brainly How would you prove that ? Vertical angles are supplementary. Corresponding angles are congruent. Vertical angles are congruent. Corresponding angles are supplementary. 1=4
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.