Question

page four: proofs! 3 pts per blank.
18.)
givens:
ae bisects bd
ac ≅ ce
statements\treasons
ae bisects bd\tgiven
\tdefinition of bisector
∠acb ≅ ∠ecd
\tgiven
ac ≅ ce
\t
△acb ≅ △ecd
\t
19.)
givens:
bc ≅ ad
bc || ad
prove:
ab ≅ dc
statements\treasons
bc ≅ ad\tgiven
bc || ad\tgiven
∠bca ≅ ∠dac
\t
\treflexive property
\t
△bca ≅ △dac\tsas
ab ≅ dc\t
choice bank. not all choices will be used
alternate interior angle theorem
corresponding angle postulate
vertical angle theorem
definition of midpoint
definition of bisector
definition of parallel
definition of perpendicular
all right angles are congruent
all triangles are congruent!
reflexive property
transitive property
given
cpctc
sss
sas
ssa
asa
aas
aaa
hl
ac ≅ ce
ac ≅ ac
ae ≅ bd
ab ≅ de
ab ≅ cd
bc ≅ bc
bc ≅ dc
∠bac ≅ ∠dec
∠bca ≅ ∠dac
∠abc ≅ ∠edc
∠abc and ∠cda are right angles

Explanation:

Step1: Fill in reasons for first proof

1. Given $\overline{AE}$ bisects $\overline{BD}$, reason is "Given".
2. By definition of bisector, we have the relevant property for the bisect - ing line segment, reason is "Definition of bisector".
3. $\angle ACB\cong\angle ECD$ because they are vertical angles, reason is "Vertical Angle Theorem".
4. Given $\overline{AC}\cong\overline{CE}$, reason is "Given".
5. $\triangle ACB\cong\triangle ECD$ by the Side - Angle - Side (SAS) congruence criterion since we have a pair of equal sides ($\overline{AC}\cong\overline{CE}$), the included equal angles ($\angle ACB\cong\angle ECD$) and another pair of equal sides from the bisection property, reason is "SAS".

Step2: Fill in reasons for second proof

1. Given $\overline{BC}\cong\overline{AD}$, reason is "Given".
2. Given $\overline{BC}\parallel\overline{AD}$, reason is "Given".
3. Since $\overline{BC}\parallel\overline{AD}$, the alternate interior angles $\angle BCA\cong\angle DAC$ by the Alternate Interior Angle Theorem.
4. $\overline{AC}\cong\overline{AC}$ by the Reflexive Property.
5. $\triangle BCA\cong\triangle DAC$ by SAS as we have $\overline{BC}\cong\overline{AD}$, $\angle BCA\cong\angle DAC$ and $\overline{AC}\cong\overline{AC}$, reason is "SAS".
6. $\overline{AB}\cong\overline{DC}$ by Corresponding Parts of Congruent Triangles are Congruent (CPCTC).

Answer:

18.

Statements	Reasons
-	Definition of bisector
$\angle ACB\cong\angle ECD$	Vertical Angle Theorem
$\overline{AC}\cong\overline{CE}$	Given
$\triangle ACB\cong\triangle ECD$	SAS

19.

Statements	Reasons
$\overline{BC}\parallel\overline{AD}$	Given
$\angle BCA\cong\angle DAC$	Alternate Interior Angle Theorem
$\overline{AC}\cong\overline{AC}$	Reflexive Property
$\triangle BCA\cong\triangle DAC$	SAS
$\overline{AB}\cong\overline{DC}$	CPCTC