Fred and Victoria provide the following proofs for vertical angles to be equal:
Fred's proof: angle 2 + angle 3 = 180° (t is a straight line)
angle 1 + angle 2 = 180° (PQ is a straight line)
Therefore, angle 1 + angle 2 = angle 2 + angle 3 (Transitive Property of Equality)
Hence, angle 1 = angle 3 (Subtraction Property of Equality)
Victoria's proof: angle 1 + angle 4 = 180° (t is a straight line)
angle 1 + angle 2 =180° (PQ is a straight line)
Therefore, angle 1 + angle 2 = angle 1 + angle 4 (Transitive Property of Equality)
Hence, angle 2 = angle 4 (Subtraction Property of Equality)
Which statement is correct?
Both Fred's and Victoria's proofs are correct.
Both Fred's and Victoria's proofs are incorrect.
Only Fred's proof is correct.
Only Victoria's proof is correct.
Question 2 (Multiple Choice Worth 4 points)
(03.05 MC)
Read the statements shown below:
If a polygon has three sides, it is a triangle.
If an angle of a triangle measures 90°, it is a right triangle.
Beth constructed a triangle in the geometry class.
Based on the given statements, which is a valid argument?
It can be concluded that Beth drew a triangle with an angle measure of 45°.
It cannot be concluded that Beth drew a right triangle.
It can be concluded that Beth drew a scalene triangle.
It cannot be concluded that Beth drew a polygon.
Question 3 (Multiple Choice Worth 5 points)
(03.04 HC)
Look at the parallelogram ABCD shown below:
The table below shows the steps to prove that if the quadrilateral ABCD is a
parallelogram, then its opposite sides are congruent:
Statement
Reasons
AB is parallel to DC
1 and AD is parallel
Definition of parallelogram
to BC
2
angle 1 = angle 2,
angle 3 = angle 4
If two parallel lines are cut by a transversal then the
corresponding angles are congruent
3 BD = BD
Reflexive Property
triangles ADB and
4
CBD are congruent
If two sides and the included angle of a triangle are
congruent to the corresponding sides and angle of another
triangle , then the triangles are congruent by SAS
postulate
5
AB = DC,
AD = BC
Corresponding parts of congruent triangles are congruent
Which statement is true about the table?
It is not correct because it provides incorrect sequence of statement 2 and
statement 4.
It is not correct because it does not provide correct reasons for statement 2 and
statement 4.
It is accurate because it provides the correct sequence of statements.
It is accurate because it provides the correct reasons for the statements.
Question 4 (Multiple Choice Worth 4 points)
(03.02 MC)
Tamara is writing statements to prove that the sum of the measures of interior angles
of triangle PQR is equal to 180o. Line m is parallel to line n.
Which is a true statement she could write?
Angle PRQ measures 30o.
Angle PRQ measures 15o.
Angle PQR measures 15o.
Angle PQR measures 45o.
Question 5 (Multiple Choice Worth 5 points)
(03.04 HC)
The figure below shows a parallelogram PQRS:
The flowchart shown below shows the sequence of steps to prove the theorem:
Opposite angles of a parallelogram are equal:
Which option best represents x and y?
x should be angle 5 is equal to angle 6 and y should be supplementary angles
x should be angle 5 is equal to angle 6 and y should be opposite angles of a
parallelogram are equal
x should be "SQ = SQ" and y should be transitive property
x should be "SQ = SQ" and y should be reflexive property
Question 6 (Multiple Choice Worth 4 points)
(03.05 LC)
Which of these is a valid conclusion?
All farmers wear straw hats.
My father wears a straw hat.
Therefore, my father is a farmer.
All farmers wear straw hats.
My father does not wear a straw hat.
Therefore, my father is a farmer.
All farmers wear straw hats.
My father is not a farmer.
Therefore, my father wears a straw hat.
All farmers wear straw hats.
My father is a farmer.
Therefore, my father wears a straw hat.
Question 7 (Multiple Choice Worth 5 points)
(03.01 HC)
Carlos draws a transversal, t, on two parallel lines AB and CD, as shown below:
He makes the following table to prove that the alternate interior angles are equal:
Statement
Justification
angle 1 = angle 5 Corresponding angles of parallel lines are congruent.
angle 1 = angle 3 ?
transitive property of equality,
angle 2 = angle 6,
angle 3 = angle 5
angle 2 = angle 4,
therefore angle 3 = angle 5
Which is the missing justification?
Opposite interior angles are congruent when parallel lines are intersected by a
transversal.
Vertical angles formed by two intersecting segments are congruent.
Adjacent angles formed by two intersecting segments are supplementary.
Opposite exterior angles are congruent when parallel lines are intersected by a
transversal.
Question 8 (Multiple Choice Worth 5 points)
(03.02 HC)
Theorem: The segment joining the midpoints of two sides of a triangle is parallel to
the third side and half its length.
A two column proof of the theorem is shown, but the statement and reasons are not in
correct order.
Statement
Justification
I
The coordinates of point D are (4, 5) and
coordinates of point E are (5, 3)
By the midpoint formula
II
Segment DE is half the length of segment AC.
By substitution
Length of segment DE is
III segment AC is
and length of
By the distance formula
.
IV Segment DE is parallel to segment AC.
V
Slope of segment DE is -2 and slope of segment
AC is -2.
Slopes of parallel lines
are equal.
By the slope formula
Which is the most logical order of statements and reasons for the proof?
II, III, I, IV, V
I, III, II, V, IV
I, III, II, IV, V
III, II, I, V, IV
Question 9 (Multiple Choice Worth 4 points)
(03.02 MC)
In the figure below, segment AC is congruent to segment AB:
Which statement is used to prove that angle ABD is congruent to angle ACD?
Triangle ACD is similar to triangle ABD.
Segment AD is congruent to segment AC.
Angle DAC is congruent to angle DAB.
Angle CAB is congruent to angle CBA.
Question 10 (Multiple Choice Worth 5 points)
(03.04 HC)
Look at the quadrilateral shown below:
Elba writes the following proof for the theorem: If the diagonals of a quadrilateral
bisect each other, the quadrilateral is a parallelogram:
Elba's proof
1. For triangles AOB and COD, angle 1 is equal to angle 2 as they are vertical angles.
2. AO = OC and BO = OD because it is given that diagonals bisect each other.
3. Therefore, the triangles AOB and COD are congruent by SAS postulate.
4. Similarly, triangles AOD and COB are congruent.
5. By CPCTC, angle ABD is equal to angle CDB and angle ADB is equal to angle
CBD.
6. As the corresponding angles are congruent, the opposite sides of quadrilateral
ABCD are parallel.
7. Therefore, ABCD is a parallelogram.
Which is the first incorrect statement in Elba's proof?
Statement 6
Statement 4
Statement 3
Statement 5
Question 11 (Multiple Choice Worth 5 points)
(03.02 HC)
Use the image shown below to answer the question that follows:
The two-column proof below is the beginning of the proof to the following theorem:
The three medians of a triangle all intersect in one point.
Statements
Reasons
Point F is a midpoint of
Point E is a midpoint of
Draw
By Construction
Statements
Reasons
Draw
Point G is the point of intersection
Intersecting Lines Postulate
between
and
By Construction
Draw
Point D is the point of intersection
Intersecting Lines Postulate
between
and
Point H lies on
such that
By Construction
and
Midsegment Theorem
and
Substitution
BCGH is a parallelogram
Properties of a Parallelogram
(diagonals bisect each other)
Which reason is missing from the proof?
Properties of a Parallelogram
Midsegment Theorem
Intersecting Lines Postulate
By Construction