Question

Given: Triangle A B C is a right triangle with altitude segment B D

Prove: triangle B D A is similar to triangle C D B




Question
Move options to the table to complete the proof.

Response area with 9 blank spaces
,begin bold,Statement,end bold,

,begin bold,Reason,end bold,

triangle A B C is a right triangle with altitude segment B D

Given

the measure of angle C equals 90 ° minus
Blank space 1 empty

Acute angles of right triangles are
Blank space 2 empty

Blank space 3 empty
equals 90 ° minus
Blank space 4 empty

Acute angles of right triangles are
Blank space 5 empty

the measure of angle C equals
Blank space 6 empty

Substitution

Blank space 7 empty
equals
Blank space 8 empty
equals 90 °

Definition of
Blank space 9 empty

triangle B D A is similar to triangle C D B

A A similarity


Answer options with 11 options.

Answer 1

Fillings:

1. Blank 1 = m∠A
2. Blank 2 = complementary
3. Blank 3 = m∠CBD
4. Blank 4 = m∠C
5. Blank 5 = complementary
6. Blank 6 = m∠A
7. Blank 7 = m∠BDA
8. Blank 8 = m∠CDB
9. Blank 9 = perpendicular

Explanation (matching the proof lines):

- In right triangle ABC the acute angles are complementary, so m∠C = 90° − m∠A. (Blanks 1,2)
- In right triangle CDB the acute angles are complementary, so m∠CBD = 90° − m∠C. (Blanks 3–5)
- By substitution m∠CBD = 90° − (90° − m∠A) = m∠A, so m∠CBD = m∠A. (Blank 6)
- BD is an altitude to AC, so BD ⟂ AC; therefore m∠BDA = m∠CDB = 90°. (Blanks 7–9)
- Triangles BDA and CDB have a right angle and a second equal acute angle, so by AA they are similar.