Given:
a.) ∠9 = 105°
b.) ∠2 = 63°
Step 1: Determine the measure of ∠10.
∠9 and ∠10 are Supplementary Angles. This means that the sum of the two angles is equal to 180°.
From this, we generate the following equation:
[tex]\text{ }\angle9\text{ + }\angle10=180^{\circ}[/tex]Let's then now proceed to find out the measure of ∠10.
[tex]\text{ }\angle9\text{ + }\angle10=180^{\circ}[/tex][tex]\text{ }105^{\circ}\text{ + }\angle10=180^{\circ}[/tex][tex]\angle10=180^{\circ}\text{ - }105^{\circ}[/tex][tex]\angle10=75^{\circ}[/tex]Step 2: Determine the measure of ∠3.
∠10 and ∠3 are Alternate Exterior Angles. Under this, the two angles must be congruent.
[tex]\text{ }\angle3\text{ = }\angle10[/tex]Therefore,
[tex]\text{ }\angle3\text{ = }\angle10[/tex][tex]\text{ }\angle3=75^{\circ}[/tex]Step 3: Determine the measure of ∠1.
∠1, ∠2 and ∠3 are also Supplementary Angles. This means that the sum of the three angles is equal to 180°.
Thus, we generate the equation below:
[tex]\text{ }\angle1\text{ + }\angle2\text{ + }\angle3=180^{\circ}[/tex]Let's now find the measure of ∠1,
[tex]\text{ }\angle1\text{ + }\angle2\text{ + }\angle3=180^{\circ}[/tex][tex]\text{ }\angle1\text{ + }63^{\circ}\text{ + }75^{\circ}=180^{\circ}[/tex][tex]\text{ }\angle1\text{ + }138^{\circ}=180^{\circ}[/tex][tex]\text{ }\angle1\text{ }=180^{\circ}\text{ - }138^{\circ}[/tex][tex]\text{ }\angle1\text{ }=42^{\circ}[/tex]Therefore, the measure of ∠1 is 42°.