What is the missing reason in steps 7 and 8 of the proof?
given: ∆abc with ab = c, bc = a, ac = b
prove: = =
statement reason
1. draw a line through b that is perpendicular to and label the point of intersection with as d. construction
2. in ∆abd, bd = c sin a definition of sine in right triangle
3. in ∆cbd, bd = a sin c definition of sine in right triangle
4. c sin a = a sin c substitution property of equality
5. = dividing throughout by sin a
sin c
6. draw a line through a that is perpendicular to and label the point of intersection with as e. construction
7. in ∆bae, ae = c sin b
8. in ∆cae, ae = b sin c
9. c sin b = b sin c substitution property of equality
10. = dividing throughout by sin b
sin c
11. = = transitive property of equality
definition of cosine in right triangle
definition of sine in right triangle
definition of tangent in right triangle
corresponding sides of congruent triangles are equal.
definition of equilateral triangle

In ∆abd, bd = c sin a
definition of cosine in right triangle :   from a  triangle . given any angle q (0 £ q £ 90°), we can find the sine or  cosine   of that angle by constructing a  right triangle   with one vertex of angle q. the sine is equal to the length of the side opposite to q, divided by the length of the  triangle's   hypotenuse.

definition of sine in right triangle:   the  sine   function, along with cosine and tangent, is one of the three most common trigonometric functions. in any  right triangle , the  sine   of an angle x is the length of the opposite side (o) divided by the length of the hypotenuse (h)

definition of tangent in right triangle:   the  tangent   function, along with sine and cosine, is one of the three most common trigonometric functions. in any  right triangle , the  tangent   of an angle is the length of the opposite side (o) divided by the length of the adjacent side (a).

if three sides of one triangle are congruent to three sides of another triangle, the triangles are congruent

in geometry, an  equilateral triangle   is a  triangle   in which all three sides are equal. in the familiar euclidean geometry,equilateral triangles   are also equiangular; that is, all three internal angles are also congruent to each other and are each 60°.

the correct answer is:

definition of sine in right triangle

explanation:

the ratio for sine is opposite/hypotenuse. this means that for the sine of angle b, we want the side opposite b, which is ae, over the hypotenuse, which is c:

sin b = ae/c

multiply both sides by c:

c sin b = (ae/c)*c

c sin b = ae

this is by definition of sine.

this also applies to step 8.

With a continuous thought process.