Question

1. (l57) from the given statement below, select the definition, property, postulate, or theorem that leads to the statement you are asked to prove.

given: $overline{bm} cong overline{dm}$; $overline{am} cong overline{cm}$
prove: $\triangle abm cong \triangle cdm$
a. corresponding parts of congruent triangles are congruent (c.p.c.t.c.).
b. angle - side - angle
c. side - angle - side
d. the diagonals of a parallelogram bisect each other.
e. vertical angles are congruent.
image of a quadrilateral with diagonals intersecting at m

1. (l54) from the given statement below, select the definition, property, postulate, or theorem that leads to the statement you are asked to prove.

given: $overline{ab} cong overline{dc}$; $overline{ad} cong overline{bc}$
prove: $abcd$ is a parallelogram
a. definition of a parallelogram
b. if one pair of opposite sides of a quadrilateral are both parallel and congruent, then it is a parallelogram.
c. if the diagonals of a quadrilateral bisect each other, then it is a parallelogram.
d. if both pairs of opposite sides of a quadrilateral are congruent, then it is a parallelogram.
e. if both pairs of opposite sides of a quadrilateral are parallel, then it is a parallelogram.
image of a quadrilateral with marked congruent sides

1. (l51) for the given statement below, select the definition, property, postulate, or theorem that leads to the statement you are asked to prove.

given: isosceles trapezoid $lijk$
prove: $angle i cong angle j$
a. the diagonals of an isosceles trapezoid are congruent.
b. the lower (and upper) base angles of an isosceles trapezoid are congruent.
c. definition of an isosceles trapezoid
d. if a quadrilateral is a parallelogram, then both pairs of opposite angles are congruent.
e. if parallel lines are cut by a transversal, then corresponding angles are congruent.
image of an isosceles trapezoid
indicate whether the statements in problems 21 and 22, respectively, are the converse, inverse, or contrapositive of the following statement:
if a quadrilateral has four right angles, then it is a rectangle.
if a quadrilateral is not a rectangle, then it does not have four right angles.

1. (l56) if a quadrilateral is not a rectangle, then it does not have four right angles.

a. inverse
b. contrapositive
c. converse
d. either a or c
e. none of the above

Explanation:

Question 18

To prove \(\triangle ABM \cong \triangle CDM\) with given \(\overline{BM} \cong \overline{DM}\), \(\overline{AM} \cong \overline{CM}\), we also note that \(\angle AMB\) and \(\angle CMD\) are vertical angles (hence congruent by vertical angles theorem). So we have two sides and the included angle congruent (Side - Angle - Side, SAS). But wait, let's check the options. Wait, the vertical angles are congruent (option E says vertical angles are congruent), and then we have two sides. Wait, the given is \(BM\cong DM\), \(AM\cong CM\), and \(\angle AMB\cong\angle CMD\) (vertical angles). So the theorem that vertical angles are congruent (option E) is needed to get the included angle. Wait, no, let's re - evaluate. The triangles have \(BM = DM\), \(AM = CM\), and the angle between them (vertical angles) is congruent. So the vertical angles theorem (option E) gives the angle congruence, which is needed for SAS. But let's check the options again. Option E: Vertical angles are congruent. So to prove the triangles congruent, we need to establish that the included angle is congruent, which is done by vertical angles theorem. So the answer is E.

Given \(\overline{AB}\cong\overline{DC}\) and \(\overline{AD}\cong\overline{BC}\), we need to prove \(ABCD\) is a parallelogram. The theorem that if both pairs of opposite sides of a quadrilateral are congruent, then it is a parallelogram (option D) matches this situation. Let's check the options:

• Option A: Definition of a parallelogram is a quadrilateral with both pairs of opposite sides parallel, not about congruent sides.
• Option B: Is about one pair of opposite sides parallel and congruent.
• Option C: Is about diagonals bisecting each other.
• Option D: Matches the given (both pairs of opposite sides congruent) to prove it's a parallelogram.
• Option E: Is about both pairs of opposite sides parallel.

So the answer is D.

Given an isosceles trapezoid \(LIJK\) and we need to prove \(\angle I\cong\angle J\). In an isosceles trapezoid, the lower (and upper) base angles are congruent (option B). Let's check the options:

• Option A: Is about diagonals, not angles.
• Option B: States that the lower (and upper) base angles of an isosceles trapezoid are congruent, which is exactly what we need to prove \(\angle I\cong\angle J\) (assuming \(IJ\) and \(LK\) are the bases).
• Option C: Definition of an isosceles trapezoid is a trapezoid with legs congruent, not directly about base angles.
• Option D: Is about parallelograms, not trapezoids.
• Option E: Is about parallel lines and transversals, not trapezoids.

So the answer is B.

Answer:

E. Vertical angles are congruent.

Question 19