Question
brainly How would you prove that ? Vertical angles are supplementary. Corresponding angles are congruent. Vertical angles are congruent. Corresponding angles are supplementary. 1=4
Answers
GPT 3.5
To prove that vertical angles are supplementary and that corresponding angles are congruent, you can use the property of parallel lines intersected by a transversal.
Let's assume we have two lines, line AB and line CD, that are intersected by a transversal line, line EF. The points of intersection are labeled as point P, Q, R, and S in the following diagram:
Q ______ R
| |
P | E F | S
|________|
To prove that vertical angles are supplementary, we need to prove that ∠APQ and ∠BPS are supplementary angles.
Proof of Vertical Angles being Supplementary:
1. Given: Line AB, line CD, and transversal line EF.
2. ∠APQ and ∠BPS are vertical angles.
3. ∠PQR and ∠QSP are corresponding angles (since they are on the same side of the transversal and between the same parallel lines).
4. ∠PQR and ∠QSP are congruent (as corresponding angles are congruent).
5. ∠APQ + ∠PQR + ∠QSP + ∠BPS = 180 degrees (the sum of all angles around a point is 360 degrees).
6. Substituting known values: ∠APQ + ∠PQR + ∠QSP + ∠BPS = ∠APQ + ∠QSP + ∠QSP + ∠BPS = 180 degrees.
7. Simplifying: 2(∠QSP) + 2(∠APQ) = 180 degrees.
8. ∠QSP + ∠APQ = 90 degrees (dividing both sides by 2).
9. Thus, ∠APQ and ∠BPS are supplementary angles.
To prove that corresponding angles are congruent, we need to prove that ∠APQ and ∠PQR are congruent angles.
Proof of Corresponding Angles being Congruent:
1. Given: Line AB, line CD, and transversal line EF.
2. ∠APQ and ∠PQR are corresponding angles (as they are on the same side of the transversal and between the same parallel lines).
3. ∠QSP and ∠PQR are vertical angles (as they are on opposite sides of the transversal and formed by intersecting lines).
4. ∠APQ and ∠QSP are congruent (as corresponding angles are congruent).
5. ∠APQ = ∠QSP (using the substitution property of equality).
6. ∠APQ = ∠PQR (using the transitive property of equality).
7. Thus, ∠APQ and ∠PQR are congruent angles.
Therefore, we have proven that vertical angles are supplementary and corresponding angles are congruent.
Let's assume we have two lines, line AB and line CD, that are intersected by a transversal line, line EF. The points of intersection are labeled as point P, Q, R, and S in the following diagram:
Q ______ R
| |
P | E F | S
|________|
To prove that vertical angles are supplementary, we need to prove that ∠APQ and ∠BPS are supplementary angles.
Proof of Vertical Angles being Supplementary:
1. Given: Line AB, line CD, and transversal line EF.
2. ∠APQ and ∠BPS are vertical angles.
3. ∠PQR and ∠QSP are corresponding angles (since they are on the same side of the transversal and between the same parallel lines).
4. ∠PQR and ∠QSP are congruent (as corresponding angles are congruent).
5. ∠APQ + ∠PQR + ∠QSP + ∠BPS = 180 degrees (the sum of all angles around a point is 360 degrees).
6. Substituting known values: ∠APQ + ∠PQR + ∠QSP + ∠BPS = ∠APQ + ∠QSP + ∠QSP + ∠BPS = 180 degrees.
7. Simplifying: 2(∠QSP) + 2(∠APQ) = 180 degrees.
8. ∠QSP + ∠APQ = 90 degrees (dividing both sides by 2).
9. Thus, ∠APQ and ∠BPS are supplementary angles.
To prove that corresponding angles are congruent, we need to prove that ∠APQ and ∠PQR are congruent angles.
Proof of Corresponding Angles being Congruent:
1. Given: Line AB, line CD, and transversal line EF.
2. ∠APQ and ∠PQR are corresponding angles (as they are on the same side of the transversal and between the same parallel lines).
3. ∠QSP and ∠PQR are vertical angles (as they are on opposite sides of the transversal and formed by intersecting lines).
4. ∠APQ and ∠QSP are congruent (as corresponding angles are congruent).
5. ∠APQ = ∠QSP (using the substitution property of equality).
6. ∠APQ = ∠PQR (using the transitive property of equality).
7. Thus, ∠APQ and ∠PQR are congruent angles.
Therefore, we have proven that vertical angles are supplementary and corresponding angles are congruent.