Question

explain whether the polygons are congruent. (see example 3)
5.
6.
yes // no why?
yes // no why?
find ( m angle 1 ). (see example 4)
7.
8.

Explanation:

5.

Step1: Analyze the number of sides and angles

The first polygon has 5 sides and specific angle - side relationships. The second polygon also has 5 sides. But when comparing corresponding sides and angles (by the markings of equal sides and right - angles etc.), they do not match. For example, the angles at \(W\) (in the first polygon) and \(L\) (in the second polygon) are not in the same relative position for congruence.

6.

Step1: Use the properties of congruent polygons (SSS - Side - Side - Side)

For the two polygons (composed of triangles in the figure), we check the side lengths. The markings on the sides show that \(XY = WZ\), \(XW=YZ\), and the other corresponding sides (formed by the common side in the middle - like the side connecting the two triangles) are equal. By the SSS (Side - Side - Side) congruence criterion for polygons (since they are made up of triangles that satisfy SSS, the whole polygons satisfy the congruence of all corresponding sides and angles).

7.

Step1: Use the angle - sum property of right - angled triangles

In a right - angled triangle, the sum of the non - right angles is \(90^{\circ}\). In \(\triangle LMN\), \(\angle L = 70^{\circ}\), \(\angle N=90^{\circ}\). In \(\triangle XYZ\) (which is congruent to \(\triangle LMN\) by some un - stated but visual congruence - assume ASA or AAS from the figure's markings), \(\angle1\) corresponds to the non - right angle in \(\triangle LMN\) other than \(\angle L\). So \(m\angle1=20^{\circ}\) (since \(90 - 70=20\)).

8.

Step1: Use the angle - sum property of triangles

In \(\triangle ABC\), using the angle - sum property of a triangle (\(\angle A+\angle B+\angle C = 180^{\circ}\)), \(\angle C=180-(80 + 45)=55^{\circ}\). Since the two triangles are congruent (by some visual congruence - assume ASA or AAS from the figure's angle markings), \(\angle1\) corresponds to \(\angle C\).

Answer:

1. NO. Corresponding sides and angles do not match.
2. YES. SSS congruence (all corresponding sides are equal).
3. \(m\angle1 = 20^{\circ}\)
4. \(m\angle1=55^{\circ}\)