Question

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Answer 1

Answer:

Therefore

1. Vertical Opposite Angle Theorem is used to prove that the triangles are similar.

2. Angle-Angle similarity statement is used.

3. x = 10 unit and y = 7.5 unit.

Step-by-step explanation:

Given:

∠A ≅ ∠E

AB = 8 , DE = 12 , BC = 5 , CE = 15

To Prove:

Δ ABC ~ ΔEDC

To Find :

x = ?

y = ?

Proof:

Vertical Angle Theorem:  

The angles opposite each other when two lines cross. They are always equal.

Here vertical opposite angles are,

∴ ∠ACB ≅ ∠ECD

Angle-Angle Similarity :

If in Two triangle Two corresponding angles are congruent then the Triangles are Similar by Angle-Angle Similarity statement.

Now,

In Δ ABC and Δ EDC  

∠A ≅ ∠E …………..{ Given }  

∠ACB ≅ ∠ECD ……….....{Vertical Opposite Angle Theorem}  

Δ ABC ~ Δ EDC ….{Angle-Angle Similarity test}  

If two triangles are similar then their sides are in proportion.  

$\dfrac{AB}{ED} =\dfrac{BC}{DC}=\dfrac{AC}{EC}\textrm{corresponding sides of similar triangles are in proportion}$  

On Substituting the given values we get

$\dfrac{8}{12} =\dfrac{5}{y}=\dfrac{x}{15}$

Therefore,

$\dfrac{8}{12} =\dfrac{5}{y}\\\\y=\dfrac{60}{8}=7.5\\\\y=7.5\ unit$

$\dfrac{8}{12} =\dfrac{x}{15}\\\\\therefore x=\dfrac{120}{12}\\\\\x=10$

Therefore

1. Vertical Opposite Angle Theorem is used to prove that the triangles are similar.

2. Angle-Angle similarity statement is used.

3. x = 10 unit and y = 7.5 unit.