Practice worksheet for calculating interior and exterior angles of triangles.

Problem Analysis:

The task involves solving for the measures of interior and exterior angles of triangles, as well as applying properties of quadrilaterals. Let's break it down step by step.

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#### Part 1: Finding the Measure of Each Angle

We are given several diagrams with labeled angles and need to find their measures using geometric properties.

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##### Question 1:
- Diagram: Triangle \( \triangle ABC \)
- Given: \( m\angle B = ? \)

Since no specific angle measures are provided in the diagram, we cannot solve this without additional information. Let’s assume this is a generic triangle where the sum of interior angles is \( 180^\circ \).

Solution:
If more details were provided (e.g., other angles or side lengths), we could use the triangle angle sum property (\( m\angle A + m\angle B + m\angle C = 180^\circ \)) to solve for \( m\angle B \). However, with the current information, we cannot determine \( m\angle B \).

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##### Question 2:
- Diagram: Triangle \( \triangle DEF \)
- Given: \( m\angle F = ? \)

From the diagram, we see that one angle is labeled as \( 20^\circ \) and another as \( 60^\circ \). Using the triangle angle sum property:

\[
m\angle D + m\angle E + m\angle F = 180^\circ
\]

Substitute the known values:

\[
20^\circ + 60^\circ + m\angle F = 180^\circ
\]

Solve for \( m\angle F \):

\[
80^\circ + m\angle F = 180^\circ
\]
\[
m\angle F = 180^\circ - 80^\circ
\]
\[
m\angle F = 100^\circ
\]

Answer: \( m\angle F = 100^\circ \)

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##### Question 3:
- Diagram: Triangle \( \triangle GHI \)
- Given: \( m\angle H = ? \)

From the diagram, we see that one angle is labeled as \( 120^\circ \). Since the sum of the interior angles of a triangle is \( 180^\circ \), the remaining two angles must sum to:

\[
180^\circ - 120^\circ = 60^\circ
\]

Without additional information about the other angles, we cannot determine \( m\angle H \) uniquely. If the triangle is isosceles or equilateral, further assumptions could be made, but we lack such details.

Solution:
More information is needed to solve for \( m\angle H \).

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##### Question 4:
- Diagram: Triangle \( \triangle JKL \)
- Given: \( m\angle L = ? \)

From the diagram, we see that one angle is labeled as \( 60^\circ \). Using the triangle angle sum property:

\[
m\angle J + m\angle K + m\angle L = 180^\circ
\]

Substitute the known value:

\[
60^\circ + m\angle K + m\angle L = 180^\circ
\]

Without the measure of \( m\angle K \), we cannot solve for \( m\angle L \) uniquely. Additional information is required.

Solution:
More information is needed to solve for \( m\angle L \).

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##### Question 5:
- Diagram: Quadrilateral \( PQRS \)
- Given: \( m\angle P = ? \)

From the diagram, we see that three angles are labeled: \( 90^\circ \), \( 70^\circ \), and \( 120^\circ \). The sum of the interior angles of a quadrilateral is \( 360^\circ \). Using this property:

\[
m\angle P + 90^\circ + 70^\circ + 120^\circ = 360^\circ
\]

Solve for \( m\angle P \):

\[
m\angle P + 280^\circ = 360^\circ
\]
\[
m\angle P = 360^\circ - 280^\circ
\]
\[
m\angle P = 80^\circ
\]

Answer: \( m\angle P = 80^\circ \)

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##### Question 6:
- Diagram: Triangle \( UVWX \)
- Given: \( m\angle X = ? \)

From the diagram, we see that one angle is labeled as \( 60^\circ \) and another as \( 70^\circ \). Using the triangle angle sum property:

\[
m\angle U + m\angle V + m\angle X = 180^\circ
\]

Substitute the known values:

\[
60^\circ + 70^\circ + m\angle X = 180^\circ
\]

Solve for \( m\angle X \):

\[
130^\circ + m\angle X = 180^\circ
\]
\[
m\angle X = 180^\circ - 130^\circ
\]
\[
m\angle X = 50^\circ
\]

Answer: \( m\angle X = 50^\circ \)

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#### Part 2: Using Knowledge of Interior and Exterior Angles

##### Question 7:
- The sum of the angle measures of a quadrilateral is \( ? \).

The sum of the interior angles of any quadrilateral is always \( 360^\circ \).

Answer: \( 360^\circ \)

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##### Question 8:
- The acute angles of a right triangle are complementary.

In a right triangle, one angle is \( 90^\circ \). The sum of the other two angles must be \( 90^\circ \) because the total sum of the angles in a triangle is \( 180^\circ \). Therefore, the two acute angles are complementary.

Answer: complementary

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##### Question 9:
- The measure of an exterior angle of a triangle is equal to the sum of the measures of its remote interior angles.

This is a fundamental property of triangles. An exterior angle is equal to the sum of the two non-adjacent (remote) interior angles.

Answer: remote interior angles

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##### Question 10:
- The angle measures of a triangle are \( 3x \), \( 4x \), and \( 5x \). Tell the measure of each angle.

The sum of the interior angles of a triangle is \( 180^\circ \). Therefore:

\[
3x + 4x + 5x = 180^\circ
\]

Simplify:

\[
12x = 180^\circ
\]

Solve for \( x \):

\[
x = \frac{180^\circ}{12} = 15^\circ
\]

Now find each angle:

\[
3x = 3 \times 15^\circ = 45^\circ
\]
\[
4x = 4 \times 15^\circ = 60^\circ
\]
\[
5x = 5 \times 15^\circ = 75^\circ
\]

Answer: \( 45^\circ, 60^\circ, 75^\circ \)

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##### Question 11:
- You know that one of the exterior angles of an isosceles triangle is \( 140^\circ \). The angle measures of the triangle could be \( ? \) or \( ? \).

An exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles. If one exterior angle is \( 140^\circ \), then the adjacent interior angle is:

\[
180^\circ - 140^\circ = 40^\circ
\]

In an isosceles triangle, two angles are equal. Let the two equal angles be \( x \) and the third angle be \( 40^\circ \). Using the triangle angle sum property:

\[
x + x + 40^\circ = 180^\circ
\]

Simplify:

\[
2x + 40^\circ = 180^\circ
\]

Solve for \( x \):

\[
2x = 140^\circ
\]
\[
x = 70^\circ
\]

Thus, the angles of the triangle are \( 70^\circ, 70^\circ, \) and \( 40^\circ \).

Alternatively, if the \( 140^\circ \) exterior angle corresponds to one of the base angles, then the adjacent interior angle is \( 40^\circ \), and the vertex angle would be:

\[
180^\circ - 2 \times 40^\circ = 100^\circ
\]

Thus, the angles of the triangle are \( 40^\circ, 40^\circ, \) and \( 100^\circ \).

Answer: \( 70^\circ, 70^\circ, 40^\circ \) or \( 40^\circ, 40^\circ, 100^\circ \)

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Final Answers:

1. Cannot be determined.
2. \( \boxed{100^\circ} \)
3. Cannot be determined.
4. Cannot be determined.
5. \( \boxed{80^\circ} \)
6. \( \boxed{50^\circ} \)
7. \( \boxed{360^\circ} \)
8. \( \boxed{\text{complementary}} \)
9. \( \boxed{\text{remote interior angles}} \)
10. \( \boxed{45^\circ, 60^\circ, 75^\circ} \)
11. \( \boxed{70^\circ, 70^\circ, 40^\circ \text{ or } 40^\circ, 40^\circ, 100^\circ} \)