40
x
(2x + 20)
70
(2x)
x
2. Triangle PQR is a right triangle with its right angle at R. An exterior angle at P measures 125. What is
the measure of the smallest interior angle of PQR?
3. In ABC, mB is four times as large as mA. An exterior angle at C measures 125. Find the measure
of A.
4. In DEF, mD = 2x + 4, mE = 6x – 58 and the degree measure of an exterior angle at F is represented by
5x. What kind of triangle is this and why?
5. a. Draw a diagram of triangle ABC and extend each side so that there is one exterior angle at each vertex.
b. If mA = a, represent the measure of an exterior angle at A in terms of a. Using b and c for the
measures of the interior angles, find expressions for the exterior angles at B and C.
c. Find the sum of the measures of the exterior angles of a triangle (one exterior angle at each vertex) in
simplest form.
l1
x
6. In the diagram at right, l1 || l2, mx = 120 and my = 135, find mz.
z
y
l2
60 40
7. Find the values of x, y and z in the diagram at right.
50
x y
z
y
65
8. Find the values of x, y and z in the diagram at right.
x
9. Triangle ABC has vertices at A(–2, 3), B(4, 7) and C(6, –1).
a. Find the coordinates of M, the midpoint of AB , and N, the midpoint of BC .
b. Show using coordinate geometry that MN || AC .
c. What is the relationship between the measures of BAC and BMN?
d. What is the relationship between the measures of ACB and MNC?
e. Given that mACB = 49.40 and mNMB = 60.26, find mABC.
60
z
Geometry HW: Triangles - 5
1. Tell whether the given lengths may be the sides of a triangle. Show how you know.
a. 6, 2, 3
b. 5, 3, 7
c. 9, 4, 5
d. 4, 6, 3
2. Which of the following could be the sides of a triangle (multiple choice)? Tell how you know.
(1) {2, 5, 50 ]
(2) {1, 3, 20 }
(3) {2, 3, 20 }
(4) {1, 5, 40 }
3. If the lengths of two sides of a triangle are 3 and 8, what are all the possible lengths of the third side
(multiple choice)? Show how you got your answer.
(1) 3 < s < 8
(2) 3 < s < 11
(3) 5 < s < 8
(4) 5 < s < 11
4. a. If the lengths of two sides of a triangle are 5 and 9, what are all the possible lengths of the third side?
b. If the lengths of two shorter sides of a triangle are 5 and 9, what are all the possible lengths of the third
side?
5. In ABC, AB = 7, BC = 9, and AC = 5. Name the smallest angle of the triangle and tell how you know.
6. In ABC, AC  BC and mB = 35. Name the shortest side of the triangle and tell how you know.
7. In ABC, mC = 58 and the measure of an exterior angle at A is 118. List the sides of the triangle in order
from smallest to greatest. Show work on a diagram.
8. In RST, RS = x + 4, ST = 2x, RT = 3x – 16 and the perimeter is 30. List the angles of the triangle in order
from smallest to largest. Show work.
9. In PQR, QR = 12 and PR = 5.
a. The smallest angle of the triangle is (multiple choice)
(1) P
(2) Q
(3) R
(4) cannot be determined.
b. The largest angle of the triangle is (multiple choice)
(1) P
(2) Q
(3) R
(4) cannot be determined.
10. In RST, an exterior angle at R measures 80 degrees. If mS > mT, name the shortest side of the
triangle.
11. The lengths of the three sides of a triangle are represented by 4x, x + 3, and 2x + 1. Which of the following
could be the perimeter of the triangle (multiple choice)? Justify your answer.
(1) 25 only
(2) 39 only
(3) both 25 and 39
(4) neither 25 nor 39
12. Hicksville and Sticksville are 14 miles apart and Slob City is 6 miles from Sticksville. Which of the
following could be the distance from Hicksville to Slob City (multiple choice)?
I. 10 miles
II. 20 miles
III. 30 miles
(1) I only
(2) II only
(3) I or II but not III
(4) I or II or III
Geometry HW: Triangles - 6
1. Find the coordinates of the centroid of PQR having vertices P(–5, 4), Q(–4, –3) , and R(6, 8).
2. Two vertices of PQR are P(2, 3) and Q(4, 8). The centroid of the triangle is C(5, 3). Find the coordinates
of R.
3. In ABC vertex A has coordinates A(2, 7) and the midpoint of side BC is M(8, –2). Find the coordinates
of the centroid of the triangle.
A
N
4. In the diagram of ABC, G is the centroid.
G
C
a. If GM = 3, what is the value of AG?
L
b. If BN = 12, what is the value of GN?
c. If CL = 15, what is the value of CG?
M
B
5. In ABC, M is the midpoint of BC and G is the centroid. If AG = x2 and GM = x + 12, find the shortest
possible length of median AM .
6. In PQR, medians PM and QN intersect at G. If GM = x, GN = 2x – y, GP = y + 6 and GQ = 6y – 12.
Find the numerical lengths of both medians.
y
7. In HAT at right, A is at the origin, T is on the x-axis, the
H
altitude measures 12, side AH measures 13 and the area is 96.
a. Find the coordinates of A.
b. Find the coordinates of T.
13
12
c. Find the coordinates of H.
d. Find the coordinates of the centroid of HAT.
A
T
x
Geometry HW: Triangles - 7
1. Triangle PQR has vertices P(–4, 2), Q(10, 0) and R(–2, 6).
a. Prove using coordinate geometry that PQR is a right triangle.
b. Prove using coordinate geometry that the length of the median to the hypotenuse of PQR is half the
length of the hypotenuse.
2. Triangle ABC has vertices A(–4, 4), B(8, 8) and C(2, 0).
a. Prove using coordinate geometry that the segment connecting the midpoints of AC and BC is parallel
to side AB .
b. Prove using coordinate geometry that the length of the segment connecting the midpoints of AC and
BC is half the length of side AB .
3. Any right triangle can be represented by the coordinates O(0, 0), A(2a, 0) and B(0, 2b) where a and b are
both positive numbers.
a. Sketch OAB. Find the coordinates of M, the midpoint of AB , and sketch median OM .
b. Prove the following theorem: The length of the median to the hypotenuse of a right triangle is half the
length of the hypotenuse.
c. Presumably, you just worked hard to prove this theorem. Now memorize it!
4. Any triangle can be represented by the coordinates O(0, 0), A(2a, 0) and B(2b, 2c) where a and c are both
positive numbers and b may be any number.
a. Sketch OAB. Find the coordinates of M and N, the midpoints of OB and AB , and sketch MN .
b. Prove the following theorem: The line segment that joins the midpoints of two sides of a triangle is
parallel to the third side of the triangle.
c. Prove the following theorem: The length of the line segment that joins the midpoints of two sides of a
triangle is half the length of the third side of the triangle.
d. Memorize these theorems!
Geometry HW: Triangles - 8
Name
1. Given: AGD , FGC , AD  FC , GD  GC
Prove: AG  FG
F
A
G
B
2. Given: BCDE , BC  DE
Prove: BD  CE
3. Given: AB  BE , FE  BE
Prove: B  E
C
D
Diagram for #1 – 3.
E
4. Given: GXF , HXE
Prove: GXH  EXF
Z
C
G
H
D
X
A
5. Given: AB || CD , GXF
Prove: DGF  AFG

6. Given: CHD , AEB , and EH ; CHE  BEH
Prove: AB || CD

7. Given: AEFB , XEF  XFE
Prove: AEX  BFX
F
E
Diagram for #4 – 7.
B
Geometry HW: Triangles - Review
1. In the diagram, l1 || l3. Determine if each of the
following is true or false:
a. 1  5
b. 9 supp to 10
c. 2 supp 8
d. 5  7
e. 5  8
f. 6  7
g. 3  6
h. 1  4
i. 4  10
j. 1  8
t1
t2
l1
1
2
3
4
6
5
l2
7
8
9
10
l3
2. Find the value of x in each of the diagrams below.
a.
b.
c.
80
70
130
t1
x
d.
x
x
e.
f.
x
(3x)
(x + 40)
(120 – x)
(3x)
g.
l3
40
h.
(x2)
130

x
(2x + 18)
(7x + 4)
3. Which set of numbers form the lengths of the sides of an isosceles triangle?
(1) {6, 5, 4}
(2) {6, 6, 13}
(3) {6, 6, 12}
(4) {6, 11, 6}
4. In quadrilateral PQRS (at right), PQ || RS , diagonal QS is drawn,
m3 = 80 and m4 = 35. With that information, it is possible to find the
measure of (choose one)
(1) 1
(2) 2
(3) Both
(4) Neither
Q
R
1 2
P
4
3
S
5. In Jellystone Park, a garbage dumpster is located 120 yards from the fishpond. Rocky Raccoon lives in a
tree 70 yards from the garbage dumpster.
a. Which could be the distance from Rocky’s tree to the pond?
(1) 30 yards
(2) 130 yards
(3) 230 yards
(4) none of these
b. Is it possible that Rocky’s tree is 190 yards from the fishpond? Justify your answer.
6. The measures of the acute angles of right triangle ABC are in the ratio 2:3. The measure of the smaller
acute angle must equal
(1) 18°
(2) 36°
(3) 54°
(4) 90°
7. The centroid of a triangle is the point of concurrency of the
(1) medians of the triangle
(2) altitudes of the triangle
(3) angle bisectors of the triangle
(4) perpendicular bisectors of the sides of the triangle
8. In ABC, AB = 10, BC = 11, and CA = 7. Which is the smallest angle of the triangle?
9. If two sides of a triangle have lengths 7 and 11, what are the possible values for the length of the third side?
10. In RST, an exterior angle at S measures 70°. Which is the longest side of RST and how do we know?
11. The degree measures of the three angles of a triangle are represented by 3x + 18, 4x + 9, and 10x. What
kind of triangle is it and why?
12. A parallelogram has sides measuring 3 feet and 8 feet. List all possible integral values for the lengths of a
diagonal of the parallelogram. (The word integral comes from the word integer. You should know both.)
13. Find the value of x in the diagram at right and determine if l1 || l2.
l1
(3x + 24)
(2x)
(x + 3y)
a
(3x – 2y)
l1
(243 – 14x) (x2 – 2x)
l2
14. Given that l1 || l2 in the diagram at left, find the ma.
15. Giselle has created a decorative glass tabletop in the shape
of a triangle. She wants it to be supported by a single
pedestal (leg) as shown in the diagram. At what
coordinates should the pedestal meet the tabletop?
l2
(21, 36)
(48, 0)
x
y
16. In ABC, the centroid is at G.
a. If AG = 9, find the length of the median from A to BC .
b. If M is the midpoint of AB and CM = 18, find the length of CG .
c. If N is the midpoint of AC , BG = x2 – 20 and GN = 3x – 10, find the numerical length of median BN .
17. In the diagram at right ABC  DEF. Find the
numerical measure of the perimeter of triangle
DEF.
B
5x – 6
3x – 1
2x + 3
A
E
7x – 10
C
D
F
18. Given: LUV || MATH , AU bisects LUT
Prove: UAT  AUT
U
L
M
A
V
H
T
K
19. Given: AT  OAK , OP  OAK , TOP  HAT
Prove: AH || OT
A
H
T
O
P
Review Answers
1. c, d, i, and j are TRUE
2a. 80
b. 50
c. 70
3. (4)
4. (1)
5a. (2)
9. 4 < s < 18 10. RT
11. Iso rt. 
14. 80
15. (23, 12) 16a. 13.5
15. Statement
1. LUV || MATH
2. LUA  UAT
3. AU bisects LUT
4. LUA TUA
5. UAT  AUT
16. Statement
1. AT  OAK , OP  OAK
2. KAT  AOP
3. TOP  HAT
4. KAH  AOT
5. AH || OT 
d. 20
e. 30
f. 50
g. 70
b. Yes
6. (2)
7. (1)
8. B
12. 6, 7, 8, 9, 10
13. x = 7 or x = 9; not ||
b. 12
c. 24
17. 22
Reason
1. Given
2. When 2 lines are ||, alt. int. s are 
3. Given
4. A bisector divides an  into 2  parts
5. Transitive Post. (2, 4)
Reason
1. Given
2.  segs form rt s and all rt s are 
3. Given
4. Subtraction (2, 3)
5. When corr. s are , lines are ||
h. 11
STUFF YOU SHOULD KNOW:
Vocabulary
Parallel
Same side interior angles
Scalene
Exterior angle
Corresponding angles
Polygon
Isosceles
Congruent triangles
Alternate interior angles
Triangle
Equilateral
Postulates and Theorems
The shortest distance between two points is a line.
Through a point not on a given line, there is exactly one line parallel to the given line.
When parallel lines are cut by a transversal, corresponding angles are congruent.
When parallel lines are cut by a transversal, alternate interior angles are congruent.
When parallel lines are cut by a transversal, same side interior angles are supplementary.
When two lines are cut by a transversal and corresponding angles are congruent, the lines are parallel.
When two lines are cut by a transversal and alternate interior angles are congruent, the lines are parallel.
When two lines are cut by a transversal and same side interior angles are supp., the lines are parallel.
The sum of the interior angles of a triangle is 180
The measure of an exterior angle of a triangle equals the sum of the measures of the remote interior angles.
If two sides of a triangle are congruent, the angles opposite those sides are also congruent (Base angles of an
isosceles triangle are congruent).
If two angles of a triangle are congruent, the sides opposite those sides are also congruent.
The longest (shortest) side of a triangle is opposite the largest (smallest) angle.
Any two sides of a triangle must add up to more than the third side.
How to:
Use angle pairs to determine if two lines are parallel.
Find angles given that two lines are parallel.
Determine the longest (shortest) side of a triangle given its angles.
Determine the largest (smallest) angle of a triangle given its sides.
Tell if three lengths can be the sides of a triangle.
Find the missing angle of a triangle given two angles.
Apply the Isosceles Triangle Theorem.
Apply the Exterior Angle Theorem.
Find the possible lengths of the third side of a triangle given the lengths of two sides.
Geometry: Triangles Answers
HW – 1
2a. Two adjacent s that form a st. line are supp.
b. Supplementary angles sum to 180
d. 3 + 2 = 180
f. 3 and 2 are supp.
3. (2)
4a. 12
b. 250
c. 54
6a. x = 13.75
b. No; alt. int. s are not 
HW – 2
1.
Statement
1. l1 || l2
2. 1  2
3. l3 || l4
4. 2  3
5. 1  3
c.
e.
g.
5.
7.
When lines are ||, corr. s are 
Substitution (3, 4)
Two angles that sum to 180 are supp.
Goofus gets full credit. Why?
(40, 20)
Reason
1. Given
2. When lines are ||, corr. s are 
3. Given
4. When lines are ||, alt. int. s are 
5. Transitive (2, 4).
2.
Statement
1. PAR , RLE
2. APE  ALR
3. ALR  LAR
4. APE  LAR
5. AL || PE
Reason
1. Given
2. Given
3. Given
4. Transitive (3, 4)
5. When 2 lines cut by a transv. and corr s are , the lines are ||.
3.
Statement
1. AM  MS , MS  SB
2. AMS  MSB
3. AMN BSR
4. NMS  MSR
5. LMN || RST
Reason
1. Given
2.  lines form rt. s which are 
3. Given
4. Subtraction (2, 3)
5. When 2 lines cut by a transv. and alt. int. s , the lines are ||.
4. 60
8. (1)
5. 30
9. (4)
6. No, alt. int. s are not 
10. (3)
11. (4)
7a. (40, 75) b. No; same side int. s not supp.
HW – 3
1. 60
2. 45
3. 24, 48, 108
4b. They are complementary
5a. See prob #1
b. Would sum to more than 180
6. 24, 39, 117
7. 4, 15, 16
8. 40
9a. 10 2  2 10
10. a + b + c = 180
s of a  sum to 180
x + y + z = 180
Same as above
a+b+c=x+ y+z
Transitive postulate
a
=x
__
Given
b+c= y+z
Subtraction postulate
b
= y_____
Given
c=z
Subtraction postulate
HW – 4
1a. 60
b. 130
c. 50
d. 35
2. 35
3. 25
5b. 180 – a, 180 – b, 180 – c
c. 360
6. 105
7. (70, 110, 30)
8a. (145, 90, 125)
9e. 70.34
HW – 5
1a. No
b. Yes
2. (3)
3. (4)
7. AB, AC , BC
10. (1)
12. (3)
4. rt. ; mF = 90
c. No
d. Yes
4a. 4 < s < 14
b. 9 < s < 14
5. B
6. AC
8. S, T, R
9a. (2)
b. (4)
10. RS
HW – 6
1. (1, 3)
2. (9, 2)
6. 15 and 18 7b. (16, 0)
3. (6, 1)
c. (5, 12)
4a. 6
d. (7, 4)
b. 4
c. 10
5. 24
HW – 7
1a. mPR  2; mQR  
1
2
b. Midpoint of PQ = M(3, 1). PQ =
200  10 2 ; RM =
50  5 2 
2a. Midpoint of AC = M(–1, 2); midpoint of BC = N(5, 4). mAB  mMN 
b. AB = 160  4 10 ; MN =
40  2 10 
3a. Midpoint of AB = M(a, b).
b. AB =
1
 PQ 
2
1
3
1
 AB 
2
4a2  4b2  2 a2  b2 ; OM =
a 2  b2 
1
 AM 
2
c. I can’t do this one for you.
4a. Midpoint of OB = M(b, c); midpoint of AB = M(a + b, c).
1
b. mOA  mMN  0
c. OA = 2a; MN = a =  OA 
2
HW – 9
1a. CBM b. RQPS
c. RTS
d. ZERO
2a. DCE  BAF, CED  AFB, EDC  FBA
3. 60
4. 120
5a. All three b. All three
d. See 3c.
b. CD  AB , DE  BF , CE  AF
HW – 10
1. Statement
1. AGD , FGC
2. AD  FC
3. GD  GC
4. AG  FG
Reason
1. Given
2. Given
3. Given
4. Subtraction (2, 3)
3. Statement
1. AB  BE , FE  BE
2. B  E
5. Statement
1. AB || CD , GXF
2. DGF  AFG
2. Statement
1. BCDE
2. BC  DE
3. CD  CD
4. BD  CE
Reason
1. Given
2.  segs form rt. s
and all rt. s are 
Reason
1. Given
2. When || lines are cut
by a transversal, alt.
s are 
7. Statement
1. AEFB
2. XEF  XFE
3. AEX is supp. to XEF
BFX is supp. to XFE
4. AEX  BFX
Reason
1. Given
2. Given
3. Reflexive Post.
4. Addition (2, 3)
4. Statement
1. GXF , HXE
2. GXH  EXF
Reason
1. Given
2. Int. lines form vert.
s which are 
6. Statement
Reason
1. CHD , AEB , EH 1. Given
2. CHE  BEH 2. When 2 lines are cut by
a transversal and alt. int.
s are , the lines are ||
Reason
1. Given
2. Given
3. When 2 adjacent s form a straight angle, they are supp.
4. When 2 s are  (2), their supplements (3) are also 