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Geometry Solution Manual | Reference Guide Unit 3 | Quadrilaterals and Their Properties
Quadrilaterals and Their Properties
Solutions
1. Quadrilateral, rectangle
___ ___ ___
9. A. SN , MS , SO; the diagonals of a parallelogram
2. Quadrilateral
3. Quadrilateral, parallelogram
B.
4. Quadrilateral, rhombus
5. Quadrilateral, trapezoid
C.
___
6. A. AD; the opposite sides of a parallelogram are
congruent.
D.
___
B. CD; the opposite sides of a parallelogram are
congruent.
___
C. PD; the diagonals of a parallelogram bisect
each other.
D. ∠ABC; the opposite angles of a parallelogram
are congruent.
E. 180°; consecutive angles of a parallelogram
are supplementary.
____
7. A. QH ; the diagonals of a parallelogram bisect
B.
C.
D.
E.
each
other.
___
FH
; the diagonals of a rectangle are congruent.
___
GH; the opposite sides of a parallelogram are
congruent.
∠ EFG, ∠ HEF, and ∠ EHG; all four vertex
angles of a rectangle are right angles.
90°; all four vertex angles of a rectangle are
right angles.
E.
bisect each other, and the diagonals of a
square are congruent.
∠MSP, ∠PSO, ∠OSN; the diagonals of a
rhombus are perpendicular and all angles are 90°.
∠ MNO, ∠ NOP, ∠ MPO; all vertex angles of
a square are right angles.
m∠ SPO = m∠ POS = m∠ SON = m∠ ONS
= m∠ SNM = m∠ NMS = m∠ SMP = 45°;
the diagonals of a rhombus bisect the vertex
angles, and all of the vertex angles in a
square are right angles.
180°; consecutive angles of a parallelogram
are supplementary.
10. A. No; a trapezoid can have only one pair of
parallel sides.
B. No; a quadrilateral in which the opposite
sides are congruent is a parallelogram and
a trapezoid cannot be a parallelogram.
C. No; a quadrilateral in which the opposite
sides are congruent is a parallelogram and
a trapezoid cannot be a parallelogram.
11. Rhombus and square
12. Trapezoid
13. Parallelogram
__ __ ___
14. Rectangle
___
15. Rhombus
8. A. IL, IJ , KL; a rhombus has four congruent sides.
B. RK ; diagonals of a parallelogram bisect each
other.
C. ∠ JRK, ∠ IRL, or ∠ KRL; the diagonals of a
rhombus are perpendicular.
D. ∠ KLR or ∠ KJR or ∠ IJR; the diagonals of
a rhombus bisect its vertices, and opposite
angles are congruent.
E. 180°; consecutive angles of a parallelogram
are supplementary.
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Geometry Solution Manual | Reference Guide Unit 3 | Quadrilaterals and Their Properties
16. A. 46.8 in.; the diagonals of a parallelogram
B.
C.
D.
E.
bisect each other. BD is 2 · BE.
24 in.; the opposite sides of a parallelogram are
congruent.
23.4 in.; the diagonals of a parallelogram
bisect each other.
49°; the opposite angles of a parallelogram
are congruent.
m∠ DAB = m∠ DCB = 120°
m∠ DCA + m∠ ACB = 120°
m∠ DCA + 71° = 120°
m∠ DCA = 49°
120°; the opposite angles of a parallelogram
are congruent.
D. 322 cm; the diagonals of a rectangle are
congruent and bisect one another.
RT = 2 · US = 2 · 161 = 322
E. 161 cm; the diagonals of a rectangle bisect
one another.
VU = US = 161
F. 144 cm; the opposite sides of a rectangle are
congruent.
19. A. 17 yd; all sides of a square are congruent.
B. 12 yd; the diagonals of a square are congruent
C.
17. A. 70.6°; the opposite angles of a parallelogram
B.
C.
D.
E.
F.
are congruent.
90°; the diagonals of a rhombus are
perpendicular bisectors.
18 ft; all sides of a rhombus are congruent.
18 ft; all sides of a rhombus are congruent.
109.4°; the consecutive angles of a
parallelogram are supplementary.
m∠GFJ = m∠ FJH = 180°
70.6° + m∠ FJH = 180°
m∠ FJH = 109.4°
54.7°; the opposite angles of a parallelogram
are congruent.
m∠ FJH = m∠ HGF = 109.4°
The diagonals of a rhombus bisect the angles.
m∠JGH = m∠HGF ÷ 2 = 109.4° ÷ 2 = 54.7°
18. A. 54°; ∠ RUV and ∠ SUT are vertical angles.
B. 126°; ∠ VUT and ∠ SUT form a linear pair.
m∠ VUT + m∠ SUT = 180°
m∠ VUT + 54° = 180°
m∠ VUT = 126°
C. 63°; the angles of a rectangle are 90°.
m∠VRT + m∠TRS = 90°
m∠VRT + 27° = 90°
m∠VRT = 63°
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D.
E.
F.
and bisect one another.
FC = EB = 12
24 yd; the diagonals of a square bisect
one another.
BD = 2 · BF = 2 · 12 = 24
90°; the angles of a square are all right angles.
90°; the diagonals of a square are perpendicular
bisectors.
45°; the diagonals of a square bisect the angles
of the square.
20. A. 59°; the base angles of an isosceles trapezoid
are congruent.
m∠ BCD = m∠CBA = 121°
m∠CBA + m∠ DAB = 180°
121° + m∠ DAB = 180°
m∠ DAB = 59°
B. 121°; the base angles of an isosceles trapezoid
are congruent.
C. 59°; the base angles of an isosceles trapezoid
are congruent.
21. A. 20 mm; the diagonals of a parallelogram bisect
each other.
VY = 2 · WY = 2 · 10 = 20
B. 16 mm; the opposite sides of a parallelogram
are congruent.
C. 35°
D. 105°; the consecutive angles of a parallelogram
are supplementary.
m∠YUV + m∠ XYU = 180°
m∠YUV + 75° = 180°
m∠YUV = 105°
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Geometry Solution Manual | Reference Guide Unit 3 | Quadrilaterals and Their Properties
E. 75°; the opposite angles of a parallelogram
24. A. 20 yd; all sides of a square are congruent.
are congruent.
F. 40°; the opposite angles of a parallelogram
are congruent.
m∠ XVU = 75°
m∠ XVY + m∠YVU = 75°
35° + m∠YVU = 75°
m∠YVU = 40°
B. 28.3 yd; the diagonals of a square are
22. A. 106°; the consecutive angles of a parallelogram
are supplementary.
m∠ ABC + m∠BCD = 180°
74° + m∠BCD = 180°
m∠BCD = 106°
B. 37°; the opposite angles of a parallelogram
are congruent.
m∠ ADC = m∠ ABC = 74°
The diagonals of a rhombus bisect the angles.
m∠ ADE = m∠ ADC ÷ 2 = 74° ÷ 2 = 37°
C. 10 cm; all sides of a rhombus are congruent.
D. 90°; the diagonals of a rhombus are
perpendicular.
23. A. 62°; the angles of a rectangle are right angles.
B.
C.
D.
E.
F.
m∠ LHF + m∠ FHG = 90°
28° + m∠ FHG = 90°
m∠ FHG = 62°
28°
124°; 180° − 28° − 28° = 124°
62°
8.5 ft; the diagonals of a rectangle are
congruent and bisectors of one another.
GL = HF = 17
HJ = HF ÷ 2 = 17 ÷ 2 = 8.5
15 ft; the opposite sides of a rectangle are
congruent.
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C.
D.
E.
F.
congruent.
90°; the diagonals of a square are perpendicular.
90°; all vertex angles of a square are
right angles.
45°; the diagonals of a square bisect the
vertex angles.
90°; the diagonals of a square are perpendicular.
25. A. 5 in.
B. 110°
26. Parallelogram, quadrilateral
27. Trapezoid
28. Square
29. Answers will vary. Sample answer: A square
has two sets of parallel sides and a trapezoid has
exactly one set of parallel sides. A square has
four congruent sides and a trapezoid can have
at most one set of congruent sides. A square has
four 90° angles and a trapezoid can have at most
two 90° angles.
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