Question

d)
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$$\begin{array}{|c|c|} \\hline \\text{angle} & \\text{reason} \\\\ \\hline & \\\\ \\hline & \\\\ \\hline & \\\\ \\hline \\end{array}$$

Explanation:

Find the measure of Angle 1

Using the Isosceles Triangle Theorem and Triangle Angle Sum Theorem
The upper triangle is a right isosceles triangle.
Let the base angles be \(x\).

$$ 90^\circ + 2x = 180^\circ \implies 2x = 90^\circ \implies x = 45^\circ $$

Thus, the measure of Angle 1 is:

$$ m\angle 1 = 45^\circ $$

Find the measure of Angle 2

Using the Vertical Angles knowledge point
Angle 2 and Angle 1 are vertical angles formed by intersecting lines.

$$ m\angle 2 = m\angle 1 = 45^\circ $$

Find the measure of Angle 3

Using the Triangle Angle Sum Theorem
In the lower-left triangle, the sum of the interior angles is \(180^\circ\).

$$ m\angle 3 + m\angle 2 + 95^\circ = 180^\circ $$

Substitute \(m\angle 2 = 45^\circ\):

$$ m\angle 3 + 45^\circ + 95^\circ = 180^\circ \implies m\angle 3 + 140^\circ = 180^\circ \implies m\angle 3 = 40^\circ $$

Answer:

Angle	Reason
\(m\angle 2 = 45^\circ\)	Vertical Angles Theorem: \(m\angle 2 = m\angle 1\)
\(m\angle 3 = 40^\circ\)	Triangle Angle Sum Theorem: \(180^\circ - (95^\circ + 45^\circ) = 40^\circ\)