Question

-.. The sum of the interior angles of a triangle equals 180° Move options to the blanks to complete the proof. Reason Gİven Alternate interlor angles are congruent. Straight angles add to 180° Substitution Statement AC | BD Z5 22 and Z6 5 mz2 +m 5 +m 26 = 180° Im2+ m F m 180 21 22 Z3 24 2

Answer 1

The transitive property of equality, we can conclude:

∠Z2 + ∠Z5 = ∠Z2 + ∠Z6 = 180°

Given:

Alternate interior angles are congruent (AC || BD)

Straight angles add to 180°

Proof:

1. ∠Z5 and ∠Z2 are alternate interior angles (AC || BD) [Given]

2. ∠Z5 ≅ ∠Z2 [Alternate interior angles are congruent]

3. ∠Z6 and ∠Z5 are supplementary (Straight angles add to 180°) [Given]

4. ∠Z6 + ∠Z5 = 180° [Definition of supplementary angles]

5. ∠Z6 + ∠Z2 = 180° [Substitution (using ∠Z5 ≅ ∠Z2)]

6. ∠Z2 + ∠Z6 = 180° [Communitive property of addition]

7. ∠Z2 + ∠Z5 = 180° [Substitution (using ∠Z5 ≅ ∠Z2)]

Now, let's consider a triangle ABC where ∠Z2, ∠Z5, and ∠Z6 are interior angles:

8. ∠Z2 + ∠Z5 + ∠Z6 = 180° [Sum of interior angles of a triangle]

9. ∠Z2 + ∠Z6 + ∠Z5 = 180° [Communitive property of addition]

Comparing equations 7 and 9, we can see that they are the same:

∠Z2 + ∠Z5 = ∠Z2 + ∠Z6

Therefore, by the transitive property of equality, we can conclude:

∠Z2 + ∠Z5 = ∠Z2 + ∠Z6 = 180°

This completes the proof that the sum of the interior angles of a triangle equals 180°.

Answer 2

Alternate interior angles:

In the given figure, ∠5 and ∠1 are alternate interior angles.

Similarly, ∠6 and ∠3 are also alternate interior angles.

`\angle5\cong\angle1\: and\: \angle6\cong\angle3`

Straight angles:

In the given figure, m∠1, m∠2, and m∠3 make a straight line angle equal to 180°.

`m\angle2+m\angle1+m\angle3=180\degree`

Substitution:

As you can see, we have already established that ∠5 = ∠1 and ∠6 = ∠3

Let us substitute this into the above equation so it becomes

`m\angle2+m\angle5+m\angle6=180\degree`