Certainly! Let's tackle the problem step by step.
### Part 1: Angles in a Parallelogram
1. Given Data:
- We are given that in parallelogram [tex]\(ABCD\)[/tex], [tex]\(\angle A = 75^\circ\)[/tex].
2. Characteristics of a Parallelogram:
- In a parallelogram, opposite angles are equal.
- Thus, [tex]\(\angle C\)[/tex] (which is opposite [tex]\(\angle A\)[/tex]) will also be [tex]\(75^\circ\)[/tex].
Therefore, [tex]\(\angle C = 75^\circ\)[/tex].
3. Sum of Adjacent Angles:
- In a parallelogram, the sum of adjacent angles is [tex]\(180^\circ\)[/tex].
This means:
[tex]\[
\angle A + \angle B = 180^\circ
\][/tex]
[tex]\[
\angle A = 75^\circ \implies \angle B = 180^\circ - 75^\circ = 105^\circ
\][/tex]
Similarly,
[tex]\[
\angle C + \angle D = 180^\circ
\][/tex]
[tex]\[
\angle C = 75^\circ \implies \angle D = 180^\circ - 75^\circ = 105^\circ
\][/tex]
Hence,
- [tex]\(\angle B = 105^\circ\)[/tex]
- [tex]\(\angle D = 105^\circ\)[/tex]
### Part 2: Properties of a Regular Polygon with an External Angle of [tex]\(20^\circ\)[/tex]
1. Given Data:
- We are given that the external angle of a regular polygon is [tex]\(20^\circ\)[/tex].
2. Number of Sides:
- The exterior angles of a regular polygon always sum to [tex]\(360^\circ\)[/tex]. Thus, the number of sides ([tex]\(n\)[/tex]) can be found using:
[tex]\[
n = \frac{360^\circ}{\text{exterior angle}}
\][/tex]
Given the exterior angle is [tex]\(20^\circ\)[/tex]:
[tex]\[
n = \frac{360^\circ}{20^\circ} = 18
\][/tex]
Therefore, the polygon has [tex]\(18\)[/tex] sides.
3. Measure of Each Interior Angle:
- The measure of an interior angle of a regular polygon can be calculated as:
[tex]\[
\text{Interior angle} = 180^\circ - \text{exterior angle}
\][/tex]
Given the exterior angle is [tex]\(20^\circ\)[/tex]:
[tex]\[
\text{Interior angle} = 180^\circ - 20^\circ = 160^\circ
\][/tex]
4. Total Measure of Interior Angles:
- The total measure of all interior angles in a regular polygon with [tex]\(n\)[/tex] sides is given by:
[tex]\[
\text{Total interior angles sum} = (n - 2) \times 180^\circ
\][/tex]
Given [tex]\(n = 18\)[/tex]:
[tex]\[
\text{Total interior angles sum} = (18 - 2) \times 180^\circ = 16 \times 180^\circ = 2880^\circ
\][/tex]
### Final Result Summary:
- In the parallelogram [tex]\(ABCD\)[/tex], the measures of the angles are:
- [tex]\(\angle B = 105^\circ\)[/tex]
- [tex]\(\angle C = 75^\circ\)[/tex]
- [tex]\(\angle D = 105^\circ\)[/tex]
- For the regular polygon with an external angle of [tex]\(20^\circ\)[/tex]:
- It has [tex]\(18\)[/tex] sides.
- Each interior angle is [tex]\(160^\circ\)[/tex].
- The total measure of its interior angles is [tex]\(2880^\circ\)[/tex].
