Which similarity statement is true for a triangle with angles R

Question

R
, S
S
, and T
T
(where m∠R=29°
m
∠
R
=
29
°
and m∠T=71°
m
∠
T
=
71
°
) and another triangle with angles C
C
, D
D
, and E
E
(where m∠D=71°
m
∠
D
=
71
°
and m∠C=80°
m
∠
C
=
80
°
)?(1 point)
Responses

△RST∼△ECD
△
R
S
T
∼
△
E
C
D
triangle upper R upper S upper T tilde triangle upper E upper C upper D

△RST≅△ECD
△
R
S
T
≅
△
E
C
D
triangle upper R upper S upper T congruent to triangle upper E upper C upper D

△RST≅△CED
△
R
S
T
≅
△
C
E
D
triangle upper R upper S upper T congruent to triangle upper C upper E upper D

△RST∼△CED

Answer 1

To determine which similarity statement is true, we need to analyze the angle measures of both triangles provided and apply the Angle-Angle (AA) criterion for triangle similarity.

1. For triangle RST:

   • m∠R = 29°
   • m∠T = 71°
   • To find m∠S, we use the fact that the sum of angles in a triangle is 180°: \[ m∠S = 180° - m∠R - m∠T = 180° - 29° - 71° = 80° \]
   • Therefore, angles in triangle RST are:
     • m∠R = 29°
     • m∠S = 80°
     • m∠T = 71°

2. For triangle CDE:

   • m∠D = 71°
   • m∠C = 80°
   • To find m∠E, we can again use the sum of angles in a triangle: \[ m∠E = 180° - m∠C - m∠D = 180° - 80° - 71° = 29° \]
   • Therefore, angles in triangle CDE are:
     • m∠C = 80°
     • m∠D = 71°
     • m∠E = 29°

Now, we can compare the angles of both triangles:

• Triangle RST has angles 29°, 80°, and 71°.
• Triangle CDE has angles 29°, 80°, and 71°.

Since both triangles have the same angle measures, we can conclude that they are similar by AA similarity criterion. The similarity statement that corresponds to this is:

\[ \triangle RST \sim \triangle CDE \]

Among the options provided, the correct choice is:

△RST∼△CED