Question

Based on the information given say whether or not △ABC∼△FED. Explain your reasoning.

m∠A=m∠B, m∠C=m∠A+30°, m∠E=m∠F=x, m∠D=2x−20°.

Answer 1

Yes, △ABC ∼ △FED by AA postulate.

Explanation:

Given:

Two triangles ABC and FED.

m∠A = m∠B

m∠C = m∠A + 30°

m∠E = m∠F =
`x`

m∠D =
`2x-20`°.

Now, let m∠A = m∠B =
`y`

So, m∠C = m∠A + 30° =
`y+30`

Now, sum of all interior angles of a triangle is 180°. Therefore,

m∠A + m∠B + m∠C = 180

`y+y+y+30=180\\3y=180-30\\3y=150\\y=(150)/(3)=50`

Therefore, m∠A = 50°, m∠B = 50° and m∠C = m∠A + 30° = 50 + 30 = 80°.

Now, consider triangle FED,

m∠D+ m∠E + m∠F = 180

`2x-20+x+x=180\\4x=180+20\\4x=200\\x=(200)/(4)=50`

Therefore, m∠F = 50°

m∠E = 50° and

m∠D =
`2x-20=2(50)-20=100-20=80\°`

So, both the triangles have congruent corresponding angle measures.

m∠A = m∠F = 50°

m∠B = m∠E = 50°

m∠C = m∠D = 80°

Therefore, the two triangles are similar by AA postulate.