Answer:
See explanation
Step-by-step explanation:
a1) Given
[tex]\angle A\cong \angle C\\ \\\angle ABD\cong \angle CDB[/tex]
Statement Reason
1. [tex]\angle A\cong \angle C[/tex] - Given
2. [tex]\angle ABD\cong \angle CDB[/tex] - Given
3. [tex]\overline{BD}\cong \overline{DB}[/tex] - Reflexive property
4. [tex]\triangle ABD\cong \triangle CDB[/tex] - AAS postulate
5. [tex]\overline {AB}\cong \overline{CD}[/tex] - Congruent triangles have congruent corresponding parts
6. [tex]AB=CD[/tex] - Definition of congruent segments
a2) Given
[tex]\overline{H O}\parallel \overline{E N}\\ \\H W=N W[/tex]
Statement Reason
1. [tex]H W=W N[/tex] - Given
2. [tex]\overline{H O}\parallel \overline{E N}[/tex] - Given
3. [tex]\angle O H W\cong \angle E N W[/tex] - Alternate interior angles theorem (two parallel lines H O and E N are cut by transversal H N)
4. [tex]\angle H W O\cong \angle N W E[/tex] - Vertical angles theorem
5. [tex]\triangle H O W\cong \triangle N E W[/tex] - ASA postulate
b) Given
[tex]PO=PR\\ \\OS=RS\\ \\\angle O\cong \angle R\\ \\m\angle P=90^{\circ}[/tex]
Statement Reason
1. [tex]m\angle P=90^{\circ}[/tex] - Given
2. [tex]\triangle OPE, \triangle RPN[/tex] are right triangles - Definition of right triangles
3. [tex]\angle O\cong \angle R[/tex] - Given
4. [tex]PO=PR[/tex] - Given
5. [tex]\triangle POE\cong \triangle PRN[/tex] - HA postulate
6. [tex]OS=RS[/tex] - Given
7. [tex]\angle O\cong \angle R[/tex] - Given
8. [tex]\angle OSN\cong \angle SRE[/tex] - Vertical angles theorem
9. [tex]\triangle SON\cong \triangle SRE[/tex] - ASA postulate
c1) Given
[tex]\overline{SA}\parallel \overline{NE}\\ \\\overline{SE}\parallel \overline{NA}[/tex]
Statement Reason
1. [tex]\overline{SA}\parallel \overline{NE}[/tex] - Given
2. [tex]\angle ENS\cong \angle ASN[/tex] - Alternate interior angles theorem (two parallel lines SA and NE are cut by transversal SN)
3. [tex]\overline{SE}\parallel \overline{NA}[/tex] - Given
4. [tex]\angle ESN\cong \angle ANS[/tex] - Alternate interior angles theorem (two parallel lines SE and NA are cut by transversal SN)
5. [tex]\overline{SN}\cong \overline{NS}[/tex] - Reflexive property
6. [tex]SN=NS[/tex] - Definition of congruent segments
7. [tex]\triangle ENS\cong \triangle ASN[/tex] - ASA postulate
8. [tex]\overline{SA}\cong \overline {NE}[/tex] - Congruent triangles have congruent corresponding parts
9. [tex]SA=NE[/tex] - Definition of congruent segments
c2) Given
[tex]\angle BTO\cong \angle IWE\\ \\\overline{WI}\cong \overline{BT}\\ \\\overline{OW}\cong \overline{ET}[/tex]
Statement Reason
1. [tex]\angle BTO\cong \angle IWE[/tex] - Given
2. [tex]\overline{WI}\cong \overline{BT}[/tex] - Given
3. [tex]\overline{OW}\cong \overline{ET}[/tex] - Given
4. [tex]\overline{WT}\cong \overline{TW}[/tex] - Reflexive property
5. [tex]WI=BT[/tex] - Definition of congruent segments
6. [tex]OW=ET[/tex] - Definition of congruent segments
7. [tex]WT=TW[/tex] - Definition of congruent segments
8. [tex]OT=OW+WT[/tex] - Segment addition postulate
9. [tex]WE=WT+TE[/tex] - Segment addition postulate
10. [tex]OW+WT=ET+WT[/tex] - Substitution property
11. [tex]OT=WE[/tex] - Substitution property
12. [tex]\triangle BOT\cong \triangle IEW[/tex] - SAS postulate
