Question

Use ΔABC shown below to answer the question that follows:

Triangle ABC with segment AD drawn from vertex A and intersecting side BC.

Which of the following must be given to prove that ΔABC is similar to ΔDBA?


a. Segment AD is an altitude of ΔABC.
b. Segment CB is a hypotenuse.
c. Segment CA is shorter than segment BA.
d. Angle C is congruent to itself.

Answer 1

this answer is a (a) segment AD is an altitude at angle ABC

Answer 2

The correct options are a and b.

Explanation:

It is given that triangle ABC with segment AD drawn from vertex A and intersecting side BC.

Two triangle are called similar triangle if their corresponding sides are proportional or the corresponding interior angle are same.

To prove ΔABC and ΔDBA are similar, we have to prove that corresponding interior angles of both triangle as same.

If segment AD is an altitude of ΔABC, then angle ADB is a right angle.

`\angle BDA=90^(\circ)`

The opposite angle of hypotenuse is right angle. If segment CB is a hypotenuse, then angle ABC is a right angle.

`\angle BAC=90^(\circ)`

In triangle ΔABC and ΔDBA

`\angle ABC\cong \angle DBA` (Reflexive property)

`\angle BAC\cong \angle BDA` (Right angles)

By AA rule of similarity ΔABC and ΔDBA are similar.

Therefore correct options are a and b.