Question

use the figure for exercises 34 - 36.

1. why are \\( \angle 1 \\) and \\( \angle 2 \\) congruent?
2. why is b the midpoint of \\( \overline{cz} \\)?
3. write a congruence statement for the 2 triangles.

Explanation:

34.

Step1: Vertical angles definition

Vertical angles are formed when two lines intersect.

Step2: Property of vertical angles

Vertical angles are congruent. Since \(∠1\) and \(∠2\) are vertical angles (formed by the intersection of lines \(AC\) and \(ZX\) at point \(B\)), they are congruent.

Step1: Congruent triangles

From the markings on the figure (side - side - side or other congruence criteria if applicable, assume triangles \(ABC\) and \(XBZ\) are congruent).

Step2: Corresponding parts of congruent triangles

If \(\triangle ABC\cong\triangle XBZ\), then \(AB = XB\) and \(CB = ZB\). By the definition of a mid - point (a point that divides a segment into two equal parts), since \(CB = ZB\), point \(B\) is the mid - point of \(CZ\).

Step1: Identify corresponding vertices

Looking at the side markings (assuming three pairs of corresponding sides are equal). Vertex \(A\) corresponds to vertex \(X\), vertex \(B\) is common in the intersection (but in terms of triangle correspondence, vertex \(C\) corresponds to vertex \(Z\)).

Step2: Write congruence statement

Using the order of corresponding vertices, the congruence statement is \(\triangle ABC\cong\triangle XBZ\)

Answer:

\(∠1\) and \(∠2\) are congruent because they are vertical angles.

35.