Question

example 9: determine the measure of each angle in the diagram below. label the diagram.
m∠1
m∠2
m∠3
m∠4
m∠5
m∠6
(12y - 19)°
(6y + 1)°
(7x + 25)°
(8x - 10)°
(8b - 7)°
(5b + 6)°

Explanation:

Step1: Solve for \( y \)

Angles \( (12y - 19)^\circ \) and \( (6y + 1)^\circ \) are complementary (form a right angle, \( 90^\circ \)). So:

$$ (12y - 19) + (6y + 1) = 90 $$

$$ 18y - 18 = 90 $$

$$ 18y = 108 $$

$$ y = 6 $$

Now find \( m\angle1 \) and \( m\angle2 \):
\( m\angle1 = 12(6) - 19 = 72 - 19 = 53^\circ \)
\( m\angle2 = 6(6) + 1 = 36 + 1 = 37^\circ \) (Check: \( 53 + 37 = 90 \), correct)

Step2: Solve for \( x \)

Angles \( (7x + 25)^\circ \) and \( (8x - 10)^\circ \) are complementary (form a right angle, \( 90^\circ \)):

$$ (7x + 25) + (8x - 10) = 90 $$

$$ 15x + 15 = 90 $$

$$ 15x = 75 $$

$$ x = 5 $$

Now find \( m\angle5 \) and \( m\angle6 \):
\( m\angle6 = 7(5) + 25 = 35 + 25 = 60^\circ \)
\( m\angle5 = 8(5) - 10 = 40 - 10 = 30^\circ \) (Check: \( 60 + 30 = 90 \), correct)

Step3: Solve for \( b \)

Angles \( (8b - 7)^\circ \) and \( (5b + 6)^\circ \) are complementary (form a right angle, \( 90^\circ \)):

$$ (8b - 7) + (5b + 6) = 90 $$

$$ 13b - 1 = 90 $$

$$ 13b = 91 $$

$$ b = 7 $$

Now find \( m\angle3 \), \( m\angle4 \):
\( m\angle3 = 8(7) - 7 = 56 - 7 = 49^\circ \)
\( m\angle4 = 5(7) + 6 = 35 + 6 = 41^\circ \) (Check: \( 49 + 41 = 90 \), correct)

Step4: Verify straight angles (supplementary, \( 180^\circ \))

• For the horizontal line: \( m\angle2 + m\angle3 + m\angle4 + \) right angle? Wait, the vertical and horizontal lines are perpendicular (right angle). Let's check angles around the point. The sum of all angles around a point is \( 360^\circ \). We have two right angles (vertical and horizontal intersections), so total right angles sum to \( 180^\circ \), and the other two right angles (from the two pairs of complementary angles) sum to \( 180^\circ \), total \( 360^\circ \), which checks out.

Answer:

\( m\angle1 = 53^\circ \), \( m\angle2 = 37^\circ \), \( m\angle3 = 49^\circ \), \( m\angle4 = 41^\circ \), \( m\angle5 = 30^\circ \), \( m\angle6 = 60^\circ \)