Download Geometry 3c Theorems and Proofs Practice questions 2. Given: 1

Geometry 3c Theorems and Proofs Practice questions
2.
Given:
1 and 2 are complementary,
3 and 4 are complementary,
1
4
Prove:
2
3
Proof:
Statements
Reasons
1. 1 and 2 are complements,
3 and 4 are complements,
1
4
1. Given
2. m 1 + m 2 = 90°
m 3 + m 4 = 90°
2. Definition of complementary
angles
3. m 1 + m 2 = m 3 + m 4
3. Transitive property of equality
4. m 1 = m 4
4.
5. m 1 + m 2 = m 3 + m 1
5. Substitution property of
equality
6. m 2 = m 3
6. Subtraction property of
equality
7. 2
7. Definition of congruent
angles
3
Which of the following reasons completes the proof?
A. Definition of complementary angles
B. Definition of congruent angles
C. Right angle congruence theorem
D. Transitive property of angle congruence
3.
Given:
Prove: p
1
2,
1 and 2 are a linear pair
q
Proof:
Statements
1.
1 and
pair
2 are a linear
2.
1 and 2 are
supplementary
Reasons
1. Given
2. Linear pair postulate
3. m 1 + m 2 = 180°
3. Definition of supplementary
angles
4.
4. Given
1
2
5. m 1 = m 2
5. Definition of congruent
angles
6. m 1 + m 1 = 180°
6. Substitution property of
equality
7. 2 · (m 1) = 180°
7. Distributive property
8. m 1 = 90°
8. Division property of equality
9.
9. Definition of a right angle
10. p
1 is a right angle
q
10.
Which of the following reasons completes the proof?
A. Congruent supplements theorem
B. Definition of parallel lines
C. Linear pair postulate
D. Definition of perpendicular lines
4.
Given: g
Prove:
h
1 and
2 are supplementary
Proof:
Statements
1. g
2.
1. Given
h
1
3
3. m 1 = m 3
4.
Reasons
2 and 3 are
supplementary
2.
3. Definition of congruent angles
4. Linear pair postulate
5. m 3 + m 2 = 180°
5. Definition of supplementary
angles
6. m 1 + m 2 = 180°
6. Substitution property of
equality
7.
1 and 2 are
supplementary
7. Definition of supplementary
angles
Which of the following reasons completes the proof?
A. Alternate interior angles theorem
B. Definition of linear pair
C. Alternate exterior angles theorem
D. Definition of congruent angles
6.
Given: ABC with exterior
4
Prove: m 4 = m 1 + m 2
Proof:
Statements
Reasons
1.
ABC with exterior
4
2.
m 4 + m 3 = 180°
Linear Pair
3.
m 1 + m 2 + m 3 = 180°
Triangle Sum Theorem
4.
5.
Given
Substitution Property
m 4=m 1+m 2
Subtraction Property
Which of the following reasons completes the proof?
A. m 4 + m 3 = m 1 + m 2 + m 4
B. m 1 + m 2 + m 4 = 180°
C. m 1 + m 3 + m 4 = 180°
D. m 4 + m 3 = m 1 + m 2 + m 3
7. Given m M = 25° and
M
P, determine the missing reason to prove MNP is obtuse.
Statements
m M = 25° and
M
Reasons
P
m P = 25°
given
transitive property of equality
m M + m N + m P = 180°
A.
25° + m N + 25° = 180°
substitution
m N = 130°
addition property of equality
N is obtuse
definition of obtuse angle
MNP is obtuse
definition of obtuse triangle
The square of the length of the third side of a triangle is equal to the sum of the squares of
the other two sides.
B. The sum of the interior angles of a triangle equals 180°.
C. Adjacent angles in a triangle are congruent.
D.
8.
The sum of the lengths of any two sides of a triangle is greater than the length of the third
side.
Given: AB DC and BC AD
Prove: ABCD is a parallelogram.
Proof:
Statements
1. AB
DC and BC
Reasons
AD
2. Draw AC and BD.
3. AC
AC and BD
2. Through any two points, there exists exactly one line.
BD
4. ABC
DBC
CDA and BDA
5. CBD
CAB
ADB and
6. BC
AD and AB
1. given
ACD
DC</span
7. ABCD is a parallelogram.
3. reflexive property of congruence
4.
5. Corresponding parts of congruent triangles are
congruent.
6. alternate interior angles converse
7. A quadrilateral is a parallelogram if opposite sides are
parallel.
Which reason completes the proof?
A. alternate interior angles
B. side-side-side congruence
C. Corresponding parts of congruent triangles are congruent.
D. angle-side-angle congruence
9.
Given: g
Prove:
h
1
3
Proof:
Statements
1. g
2.
1. Given
h
2
Reasons
3
2.
3. m 2 = m 3 3. Definition of congruent angles
4.
1
2
4. Vertical angles theorem
5. m 1 = m 2 5. Definition of congruent angles
6. m 1 = m 3 6. Transitive property of equality
7.
1
3
7. Definition of congruent angles
Which of the following reasons completes the proof?
A. Definition of complementary angles
B. Corresponding angles postulate
C. Definition of parallel lines
D. Consecutive interior angles theorem
10.
Given: Point D is the midpoint of AB and point E is the midpoint of segment AC.
Prove: DE
Proof:
BC
Statements
Reasons
1.
D is the midpoint of AB and E is the midpoint
of AC
Given
2.
AD = DB and AE = EC
Definition of Midpoint
3.
Division Property of Equality
4.
Transitive Property of Equality
5.
Addition Property of Equality
6.
Substitution Property of Equality
7.
simplify
8.
Segment Addition Postulate
9.
A=
10.
ADE ∼ ABC
11.
ADE ≅
12. DE
A
Reflexive Property of Equality
ABC
BC
Which of the following reasons complete the proof?
Angle-Angle similarity postulate
A.
B.
C.
D.
11.
Side-Side-Side similarity theorem
Side-Angle-Side similarity theorem
Triangle Midsegment Theorem
Definition of Similarity
Converse of Corresponding Angles
Postulate
Given: ABCD and FECG are parallelograms.
1
3
Prove:
Proof:
Statements
Reasons
1. ABCD and FECG are parallelograms.
1. Given
2.
1
2 and
2.
3.
1
3
2
3
3. transitive property of congruence
Which reason completes the proof?
A. alternate interior angles converse
B. Opposite angles of a parallelogram are congruent.
C. alternate interior angles theorem
D. Opposite sides of a parallelogram are congruent.
12.
Given:
is the perpendicular bisector of
.
Prove: Point S is equidistant from points P and R.
Proof:
Statements
1.
Reasons
is the
perpendicular Given
bisector of
.
2.
3.
PQS and
RQS are
right angles.
A perpendicular bisector forms right angles with the segment it bisects.
4.
PQS
RQS
Right angles are congruent.
5.
reflexive property
6.
Draw
Two points determine a segment.
7.
Draw
Two points determine a segment.
8.
SPQ
SRQ.
Side-Angle-Side
9.
Corresponding parts of congruent triangles are congruent.
Point S is
equidistant
The shortest distance between two points is the length of the segment
10.
from points P joining them.
and R.
Which of the following reasons completes the proof?
A. A bisector divides a segment into two congruent segments.
B. Perpendicular bisectors form congruent right angles.
C. Perpendicular segments form angles of equal measure.
D. symmetric property
13. In the triangles below, NP
RP, and PM
PQ.
Determine the missing reason to prove that MNP
QRP.
Statement
NP
RP, PM
PQ
Reason
given
m NPM = m RPQ
MNP
QRP
SAS
A. Vertical angles are congruent.
B. An angle is congruent to itself.
C. Alternate interior angles are congruent.
D. Alternate exterior angles are congruent.
14.
Given: Rectangle PQRS with diagonals PR and QS intersecting at point T.
Prove: PR
QS
Proof:
Statements
Reasons
1. PQRS is a rectangle.
Given
2. PQRS is a parallelogram.
A rectangle is a parallelogram.
3. QR
SP
Opposite sides of a parallelogram are congruent.
4. RS
SR
Reflexive property of congruence
5. m PSR = 90°
Definition of rectangle
m QRS = 90°
6. m PSR = m QRS
PSR
QRS
7. PRS
8. PR
Transitive property
QSR
Corresponding parts of congruent triangles are
congruent.
QS
Which of the following reasons completes the proof?
A. ASA
B. SSS
C. AAS
D. SAS
15.
Given: ABC and EDC
Prove: m BCA = m DCE
Proof:
Statements
Reasons
1.
Given
2.
ABC and EDC
m ACD + m DCE = 180°
Linear Pair
3.
m ACD + m BCA = 180°
4.
[180° - m BCA] + m DCE = 180°
5.
m BCA = m DCE
Which of the following reasons completes the proof?
A. Triangle Sum Theorem
B. Substitution Property
C. Corresponding Angles Postulate
D. Linear Pairs Theorem
Linear Pair
Combine Like Terms
and Addition Property