Preparing for the Regents Examination Geometry, AK

10-6 Squares
(pages 232–233)
1 (4) A rectangle is a square.
2 (3) sides and angles are congruent
3 (4) A trapezoid is a parallelogram.
4 (3) −−
AC −−−
DC
5 (1) congruent and bisect the angles to which
they are drawn
6 (2) x √ 2
7 Slope of −−−
DA slope of −−
VE 3 _ . Slope of
4 −−
ED slope of −−
AV 4 _ . A parallelogram
3
has two pairs of opposite sides that are parallel.
A rectangle is a parallelogram in which
consecutive sides have slopes that are negative
reciprocals. Slope of −−−
DV 1 _ . Slope
7
of −−
AE 7. Slopes of the diagonals are negative
reciprocals, thus they are perpendicular.
Therefore, DAVE is a square.
8 Slope of −−−
MA slope of −−
TH 4 _ . Slope of
3 −−−
HM slope of −−
AT 3 _ . A parallelogram has
4
two pairs of opposite sides that are parallel.
A rectangle is a parallelogram in which consecutive
sides have slopes that are negative
reciprocals. Slope of −−−
MT 1 _ . Slope of
7 −−−
AH 7. Slopes of the diagonals are negative
reciprocals, thus they are perpendicular.
Therefore, MATH is a square.
9 The diagonals bisect the vertex angles, creating
four congruent isosceles triangles. The
bisected angles (the base angles of the triangles)
measure 45. The vertex angles thus
measure 90. The diagonals cross, forming
90 angles.
10 a Slope of −−
PQ slope of −−
RS 0. −−−
QR and −−
SP
have no slope. A parallelogram has two
pairs of opposite sides that are parallel.
A rectangle is a parallelogram in which
consecutive sides are perpendicular.
b PQ RS 20. QR PS 7. Since all
sides are not congruent to each other,
PQRS is not a square.
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
11 a Given ABCD is a square. BFE is a right
angle because −−
EF −−
BD . mEBF 45
because diagonal −−
BD bisects right angle
ABC. mFEB 45. Therefore, −−
BF −−
EF .
b 1. ABCD is a square.
2. −−
EF −−
BD
3. EFD is a right angle.
4. DAE is a right ( Definition of
angle. a square)
5. DAE and DFE are right triangles.
6. (Draw −−
DE ). −−
DE −−
DE
7. DAE DFE (HL HL)
8. −−
EF −−
EA (CPCTC)
12 1. Rhombus ABCD with diagonals −−
AC
and −−
BD intersecting at E.
2. 1 2
3. −−
BE −−
CE
4. E bisects −−
AC and −−
BD .
5. −−
BD −−
AC
6. ABCD is a rectangle.
7. ABCD has all right angles.
8. ABCD is a square. (A square is a
rhombus in which
all angles are right
angles.)
10-7 Trapezoids
(pages 236–238)
1 (2) They are congruent.
2 (3) ADC ABC
3 a x z 70; y 110
b x 73; y z 107
c x 40; y 108; z 32
d x 70; y 44; z 66
e x 130; y 20; z 30
f x 82; y z 41
4 Slope of −−−
MA slope of −−
TH 1 _ . These legs
2
are parallel. Slope of −−
AT 3 _ HM has
4 . −−−
no slope. These legs are not parallel.
AT HM 10. Therefore, MATH is an
isosceles trapezoid.
10-7 Trapezoids 63