Question

1. understand given kite abcd and trapezoid wacz, complete a two - column proof to show that wacz is an isosceles trapezoid.

statement	reason
2. $overline{ad} \cong \overline{cd}$	2.
3.	3. diagonals of a kite are $\perp$.
4. $overline{ed} \cong \overline{ed}$	4.
5.	5. hl thm.
6. $\angle dae \cong \angle dce$	6.
7.	7. alt. int. $\angle$s thm.
8.	8. transitive prop. of $\cong$
9. $\triangle adw \cong \triangle cdz$	9.
10. $\angle awd \cong \angle czd$	10.
11. wacz is an isosceles trapezoid.	11.

Explanation:

Step1: Analyze Statement 2

In a kite, two distinct pairs of adjacent sides are congruent. Also, by the property of a kite, the diagonal that connects the vertices between the unequal sides (or the axis of symmetry) bisects the other diagonal and the angles. But for \( \overline{AD} \cong \overline{CD} \), the reason is that in kite \( ABCD \), adjacent sides? Wait, no, actually, in a kite \( ABCD \) with \( AB = BC \) and \( AD=CD \) (assuming the kite is \( AB=BC \) and \( AD = CD \)), so the reason for \( \overline{AD}\cong\overline{CD} \) is "Definition of a kite (two distinct pairs of adjacent sides are congruent, so \( AD = CD \) as one pair)".

Step2: Statement 3

Given that diagonals of a kite are perpendicular, so the statement should be \( \overline{AC}\perp\overline{BD} \) (i.e., \( \angle AED\) and \( \angle CED \) are right angles), so Statement 3: \( \angle AED\cong\angle CED = 90^\circ \) (or \( \overline{AC}\perp\overline{BD} \)).

Step3: Statement 4

\( \overline{ED}\cong\overline{ED} \) is the Reflexive Property of Congruence (a segment is congruent to itself).

Step4: Statement 5

Using HL (Hypotenuse - Leg) Theorem, we need two right triangles with hypotenuse and one leg congruent. We have \( \overline{AD}\cong\overline{CD} \) (hypotenuse), \( \overline{ED}\cong\overline{ED} \) (leg), and right angles \( \angle AED\cong\angle CED \). So the triangles are \( \triangle AED\cong\triangle CED \) by HL. So Statement 5: \( \triangle AED\cong\triangle CED \).

Step5: Statement 6

Since \( \triangle AED\cong\triangle CED \), corresponding parts of congruent triangles are congruent (CPCTC), so \( \angle DAE\cong\angle DCE \). So Reason 6: CPCTC.

Step6: Statement 7

Alt. Int. \( \angle s \) Thm: Alternate Interior Angles Theorem. So we need two parallel lines cut by a transversal. Looking at trapezoid \( WACZ \), \( WZ \) and \( AC \) are the bases? Wait, no, \( WA \) and \( CZ \) are the legs? Wait, to apply Alternate Interior Angles, we need a transversal cutting two parallel lines. Suppose \( AC \parallel WZ \)? No, maybe \( AD \) and \( CD \) are not. Wait, actually, in the diagram, \( WZ \) has markings of congruent segments (mid - segment?), and \( WA \) and \( CZ \) are the legs. Wait, the Alternate Interior Angles Theorem would apply if \( AC \) is parallel to \( WZ \), but maybe the transversal is \( AD \) or \( CD \). Wait, the angle \( \angle DAE \) and \( \angle ADW \) (or \( \angle DCE \) and \( \angle CDZ \))? Wait, maybe Statement 7: \( \angle DAE\cong\angle ADW \) (if \( AC \parallel WZ \), then \( \angle DAE \) and \( \angle ADW \) are alternate interior angles). So Statement 7: \( \angle DAE\cong\angle ADW \) (and similarly \( \angle DCE\cong\angle CDZ \)).

Step7: Statement 8

By Transitive Property of Congruence, since \( \angle DAE\cong\angle DCE \) (Statement 6) and \( \angle DAE\cong\angle ADW \), \( \angle DCE\cong\angle CDZ \) (Statement 7), then \( \angle ADW\cong\angle CDZ \).

Step8: Statement 9

To prove \( \triangle ADW\cong\triangle CDZ \), we have \( \overline{AD}\cong\overline{CD} \) (Statement 2), \( \angle ADW\cong\angle CDZ \) (Statement 8), and \( \overline{DW}\cong\overline{DZ} \) (from the diagram, \( WZ \) is marked with congruent segments, so \( D \) is the mid - point? Wait, in the kite, diagonal \( BD \) bisects \( AC \), and if \( WZ \) is parallel to \( AC \) and \( D \) is on \( WZ \), then \( DW = DZ \) (since \( BD \) is the axis of symmetry). So using SAS (Side - Angle - Side) Congruence: \( \overline{AD}\cong\overline{CD} \), \( \angle ADW\cong\angle CDZ \), \( \overline{DW}\cong\overline{DZ} \), so \( \triangle ADW\cong\triangle CDZ \) by SAS. So Reason 9: SAS.

Step9: Statement 10

Since \( \triangle ADW\cong\triangle CDZ \), corresponding parts of congruent triangles are congruent (CPCTC), so \( \angle AWD\cong\angle CZD \). Reason 10: CPCTC.

Step10: Statement 11

A trapezoid with base angles congruent (or with the non - parallel sides congruent) is isosceles. Since \( \angle AWD\cong\angle CZD \), and \( WZ \) is the base, \( WA \) and \( CZ \) are the legs. If the base angles are congruent, then the trapezoid is isosceles. Also, since \( \triangle ADW\cong\triangle CDZ \), \( WA = CZ \) (CPCTC), so a trapezoid with congruent legs is isosceles. So Reason 11: "A trapezoid with congruent legs (or congruent base angles) is an isosceles trapezoid" (Definition of an isosceles trapezoid: a trapezoid with the non - parallel sides congruent, or base angles congruent).

Filling the two - column proof:

| Statement | Reason |
| --- | --- |
| 1. Kite \( ABCD \) and trapezoid \( WACZ \) | 1. Given |
| 2. \( \overline{AD}\cong\overline{CD} \) | 2. Definition of a kite (two distinct pairs of adjacent sides are congruent, so \( AD = CD \)) |
| 3. \( \angle AED\cong\angle CED = 90^\circ \) (or \( \overline{AC}\perp\overline{BD} \)) | 3. Diagonals of a kite are \( \perp \) |
| 4. \( \overline{ED}\cong\overline{ED} \) | 4. Reflexive Property of Congruence |
| 5. \( \triangle AED\cong\triangle CED \) | 5. HL Thm. ( \( \angle AED=\angle CED = 90^\circ \), \( \overline{AD}\cong\overline{CD} \), \( \overline{ED}\cong\overline{ED} \)) |
| 6. \( \angle DAE\cong\angle DCE \) | 6. CPCTC (Corresponding Parts of Congruent Triangles are Congruent) |
| 7. \( \angle DAE\cong\angle ADW \), \( \angle DCE\cong\angle CDZ \) | 7. Alt. Int. \( \angle s \) Thm. (if \( AC\parallel WZ \), transversal \( AD \) and \( CD \)) |
| 8. \( \angle ADW\cong\angle CDZ \) | 8. Transitive Prop. of \( \cong \) (from \( \angle DAE\cong\angle DCE \), \( \angle DAE\cong\angle ADW \), \( \angle DCE\cong\angle CDZ \)) |
| 9. \( \triangle ADW\cong\triangle CDZ \) | 9. SAS ( \( \overline{AD}\cong\overline{CD} \), \( \angle ADW\cong\angle CDZ \), \( \overline{DW}\cong\overline{DZ} \)) |
| 10. \( \angle AWD\cong\angle CZD \) | 10. CPCTC |
| 11. \( WACZ \) is an isosceles trapezoid | 11. A trapezoid with congruent base angles (or congruent legs) is isosceles (since \( \angle AWD\cong\angle CZD \), so \( WA = CZ \) or base angles are equal) |

Answer:

The completed two - column proof is as above. The key reasons and statements are filled based on properties of kites, congruence theorems (HL, SAS, CPCTC), and properties of isosceles trapezoids.