Question

3. Angle 1, angle 2, and angle 3 are adjacent angles, with angle 1 supplementary to angle 2 and angle 2 complementary
to angle 3. If m(angle)1 = (8x + 3) and m(angle)2 = (5x-18), find m(angle)3.
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Answer 1

Answer:

m∠3 = 33°

Step-by-step explanation:

As angles 1, 2 and 3 are adjacent angles, angles 1 and 2 share a common vertex and a common side but do not overlap, as do angles 2 and 3.

Supplementary angles are two angles whose measures sum to 180°. Therefore, if angles 1 and 2 are supplementary, then:

$\sf m\angle 1 + m\angle 2 = 180^{\circ}$

Complementary angles are two angles whose measures sum to 90°. Therefore, if angles 3 and 3 are complementary , then:

$\sf m\angle 2 + m\angle 3 = 90^{\circ}$

Given that m∠1 = (8x + 3)° and m∠2 = (5x - 18)°, to find the measure of angle 3, we first need to find the value of x by substituting the angle expressions into m∠1 + m∠2 = 180° and solving for x:

$\sf m\angle 1 + m\angle 2 = 180^{\circ} \\\\ (8x + 3)^{\circ} + (5x - 18)^{\circ} = 180^{\circ} \\\\ 8x + 3 + 5x - 18 = 180 \\\\ 13x -15 = 180 \\\\ 13x = 195 \\\\ x = 15$

Now, substitute x = 15 into the expression for angle 2:

$\sf m\angle 2 = (5(15) - 18)^{\circ} \\\\ m\angle 2 = (75 - 18)^{\circ} \\\\ m\angle 2 = 57^{\circ}$

Finally, substitute m∠2 = 57° into m∠2 + m∠3 = 90° to find the measure of angle 3:

$\sf m\angle 2 + m\angle 3 = 90^{\circ} \\\\ 57^{\circ} + m\angle 3 = 90^{\circ} \\\\ m\angle 3 = 90^{\circ} - 57^{\circ} \\\\ m\angle 3 = 33^{\circ}$

Therefore, the measure of angle 3 is:

$\LARGE\boxed{\boxed{m\angle 3 = 33^{\circ}}}$