Question

1. (1.06) if a quadrilateral does not have four right angles, then it is not a rectangle.

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

1. select the correct proof from the options listed.

given: rectangle (abcd); (m) is the midpoint of (overline{ad})
prove: (\triangle bmc) is isosceles
image of rectangle (abcd) with (m) as midpoint of (ad), and diagonals (bm) and (cm) drawn
a.

statements	reasons
2. (angle a) and (angle d) are right angles.	2. definition of a right triangle
3. (angle a cong angle d)	3. all right angles are congruent.
4. (m) is the midpoint of (overline{ad}).	4. given
5. (overline{am} cong overline{dm})	5. definition of a midpoint
6. (overline{ab} cong overline{dc})	6. if a quadrilateral is a rectangle/parallelogram, then both pairs of opposite sides are congruent.
7. (\triangle bam cong \triangle cdm)	7. side - angle - side
8. (overline{bm} cong overline{cm})	8. c.p.c.t.c.
9. (\triangle bcm) is isosceles.	9. definition of an isosceles triangle

b.

statements	reasons
2. (angle a) and (angle d) are right angles.	2. definition of a right triangle
3. (angle a cong angle d)	3. all right angles are congruent.
4. (m) is the midpoint of (overline{ad}).	4. given
5. (overline{am} cong overline{dm})	5. definition of a right point
6. (overline{ab} cong overline{dc})	6. if a quadrilateral is a rectangle/parallelogram, then both pairs of opposite sides are congruent.
7. (\triangle bam cong \triangle cdm)	7. side - angle - side
8. (overline{bm} cong overline{cm})	8. c.p.c.t.c.
9. (\triangle bcm) is isosceles.	9. definition of an equilateral triangle

c.

statements	reasons
2. (overline{ab} cong overline{dc})	2. diagonals of a rectangle are congruent.
3. (\triangle bcm) is isosceles.	3. definition of an isosceles triangle

Explanation:

Step1: Analyze Option B

• In step 5 of Option B, the reason is "Definition of a right point", which is incorrect (should be "Definition of a midpoint").
• In step 9, the reason is "Definition of an equilateral triangle", but we need to prove it's isosceles, so this is wrong.

Step2: Analyze Option C

• Option C has only 3 statements. The reasoning is incomplete (it skips proving \( \overline{BM} \cong \overline{CM} \) properly) and the second reason ("Diagonals of a rectangle are congruent") is not relevant to showing \( \overline{AB} \cong \overline{DC} \) directly for this proof structure.

Step3: Analyze Option A

• Step 2: "∠A and ∠D are right angles" follows from the definition of a rectangle (not right triangle, but this is a minor wording issue, and the concept is correct as rectangles have right angles).
• Step 5: "Definition of a midpoint" correctly justifies \( \overline{AM} \cong \overline{DM} \).
• Step 7: Uses SAS ( \( \overline{AB} \cong \overline{DC} \), \( \angle A \cong \angle D \), \( \overline{AM} \cong \overline{DM} \)) to prove \( \triangle BAM \cong \triangle CDM \).
• Step 8: CPCTC (Corresponding Parts of Congruent Triangles are Congruent) gives \( \overline{BM} \cong \overline{CM} \).
• Step 9: Definition of isosceles triangle (a triangle with two congruent sides) correctly concludes \( \triangle BMC \) is isosceles.

Answer:

A.

1. Rectangle \( ABCD \) - Given
2. \( \angle A \) and \( \angle D \) are right angles. - Definition of a rectangle (note: wording "right triangle" is a typo, but concept holds)
3. \( \angle A \cong \angle D \) - All right angles are congruent.
4. \( M \) is the midpoint of \( \overline{AD} \). - Given
5. \( \overline{AM} \cong \overline{DM} \) - Definition of a midpoint
6. \( \overline{AB} \cong \overline{DC} \) - If a quadrilateral is a rectangle/parallelogram, then both pairs of opposite sides are congruent.
7. \( \triangle BAM \cong \triangle CDM \) - Side - Angle - Side
8. \( \overline{BM} \cong \overline{CM} \) - C.P.C.T.C.
9. \( \triangle BCM \) is isosceles. - Definition of an isosceles triangle