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.
[tex]\dfrac{AB}{ED} =\dfrac{BC}{DC}=\dfrac{AC}{EC}\textrm{corresponding sides of similar triangles are in proportion}\\[/tex]
On Substituting the given values we get
[tex]\dfrac{8}{12} =\dfrac{5}{y}=\dfrac{x}{15}[/tex]
Therefore,
[tex]\dfrac{8}{12} =\dfrac{5}{y}\\\\y=\dfrac{60}{8}=7.5\\\\y=7.5\ unit[/tex]
[tex]\dfrac{8}{12} =\dfrac{x}{15}\\\\\therefore x=\dfrac{120}{12}\\\\\x=10[/tex]
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.
