Question

1. find the measure of each lettered angle in the diagram below.

Explanation:

Step1: Solve for angle $a$

The angle $a$ forms a linear pair with $105^\circ$, so:
$a = 180^\circ - 105^\circ = 75^\circ$

Step2: Solve for angle $b$

Triangle with $a$, $b$, right angle: sum to $180^\circ$.
$b = 180^\circ - 90^\circ - a = 180^\circ - 90^\circ - 75^\circ = 15^\circ$

Step3: Solve for angle $c$

Vertical angle to $b$, so:
$c = b = 15^\circ$

Step4: Solve for angle $d$

Congruent to $a$ (marked segments), so:
$d = a = 75^\circ$

Step5: Solve for angle $e$

Isosceles triangle base angle:
$e = \frac{180^\circ - 105^\circ}{2} = 37.5^\circ$

Step6: Solve for angle $f$

Linear pair with $138^\circ$:
$f = 180^\circ - 138^\circ = 42^\circ$

Step7: Solve for angle $g$

Triangle with $f$, right angle:
$g = 180^\circ - 90^\circ - f = 180^\circ - 90^\circ - 42^\circ = 48^\circ$

Step8: Solve for angle $h$

Congruent to $f$ (marked segments), so:
$h = f = 42^\circ$

Answer:

$a=75^\circ$
$b=15^\circ$
$c=15^\circ$
$d=75^\circ$
$e=37.5^\circ$
$f=42^\circ$
$g=48^\circ$
$h=42^\circ$