Question

1. find the value of x.

d (6x + 11)°
e (10x + 3)°
h (13x - 5)°
g
m∠e =
x =

Explanation:

Step1: Use the property of similar triangles

Since the triangles are similar (by AA similarity, as vertical angles are equal and right angles are equal), the corresponding angles are equal. So, \(10x + 3=13x - 5\) (corresponding angles).

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This is wrong. Let's use the angle - sum property of a triangle.

In \(\triangle DFG\) and \(\triangle EFG\) (by AA similarity, \(\angle D=\angle E\) (vertical angles' property in a way of triangle similarity). Also, in a right - triangle, the sum of angles is \(180^{\circ}\).

For \(\triangle DFG\): \(\angle D=(6x + 11)^{\circ}\), and for \(\triangle EFG\): \(\angle E=(10x + 3)^{\circ}\)

Since the two triangles (the two right - triangles) are similar (by AA, as \(\angle F\) is common and both have a right - angle), we can use the fact that in a right - triangle \(\angle D+\angle H = 90^{\circ}\) (complementary angles in a right - triangle)

\((6x + 11)+(13x-5)=90\)

Step2: Solve the equation for \(x\)

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Step3: Find \(m\angle E\)

Substitute \(x = 4\) into \(\angle E=(10x + 3)^{\circ}\)
\(m\angle E=10\times4+3=43^{\circ}\)

Answer:

\(x = 4\), \(m\angle E=43^{\circ}\)