Question

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Prove ⤵️

Diagram *

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If two parallel lines are cut by a transversal,

then each pair of corresponding angles is congruent.,

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Answer 1

Theorem 10.7: If two lines are cut by a transversal so that the corresponding angles are congruent, then these lines are parallel.

A drawing of this situation is shown in Figure 10.8. Two lines, l and m are cut by a transversal t, and ∠1 and ∠2 are corresponding angles.

Given: l and m are cut by a transversal t, l ‌/‌ m.

Prove: ∠1 and ∠2 are not congruent (∠1 ~/= ∠2).

Proof: Assume that l ‌/‌ m. Because l and m are cut by a transversal t, m and t must intersect. You might call the point of intersection of m and t the point O. Because l is not parallel to m, we can find a line, say r, that passes through O and is parallel to l. I've drawn this new line in Figure 10.9. In this new drawing, ∠3 and ∠2 are corresponding angles, so by Postulate 10.1, they are congruent. But wait a minute! If ∠2 ~= ∠3, and m∠3 + m∠4 = m∠1 by the Angle Addition Postulate, m∠2 + m∠4 = m∠1. Because m∠4 > 0 (by the Protractor Postulate), this means that m∠2 < m∠1, and ∠1 ~/= ∠2. Let's put this all down in two columns.

Statements Reasons

1. l and m are two lines cut by a transversal t, with 1 |/| m Given

2. Let r be a line passing through O which is parallel to l Euclid's 5th postulate

3. ∠3 and ∠2 are corresponding angles Definition of corresponding angles

4. ∠2 ~= ∠3 Postulate 10.1

5. m∠2 = m∠3 Definition of ~=

6. m∠3 + m∠4 = m∠1 Angle Addition Postulate

7. m∠2 + m∠4 = m∠1 Substitution (steps 5 and 6)

8. m∠4 > 0 Protractor Postulate

9. m∠2 < m∠1 Definition of inequality

10. ∠4 ~/= ∠8 Definition of ~=

Answer 2

HERE IS YOUR ANSWER. .

Suppose you have two parallel lines cut by a transversal.

Due to the straight angle (linear pair) theorem, we know that

{m∠2+m∠3=180˚m∠5+m∠6=180˚

Thought the transitive property, we can say that

m∠2+m∠3=m∠5+m∠6×× (1)

Though the alternate interior angles theorem, we know that

m∠3=m∠5

Use substitution in (1):

∠2+m∠3=m∠3+m∠6

Subtract

m∠3 from both sides of the equation

m∠2=m∠6

∴

∠2≅∠6

Thus

∠2 and ∠6 are corresponding angles and have proven to be congruent.

.

PROVED