5. Reasoning in Algebra and Geometry - 2. Reasoning and Proof

We are given the following diagram.

As we can see, there are two intersecting segments that form four angles. Let's first analyze the segments and then we will consider the angles.

Segments

On the diagram, two segments intersect each other, and we can see two red marks on ED.

This means that BD and EB are congruent segments. Therefore, AC bisects DE.

Angles

Analyzing the diagram, we can see that there are two pairs of vertical angles — ∠ 1 and ∠ CBD, as well as ∠ 2 and ∠ ABD. By the Vertical Angles Theorem, we can conclude that the angles in each pair are congruent.

Therefore, the following statements are true. Vertical Angles ∠ 1 ≅ ∠ CBD and ∠ 2 ≅ ∠ ABD Also, there are four pairs of angles that form linear pairs in the diagram.

Let's recall the Linear Pair Postulate.

Linear Pair Postulate
If two angles form a linear pair, then they are supplementary.

According to this postulate, the angles from each pair are supplementary angles.

	Supplementary Angles
Pair 1	∠ 1 and ∠ 2
Pair 2	∠ 2 and ∠ CBD
Pair 3	∠ CBD and ∠ ABD
Pair 4	∠ ABD and ∠ 1

Conclusion

Let's list all the statements that we can make based on the given diagram.

BD and EB are congruent segments.
AC bisects DE.
∠ 1 and ∠ 2, ∠ 2 and ∠ CBD, ∠ CBD and ∠ ABD, and ∠ ABD and ∠ 1 form linear pairs.
∠ 1 and ∠ 2, ∠ 2 and ∠ CBD, ∠ CBD and ∠ ABD, and ∠ ABD and ∠ 1 are supplementary angles.
∠ 1 and ∠ CBD, as well as ∠ 2 and ∠ ABD, are vertical angles.
∠ 1 and ∠ CBD, as well as ∠ 2 and ∠ ABD, are are congruent angles.

Exercise 20 Page 118

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Segments

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