Question

given: ( ad perp ba ); ( cd perp bc ); ( ad cong cd ) prove: ( overline{bd} ) bisects ( angle abc ) statements 1. ( overline{ad} perp overline{ba} ); ( overline{cd} perp overline{bc} ); ( overline{ad} cong overline{cd} ) 2. ( angle bad ) and ( angle bcd ) are right ( angle ) s 3. ( \triangle bad ) and ( \triangle bcd ) are right ( \triangle ) s 4. 5. 6. ( angle abd cong angle cbd ) 7. ( overline{bd} ) bisects ( angle abc ) reasons 1. given 2. def. of 3. def. of 4. reflex. prop. of ( cong ) 5. hl 6. 7. def. of ( angle ) bisector a. ( perp ) b. right ( angle ) c. ( angle bad cong angle bcd ) d. cpctc e. right ( \triangle ) f. ( \triangle bad cong \triangle bcd ) g. ( overline{bd} cong overline{bd} )

Explanation:

Step1: Determine the reason for statement 2

Since \(AD\perp BA\) and \(CD\perp BC\), by the definition of perpendicular (\(\perp\)), \(\angle BAD\) and \(\angle BCD\) are right angles. So, the reason for statement 2 is "Def. of \(\perp\)" (option a).

Step2: Determine the reason for statement 3

A triangle with a right angle is a right triangle. Since \(\angle BAD\) and \(\angle BCD\) are right angles (from statement 2), \(\triangle BAD\) and \(\triangle BCD\) are right triangles. So, the reason for statement 3 is "Def. of right \(\triangle\)" (option e).

Step3: Determine statement 4

By the Reflexive Property of \(\cong\), \(\overline{BD}\cong\overline{BD}\) (option g).

Step4: Determine statement 5

We have \(AD\cong CD\) (given), \(\overline{BD}\cong\overline{BD}\) (from statement 4), and \(\triangle BAD\) and \(\triangle BCD\) are right triangles (from statement 3). By the Hypotenuse - Leg (HL) congruence criterion, \(\triangle BAD\cong\triangle BCD\) (option f).

Step5: Determine the reason for statement 6

Since \(\triangle BAD\cong\triangle BCD\), by Corresponding Parts of Congruent Triangles are Congruent (CPCTC), \(\angle ABD\cong\angle CBD\). So, the reason for statement 6 is "CPCTC" (option d).

Answer:

1. a. \(\perp\)
2. e. right \(\triangle\)
3. g. \(\overline{BD}\cong\overline{BD}\)
4. f. \(\triangle BAD\cong\triangle BCD\)
5. d. CPCTC