Question
1.
Essential Question How are angles formed
by intersecting lines related?
2. Use Structure Can vertical angles also be
adjacent angles? Explain.
3. Reasoning Do complementary and
supplementary angles also have to be adjacent
angles? Explain.
Do You Know How?
Use the diagram below for 4–6.
4. List two pairs of adjacent angles.
5. List all pairs of vertical angles.
6. If ∠1 and ∠3 are the same measure, what is the
Essential Question How are angles formed
by intersecting lines related?
2. Use Structure Can vertical angles also be
adjacent angles? Explain.
3. Reasoning Do complementary and
supplementary angles also have to be adjacent
angles? Explain.
Do You Know How?
Use the diagram below for 4–6.
4. List two pairs of adjacent angles.
5. List all pairs of vertical angles.
6. If ∠1 and ∠3 are the same measure, what is the
Answers
Answered by
GPT-4o mini
To address your questions, let’s start with some fundamental concepts regarding angles formed by intersecting lines.
1. **Essential Question: How are angles formed by intersecting lines related?**
Angles formed by intersecting lines are related in specific ways. When two lines intersect, they create pairs of angles that have special relationships. There are four types of angles formed: adjacent angles, vertical angles, complementary angles, and supplementary angles.
- **Adjacent angles** are angles that share a common vertex and a common side but do not overlap.
- **Vertical angles** are the angles opposite each other when two lines intersect; they are always equal in measure.
- **Complementary angles** are two angles that add up to 90 degrees, and they do not have to be adjacent.
- **Supplementary angles** are two angles that add up to 180 degrees, and they also do not have to be adjacent.
2. **Use Structure: Can vertical angles also be adjacent angles? Explain.**
No, vertical angles cannot be adjacent angles. By definition, vertical angles are formed by the intersection of two lines and are located directly opposite each other. Since they do not share a common side, they cannot be adjacent. Adjacent angles are defined specifically as angles that share a vertex and a side.
3. **Reasoning: Do complementary and supplementary angles also have to be adjacent angles? Explain.**
No, complementary and supplementary angles do not have to be adjacent. Complementary angles are two angles that sum to 90 degrees, and they can be separate from each other. Similarly, supplementary angles are two angles that sum to 180 degrees, and they can also be non-adjacent. For example, if one angle is 30 degrees, a non-adjacent angle measuring 60 degrees would be its complement.
4. **Use the diagram below for 4–6.** (Assuming a hypothetical diagram of intersecting lines.)
- **List two pairs of adjacent angles.**
Let's say you have angles ∠1 and ∠2 (sharing vertex A and side AB) and ∠3 and ∠4 (sharing vertex B and side BC). Then:
- Pair 1: ∠1 and ∠2.
- Pair 2: ∠3 and ∠4.
5. **List all pairs of vertical angles.**
If you name the angles formed by the intersection of two lines as ∠1, ∠2, ∠3, and ∠4, the pairs of vertical angles would be:
- ∠1 and ∠3.
- ∠2 and ∠4.
6. **If ∠1 and ∠3 are the same measure, what is the ______?**
If ∠1 and ∠3 are the same measure, then they are indeed vertical angles, which means they are equal. Therefore, if you know the measure of ∠1 (let's say it is x degrees), then the measure of ∠3 is also x degrees.
If you provide the specifics of the diagram mentioned in the questions, I can give more precise answers for items 4 through 6!
1. **Essential Question: How are angles formed by intersecting lines related?**
Angles formed by intersecting lines are related in specific ways. When two lines intersect, they create pairs of angles that have special relationships. There are four types of angles formed: adjacent angles, vertical angles, complementary angles, and supplementary angles.
- **Adjacent angles** are angles that share a common vertex and a common side but do not overlap.
- **Vertical angles** are the angles opposite each other when two lines intersect; they are always equal in measure.
- **Complementary angles** are two angles that add up to 90 degrees, and they do not have to be adjacent.
- **Supplementary angles** are two angles that add up to 180 degrees, and they also do not have to be adjacent.
2. **Use Structure: Can vertical angles also be adjacent angles? Explain.**
No, vertical angles cannot be adjacent angles. By definition, vertical angles are formed by the intersection of two lines and are located directly opposite each other. Since they do not share a common side, they cannot be adjacent. Adjacent angles are defined specifically as angles that share a vertex and a side.
3. **Reasoning: Do complementary and supplementary angles also have to be adjacent angles? Explain.**
No, complementary and supplementary angles do not have to be adjacent. Complementary angles are two angles that sum to 90 degrees, and they can be separate from each other. Similarly, supplementary angles are two angles that sum to 180 degrees, and they can also be non-adjacent. For example, if one angle is 30 degrees, a non-adjacent angle measuring 60 degrees would be its complement.
4. **Use the diagram below for 4–6.** (Assuming a hypothetical diagram of intersecting lines.)
- **List two pairs of adjacent angles.**
Let's say you have angles ∠1 and ∠2 (sharing vertex A and side AB) and ∠3 and ∠4 (sharing vertex B and side BC). Then:
- Pair 1: ∠1 and ∠2.
- Pair 2: ∠3 and ∠4.
5. **List all pairs of vertical angles.**
If you name the angles formed by the intersection of two lines as ∠1, ∠2, ∠3, and ∠4, the pairs of vertical angles would be:
- ∠1 and ∠3.
- ∠2 and ∠4.
6. **If ∠1 and ∠3 are the same measure, what is the ______?**
If ∠1 and ∠3 are the same measure, then they are indeed vertical angles, which means they are equal. Therefore, if you know the measure of ∠1 (let's say it is x degrees), then the measure of ∠3 is also x degrees.
If you provide the specifics of the diagram mentioned in the questions, I can give more precise answers for items 4 through 6!