m∠1 = 30° (by Vertical angle theorem)
m∠A = 80° (by Triangle sum theorem)
m∠D = 80° (by Triangle sum theorem)
The value of x is 7.5 and y is 9.
Solution:
∠ACB and ∠DCE are vertically opposite angles.
Vertical angle theorem:
If two lines are intersecting, then vertically opposite angles are congruent.
⇒ m∠DCE = m∠ACB
⇒ m∠1 = 30° (by Vertical angle theorem)
In triangle ACD,
Triangle sum property:
Sum of the interior angles of the triangle = 180°
⇒ m∠A + m∠C + m∠B = 180°
⇒ m∠A + 30° + 70° = 180°
⇒ m∠A + 100° = 180°
⇒ m∠A = 100° – 180°
⇒ m∠A = 80° (by Triangle sum theorem)
Similarly, m∠D = 80° (by Triangle sum theorem)
In ΔACD and ΔDCE,
All the angles are congruent, so ΔACD and ΔDCE are similar triangles.
In similar triangle corresponding sides are in the same ratio.
[tex]$\frac{9}{12}=\frac{x}{10}[/tex]
Do cross multiplication.
90 = 12x
7.5 = x
Now, to find y:
[tex]$\frac{9}{12}=\frac{6}{y}[/tex]
Do cross multiplication.
9y = 72
Divide by 9, we get
y = 8
Hence the value of x is 7.5 and y is 9.
