Question

orange

1. solve for x.

57°
7x + 5
69°
purple

1. solve for x.

8x + 1
88°
51°
green

1. find the missing angle measure.

75°
?
black

1. find the missing angle measure.

113°
?
brown

1. find the missing angle measure.

68°
?
orange

1. find the missing angle measure.

?
123°
brown
purple
purple

Explanation:

Step1: Use the triangle angle sum theorem

The sum of angles in a triangle is \(180^{\circ}\). For the orange problem (4), we have \((7x + 5)+57 + 69=180\).
Simplify the left - hand side: \(7x+5+57 + 69=7x+(5 + 57+69)=7x + 131\).
So, \(7x+131 = 180\).
Subtract 131 from both sides: \(7x=180 - 131\), \(7x = 49\).
Divide both sides by 7: \(x=\frac{49}{7}=7\).

Step2: Use the triangle angle sum theorem for purple problem (5)

The sum of angles in a triangle is \(180^{\circ}\). So, \((8x + 1)+88+51 = 180\).
Simplify the left - hand side: \(8x+1+88 + 51=8x+(1 + 88+51)=8x + 140\).
Then, \(8x+140 = 180\).
Subtract 140 from both sides: \(8x=180 - 140\), \(8x = 40\).
Divide both sides by 8: \(x=\frac{40}{8}=5\).

Step3: Use the alternate - interior angles for green problem (6)

When two parallel lines are cut by a transversal, alternate - interior angles are equal. The angle is \(75^{\circ}\).

Step4: Use the corresponding angles for black problem (7)

When two parallel lines are cut by a transversal, corresponding angles are equal. The angle is \(113^{\circ}\).

Step5: Use the corresponding angles for brown problem (8)

When two parallel lines are cut by a transversal, corresponding angles are equal. The angle is \(68^{\circ}\).

Step6: Use the supplementary angles for orange problem (9)

If two angles are on the same side of a transversal and between two parallel lines, they are supplementary. Let the missing angle be \(y\). Then \(y + 123=180\), \(y=180 - 123=57^{\circ}\).

Answer:

1. \(x = 7\)
2. \(x = 5\)
3. \(75^{\circ}\)
4. \(113^{\circ}\)
5. \(68^{\circ}\)
6. \(57^{\circ}\)