1. Quadrilateral ABCD has vertices A(6,0), B(3,9), C(-3,7) and D(0,-2). Prove that ABCD is a rectangle.
Z
2.
Given:
Prove:
WX ⊥ XY
ZY ⊥ XY
m W=m
W
Z
WX = ZY
Y
X
3. Complete the given partial proof by providing the missing statements and/or reasons.
STATEMENTS
U
1
P
3
Q
4
R
V
Given:
Prove:
PQRST
R is the midpoint of QS
1≅ 2
U≅ V
S T
2
REASONS
1. PQRST, R is the midpoint of QS,
1≅ 2
2. QR ≅ RS
1. Given
3.
3.
1 supplementary to
2 supplementary to
4.
5.
4. If 2 angles are supp. to the same
angle or congruent angles, they
are congruent to each other.
URQ ≅
6.
7.
3
4
2.
SRV
5.
6. a.s.a. ≅ a.s.a.
U≅ V
7.
ANSWERS
1. Quadrilateral ABCD has vertices A(6,0), B(3,9), C(-3,7) and D(0,-2). Prove that ABCD is a rectangle.
y2 - y1
x2 - x 1
Use slope formula:
9
AB = 9 - 0 =
= -3
-3
3-6
1
BC =
-9
CD = 2 - 7 =
= -3
3
0 - (-3)
1
7-9
-2
=
= 1
-3 - 3
-6
3
AD =
-2 - 0
-2
=
= 1
0-6
-6
3
Z
2.
Given:
Prove:
WX ⊥ XY
ZY ⊥ XY
m W=m
W
Z
WX = ZY
Y
X
STATEMENTS
REASONS
1. WX ⊥ XY, ZY ⊥ XY, m W = Z
2. WX = ZY
3. WXY and ZYX are right angles
4. WXY = ZYX
5. XY = XY
6. ∆ZYX ≅ ∆WXY
7. m W = m Z
8. WX = ZY
1. Given
2. Assumption
3. Perpendicular lines form right angles
4. All right angles are equal
5. Reflexive property
6. s.a.s. ≅ s.a.s.
7. Corresponding parts of congruent triangles
are congruent
8. Contradiction (steps 1, 7)
3. Complete the given partial proof by providing the missing statements and/or reasons.
STATEMENTS
U
1
P
3
Q
4
R
S T
2
1. PQRST, R is the midpoint of QS,
1≅ 2
2. QR ≅ RS
1. Given
3.
1 supplementary to
2 supplementary to
3. Two adjacent angles on a straight
line are supplementary
4.
3≅ 4
5.
URQ ≅
V
Given:
Prove:
PQRST
R is the midpoint of QS
1≅ 2
U≅ V
REASONS
3
4
2. A midpoint divides a line segment
into two congruent line segments
4. If 2 angles are supp. to the same
angle or congruent angles, they
are congruent to each other.
SRV
5. Vertical angles are congruent
6. ∆URQ ≅ ∆SRV
6. a.s.a. ≅ a.s.a.
7.
7. Corresponding parts of congruent
triangles are congruent.
U≅ V