I see the image you've uploaded. I'll work through each problem step by step.

Problem 21

• Given: In the diagram, $\angle B = 35^\circ$, $\angle C = 75^\circ$.
• To Find: The degree measure of $\angle A$.

Solution: The sum of the angles in any triangle is $180^\circ$. Therefore, we have:

$\angle A + \angle B + \angle C = 180^\circ$

Substituting the given values:

$\angle A + 35^\circ + 75^\circ = 180^\circ$

$\angle A + 110^\circ = 180^\circ$

$\angle A = 180^\circ - 110^\circ = 70^\circ$

Answer: $\angle A = 70^\circ$

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Problem 22

• Given: $\overline{AB}$ and $\overline{CD}$ are two lines intersecting at point $O$. $\angle AOD = 72^\circ$.
• To Find: The measure of $\angle BOC$.

Solution: In a pair of intersecting lines, opposite angles (also called vertically opposite angles) are equal. Thus:

$\angle BOC = \angle AOD = 72^\circ$

Answer: $\angle BOC = 72^\circ$

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Problem 23

• Given: A diagram of a pentagon (5-sided polygon) where all sides and angles are not equal, with one angle labeled as $145^\circ$, and the other internal angles to be found.
• To Find: The sum of the interior angles of the pentagon.

Solution: The formula for finding the sum of the interior angles of a polygon is:

$\text{Sum of interior angles} = (n-2) \times 180^\circ$

For a pentagon ($n = 5$):

$\text{Sum of interior angles} = (5-2) \times 180^\circ = 3 \times 180^\circ = 540^\circ$

Answer: The sum of the interior angles of the pentagon is $540^\circ$.

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Problem 24

• Given: $\overline{AB} \parallel \overline{CD}$ and $\overline{EF}$ is a transversal cutting them at points $G$ and $H$ respectively. $\angle AGH = 110^\circ$ and $\angle GHE = x^\circ$.
• To Find: The measure of $\angle GHE$.

Solution: Since $\overline{AB} \parallel \overline{CD}$ and $\overline{EF}$ is a transversal, the angles $\angle AGH$ and $\angle GHE$ are supplementary because they form a pair of co-interior angles.

$\angle AGH + \angle GHE = 180^\circ$

Substituting the given value:

$110^\circ + x^\circ = 180^\circ$

$x^\circ = 180^\circ - 110^\circ = 70^\circ$

Answer: $x = 70^\circ$

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Problem 25

• Given: The ratio of the angles in a quadrilateral is $2:3:5:8$.
• To Find: The degree measure of each angle.

Solution: The sum of the angles in a quadrilateral is $360^\circ$.

Let the angles be $2x$, $3x$, $5x$, and $8x$.

$2x + 3x + 5x + 8x = 360^\circ$

$18x = 360^\circ$

$x = \frac{360^\circ}{18} = 20^\circ$

Thus, the angles are:

• $2x = 2 \times 20^\circ = 40^\circ$
• $3x = 3 \times 20^\circ = 60^\circ$
• $5x = 5 \times 20^\circ = 100^\circ$
• ( 8x = 8 \times 20^\circ =