Download Mathematics 350 CW Solutions Section 3.4 CW 1. Parallelograms

Mathematics 350 CW Solutions Section 3.4
CW 1. Parallelograms:
(a) The diagonals are congruent. FALSE: #1, 5, 10
(b) The diagonals are perpendicular. FALSE: #1, 5, 6, 7, 10, 16
(c) The diagonals bisect each other (cut each other in half). TRUE
(d) The diagonals bisect the interior angles of the quadrilateral. FALSE; #1, 6, 7, 10, 16
A
CW 2: Given a parallelogram ABCD as shown, draw the
diagonals AC and BD . Let E be their point of intersection.
(i) Show that triangles ADE and CBE are congruent. (You may
E
apply Theorem 3.3. Also, be sure you give the explicit
correspondence of this congruence, identify which
C
D
corresponding parts of the triangles are know to be congruent
and why, and identify which Triangle Congruence Postulate is used to conclude the triangles are
congruent.)
By Theorem 3.3 we know that opposite sides AD and BC are congruent and parallel. When
diagonal AC crosses them alternate interior angles DAC and BCA are created, so they are
congruent by Theorem 1.2. Analogously when diagonal BD crosses those sides the alternate
interior angles ADB and CBD are created, so they are congruent. Now Triangle Congruence
Postulate ASA may be applied to show triangles ADE and CBE are congruent.
(ii) Once that triangle congruence is established, argue why we can conclude that AE and EC
are congruent, and BE and DE are congruent.
These segments are corresponding sides in the triangle congruence.
CW 3: For a rhombus:
(a) The diagonals are congruent. FALSE: #5
(b) The diagonals are perpendicular. TRUE
(c) The diagonals bisect each other (cut each other in half). TRUE
(d) The diagonals bisect the interior angles of the quadrilateral. TRUE
CW 4: For a rectangle:
(a) The diagonals are congruent. TRUE
(b) The diagonals are perpendicular. FALSE; #6, 7, 16
(c) The diagonals bisect each other (cut each other in half). TRUE
(d) The diagonals bisect the interior angles of the quadrilateral. FALSE; #6, 7, 16
B
CW 5: Show how to use two previous theorems to prove the diagonals of a rectangle bisect each
other.
By Theorem 3.7 a rectangle is a parallelogram, and by Theorem 3.8 the diagonals of a
parallelogram bisect each other.
CW 6. Show that triangles ACD and BDC are congruent. Be sure
you give the explicit correspondence of this congruence, identify
which corresponding parts of the triangles are know to be
congruent and why, and identify which Triangle Congruence
Proposition is used to conclude the triangles are congruent.
A
Proof: We set up correspondence ACD ↔ BDC and show it is a D
congruence.
Side DC in one triangle corresponds to side CD in the other, and these are congruent.
∠ADC and ∠BCD correspond and are congruent since both are right angles. Finally, side AD
in one triangle corresponds to side BC in the other, and these are congruent since they are the
opposite sides of a parallelogram (See Property 3 of this section). So we have established the
three congruences necessary to apply the SAS triangle Congruence Proposition to the triangles,
and can conclude that triangles ACD and BDC are congruent.
Why does the congruence of these triangles mean that the diagonals are congruent?
A: Because the diagonals are corresponding sides under this congruence.
CW 7: Use the sheet of quadrilaterals to answer the TRUE OR FALSE questions below about
the diagonals of convex kites. (If the kite is not convex, it only has one diagonal!) TRUE means
that the statement is true about all the examples on the sheet; FALSE means you are to give the
number of a quadrilateral on our sheet that is a counterexample.
(a) The diagonals of a convex kite are congruent. FALSE; #2, 5, 14
(b) The diagonals of a convex kite are perpendicular. TRUE
(c) The diagonals of a convex kite bisect each other (cut each other in half).
FALSE; #2, 5, 14
(d) The diagonals of a convex kite bisect the interior angles of the convex kite.
FALSE; #2, 5, 14
CW 8: In Classwork 7 you should have found that (c) and (d) are FALSE. How could you
change their statements so that the new statements are true about all the examples on the sheet?
(c) could read: The longer diagonal bisects the shorter diagonal; if the diagonals are the
same length, they bisect each other.
(d) Could read: The longer diagonal bisects its interior angles.
B
C
CW 9. Suppose ABCD is a convex kite with AB and BC
congruent and CD and DA congruent. Draw the shorter
diagonal AC and the longer diagonal BD , and let E be their
intersection. Show triangles ABD and CBD are congruent.
A
D
E
Proof: Since the quadrilateral is a kite we are given that two
C
pairs of adjacent sides are congruent. In the drawing these
adjacent sides are AD ≅ DC and BA ≅ BC . Also the common side BD is congruent to itself. So
we have established the congruence of each pair of corresponding sides of the two triangles. By
the SSS Triangle Congruence Proposition, triangles ABD and CBD are congruent.
CW 10: In the previous diagram, why can we conclude that ∠AEB and ∠CEB are right angles?
Answer: These angles correspond in the triangle congruence proved in the previous part (i):
triangles ABE and CBE are congruent. So ∠AEB and ∠CEB are congruent. Also, together they
create the straight angle AEC, so each must measure half of 180 degrees. This means each is a
right angle.
B