Interior and Exterior Angles with Quadrilaterals worksheet | Live ... - Free Printable

To solve the missing angles in the given quadrilaterals, we will use the fact that the sum of the interior angles of any quadrilateral is always 360°. Let's go through each problem step by step.

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Problem 1:

The quadrilateral has three known angles: 105°, 115°, and 55°. We need to find the fourth angle.

#### Steps:
1. The sum of the interior angles of a quadrilateral is 360°.
2. Let the missing angle be \( x \).
3. Write the equation:
\[
105^\circ + 115^\circ + 55^\circ + x = 360^\circ
\]
4. Simplify the left side:
\[
275^\circ + x = 360^\circ
\]
5. Solve for \( x \):
\[
x = 360^\circ - 275^\circ = 85^\circ
\]

#### Answer:
\[
\boxed{85^\circ}
\]

---

Problem 2:

The quadrilateral (a rhombus) has three known angles: 95°, 145°, and 67°. We need to find the fourth angle.

#### Steps:
1. The sum of the interior angles of a quadrilateral is 360°.
2. Let the missing angle be \( x \).
3. Write the equation:
\[
95^\circ + 145^\circ + 67^\circ + x = 360^\circ
\]
4. Simplify the left side:
\[
307^\circ + x = 360^\circ
\]
5. Solve for \( x \):
\[
x = 360^\circ - 307^\circ = 53^\circ
\]

#### Answer:
\[
\boxed{53^\circ}
\]

---

Problem 3:

The quadrilateral has three known angles: 46°, 52°, and 116°. We need to find the fourth angle.

#### Steps:
1. The sum of the interior angles of a quadrilateral is 360°.
2. Let the missing angle be \( x \).
3. Write the equation:
\[
46^\circ + 52^\circ + 116^\circ + x = 360^\circ
\]
4. Simplify the left side:
\[
214^\circ + x = 360^\circ
\]
5. Solve for \( x \):
\[
x = 360^\circ - 214^\circ = 146^\circ
\]

#### Answer:
\[
\boxed{146^\circ}
\]

---

Problem 4:

The quadrilateral has three known angles: 83°, 45°, and 34°. We need to find the fourth angle.

#### Steps:
1. The sum of the interior angles of a quadrilateral is 360°.
2. Let the missing angle be \( x \).
3. Write the equation:
\[
83^\circ + 45^\circ + 34^\circ + x = 360^\circ
\]
4. Simplify the left side:
\[
162^\circ + x = 360^\circ
\]
5. Solve for \( x \):
\[
x = 360^\circ - 162^\circ = 198^\circ
\]

#### Answer:
\[
\boxed{198^\circ}
\]

---

Problem 5:

The quadrilateral has three known angles: 54°, 83°, and 43°. We need to find the fourth angle.

#### Steps:
1. The sum of the interior angles of a quadrilateral is 360°.
2. Let the missing angle be \( x \).
3. Write the equation:
\[
54^\circ + 83^\circ + 43^\circ + x = 360^\circ
\]
4. Simplify the left side:
\[
180^\circ + x = 360^\circ
\]
5. Solve for \( x \):
\[
x = 360^\circ - 180^\circ = 180^\circ
\]

#### Answer:
\[
\boxed{180^\circ}
\]

---

Problem 6:

The quadrilateral has three known angles: 87°, 99°, and 67°. We need to find the fourth angle.

#### Steps:
1. The sum of the interior angles of a quadrilateral is 360°.
2. Let the missing angle be \( x \).
3. Write the equation:
\[
87^\circ + 99^\circ + 67^\circ + x = 360^\circ
\]
4. Simplify the left side:
\[
253^\circ + x = 360^\circ
\]
5. Solve for \( x \):
\[
x = 360^\circ - 253^\circ = 107^\circ
\]

#### Answer:
\[
\boxed{107^\circ}
\]

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Final Answers:

1. \( \boxed{85^\circ} \)
2. \( \boxed{53^\circ} \)
3. \( \boxed{146^\circ} \)
4. \( \boxed{198^\circ} \)
5. \( \boxed{180^\circ} \)
6. \( \boxed{107^\circ} \)