Geo_Book_Answers

Answers to Exercises
LESSON 5.6
1. Sometimes true; it is true only if the
parallelogram is a rectangle.
2. Always true; by the definition of rectangle, all
the angles are congruent. By the Quadrilateral Sum
Conjecture and division, they all measure 90°, so
any two angles in a rectangle, including consecutive
angles, are supplementary.
3. Always true by the Rectangle Diagonals
Conjecture.
4. Sometimes true; it is true only if the rectangle is
a square.
true
true
5. Always true by the Square Diagonals Conjecture.
6. Sometimes true; it is true only if the rhombus is
equiangular.
true
false
false
false
7. Always true; all squares fit the definition of
rectangle.
8. Always true; all sides of a square are congruent
and form right angles, so the sides become the legs
of the isosceles right triangle and the diagonal is
the hypotenuse.
9. Always true by the Parallelogram Opposite
Angles Conjecture.
10. Sometimes true; it is true only if the
parallelogram is a rectangle. Consecutive angles of
a parallelogram are always supplementary, but are
congruent only if they are right angles.
11. 20
12. 37°
13. 45°, 90°
14. DIAM is not a rhombus because it is not
equilateral and opposite sides are not parallel.
15. BOXY is a rectangle because its adjacent sides
are perpendicular.
16. Yes. TILE is a rhombus, and a rhombus is a
parallelogram.
68 ANSWERS TO EXERCISES
17.
18. Constructions will vary.
B
A
19. one possible construction:
I
E V
L O
E
K
P S
E
20. Converse: If the diagonals of a quadrilateral
are congruent and bisect each other, then the
quadrilateral is a rectangle.
Given: Quadrilateral ABCD with diagonals
AC BD. AC and BD bisect each other
Show: ABCD is a rectangle
A
8 E
B
D
C
1 2 3
7
6 5
4
B
A
E
Because the diagonals are congruent and bisect
each other, AE BE DE EC.Using the
Vertical Angles Conjecture, AEB CED and
BEC DEA.So AEB CED and AED
CEB by SAS. Using the Isosceles Triangle
Conjecture and CPCTC, 1 2 5 6,
and 3 4 7 8. Each angle of the
quadrilateral is the sum of two angles, one from
each set, so for example, mDAB m1 m8.
By the addition property of equality, m1
m8 m2 m3 m5 m4 m6
m7. So mDAB mABC mBCD
mCDA. So the quadrilateral is equiangular. Using
1 5 and the Converse of AIA, AB CD.
Using 3 7 and the Converse of AIA,
BC AD . Therefore ABCD is an equiangular
parallelogram, so it is a rectangle.
21. If the diagonals are congruent and bisect each
other, then the room is rectangular (converse of the
Rectangle Diagonals Conjecture).
K