Download Geometry 1 – Unit Two: Reasoning and Geometric Proof, Practice

Geometry 1 – Unit Two: Reasoning and Geometric Proof, Practice
Use the diagram to the right to answer Questions #1 – #5.
1.
T or F: Another name for plane P is plane CDF.
2.
T or F: The intersection of the horizontal plane and
the vertical plane is point D.
3.
T or F: Points G and F are coplanar.
4.
T or F: Only one line can pass through point C.
5.
HJJG
HJJG
T or F: CE and AB intersect at point D.
B
F
C
D
E
P
A
G
In Questions #6 – #13, identify the algebra property illustrated.
6.
3( x + 4) = 3x + 12
7.
If x − 7 = 1, then x = 8.
8.
If AB ≅ CD, then CD ≅ AB.
9.
If 2 x = 24, then x = 12.
10.
If 3x + 1 = −8, then 3x = −9.
11.
∠1≅∠1
12.
If
13.
If x + y = 10 and y = 2, then x + 2 = 10.
x
= 10, then x = 40.
4
In Question #14, solve the equation and write a reason for each step.
14.
2( x − 4) + 3x = 4 x − 10
Original equation
______________________________
______________________________
______________________________
Combine like terms.
______________________________
______________________________
______________________________
______________________________
In Questions #15 – #20, identify the postulate, definition, or theorem illustrated.
15.
If ∠ 1 and ∠ 2 are vertical angles, then ∠ 1 ≅ ∠ 2.
16.
If ∠ A is a right angle, then ∠ A = 90º.
17.
If ∠ 3 and ∠ 4 are complementary, then ∠ 3 + ∠ 4 = 90º.
18.
If point R lies between points S and T on ST , then SR + RT = ST .
19.
∠ 1 and ∠ 2 are supplementary. ∠ 2 and ∠ 3 are supplementary. Therefore, ∠ 1 ≅ ∠ 3.
20.
If ∠ J and ∠ K form a linear pair, then ∠ J and ∠ K are supplementary.
In Question #21, fill in the blanks to complete the following proof, using the diagram
to the right.
21.
GIVEN: ∠ 1 = ∠ 3
B
PROVE: ∠ BAD = ∠ CAE
C
1
D
2
3
A
STATEMENT
E
1) ∠ 1 = ∠ 3
REASON
1) ________________________________
2) ∠ 1 + ∠ ______ = ∠ 3 + ∠ ______
2) Addition Property
3) ∠ 1 + ∠ 2 = ∠ ______
3) Angle Addition Property
4) ∠ ______ + ∠ ______ = ∠ ______
4) ________________________________
5) ∠ BAD = ∠ CAE
5) ________________________________
In Question #22, fill in the blanks to complete the following proof, using the diagram
to the right.
22.
GIVEN: WY = XZ
W
X
PROVE: WX = YZ
STATEMENT
Y
1) WY = XZ
REASON
1) ________________________________
2) WY = ______ + ______
2) Segment Addition Postulate
3) ______ = XY + YZ
3) ________________________________
4) WX + XY = ______ + ______
4) Substitution Property
5) ________________________________
5) ________________________________
Z
In Question #23, fill in the blanks to complete the following proof, using the diagram
to the right.
23.
GIVEN: ∠ LON is a right angle
L
PROVE: ∠ 4 and ∠ 5 are complementary
M
4
5
O
STATEMENT
1) ∠ LON is a right angle
REASON
1) ________________________________
2) ∠ LON = ______º
2) ________________________________
3) ∠ LON = ∠ ______ + ∠ ______
3) ________________________________
4) ∠ ______ + ∠ ______ = ______º
4) Substitution Property
5) ________________________________
5) ________________________________
N
**************************ANSWERS**************************
1.
True
Another way to name a plane is to identify three points that are contained in the plane.
2.
False
HJJG
The intersection of two planes is a line. To make the statement true, replace point D with AB.
3.
False
Coplanar means part of the same plane. Point G is part of the vertical plane, while point F is
part of the horizontal plane.
4.
False
Although there is only one line pictured that is passing through point C, there are actually an
infinite number of lines that can pass through point C.
5.
True
A
HJJpostulate
G
HJJG states “If two lines intersect, then their intersection is exactly one point.”
CE and AB intersect at point D.
6.
Distributive Property – a ( b + c ) = a ⋅ b + a ⋅ c → 3 ( x + 4 ) = 3 ⋅ x + 3 ⋅ 4 = 3 x + 12
7.
Addition Property – If a = b, then a + c = b + c → x − 7 = 1 → x − 7 + 7 = 1 + 7 → x = 8
8.
Symmetric Property – If a = b, then b = a → If AB ≅ CD, then CD ≅ AB.
9.
Division Property – If a = b and c ≠ 0, then a / c = b / c → 2 x = 24 → 2 x / 2 = 24 / 2 → x = 12
10.
Subtraction Property – If a = b, then a − c = b − c → 3 x + 1 = −8 → 3 x + 1 − 1 = −8 − 1 → 3 x = −9
11.
Reflexive Property – a = a → ∠ 1 ≅ ∠ 1
12.
Multiplication Property – If a = b, then a ⋅ b = b ⋅ c →
13.
Substitution Property – If a = b, then b may be substituted for a in any equation or inequality
If x + y = 10 and y = 2, then x + 2 = 10 → In the original equation “x + y = 10,” the variable y
has been replaced with the number 2.
14.
2( x − 4) + 3 x = 4 x − 10 -- Original Equation
2 ⋅ x − 2 ⋅ 4 + 3x = 4 x − 10 → 2 x − 8 + 3 x = 4 x − 10 -- Distributive Property
5 x − 8 = 4 x − 10 -- Combine Like Terms
5 x − 8 − 4 x = 4 x − 10 − 4 x → x − 8 = −10 -- Subtraction Property
x − 8 + 8 = −10 + 8 → x = −2 -- Addition Property
15.
Vertical angles are congruent
x
x
= 10 → ⋅ 4 = 10 ⋅ 4 → x = 40
4
4
16.
Definition of a Right Angle → A right ∠ is an ∠ with a measure of 90°
17.
Definition of Complementary Angles → Complementary ∠s are 2 ∠s whose sum is 90°
18.
Segment Addition Postulate
19.
If two ∠s are supplementary to the same ∠, then they are congruent. → ∠ 1 and ∠ 3 are both
supplementary to ∠ 2
20.
Linear Pair Postulate → If two ∠s form a linear pair, then they are supplementary
21.
1) ∠ 1 = ∠ 3 -- Given
2) ∠ 1 + ∠ 2 = ∠ 3 + ∠ 2 -- Addition Property
Added ∠ 2 to both sides of equation in Step #1
3) ∠ 1 + ∠ 2 = ∠ BAD -- Angle Addition Postulate
4) ∠ 3 + ∠ 2 = ∠ CAE -- Angle Addition Postulate
5) ∠ BAD = ∠ CAE -- Substitution Property
Replaced ∠ 1 + ∠ 2 with ∠ BAD in Step #2, replaced ∠ 3 + ∠ 2 with ∠ CAE in Step #2
22.
1) WY = XZ -- Given
2) WY = WX + XY -- Segment Addition Postulate
3) XZ = XY + YZ -- Segment Addition Postulate
4) WX + XY = XY + YZ -- Substitution Property
Replaced WY with WX + XY in Step #1, replaced XZ with XY + YZ in Step #1
5) WX = YZ -- Subtraction Property
Subtracted XY from both sides of equation in Step #4
23.
1) ∠ LON is a right angle -- Given
2) ∠ LON = 90º -- Definition of a Right Angle
3) ∠ LON = ∠ 4 + ∠ 5 -- Angle Addition Postulate
4) ∠ 4 + ∠ 5 = 90º -- Substitution Property
Replaced ∠ LON with ∠ 4 + ∠ 5 in Step #2
5) ∠ 4 and ∠ 5 are complementary -- Definition of Complementary Angles