Question

fill in the blanks to the proof that segment ad is congruent to segment bc. segment bc and segment ad are parallel, and segment ae is congruent to segment segment ce because is is given in the diagram. angle ceb is congruent to angle \boxed{} because they are vertical angles. angle dae is congruent to angle \boxed{} because they are alternate interior angles. triangle bec is congruent to triangle \boxed{} by the angle - side - angle triangle congruence theorem. therefore, segment ad is congruent to bc because they are corresponding parts to congruent triangles. \boxed{aed} \boxed{bce} \boxed{ebc} \boxed{eda} \boxed{dea}

Explanation:

Step1: Analyze Vertical Angles

Angle CEB and angle AED are vertical angles, so they are congruent. So the first blank (angle congruent to CEB) is angle AED? Wait, no, wait. Wait, the first blank: "Angle CEB is congruent to angle ____ because they are vertical angles." Vertical angles: angle CEB and angle AED? Wait, no, let's look at the diagram. Points: A, B, C, D, E (intersection of diagonals). So lines AC and BD intersect at E. So angle CEB and angle AED are vertical angles? Wait, no, angle CEB and angle AED? Wait, maybe angle CEB and angle AED? Wait, no, let's check the options. The options are AED, BCE, EBC, EDA, DEA. Wait, vertical angles: when two lines intersect, opposite angles are vertical. So if AC and BD intersect at E, then angle CEB (at E, between C, E, B) and angle AED (at E, between A, E, D) are vertical angles. So angle CEB ≅ angle AED? Wait, no, maybe angle CEB and angle AED? Wait, the first blank: "Angle CEB is congruent to angle ____ because they are vertical angles." So the answer here should be angle AED? Wait, no, the options include DEA? Wait, DEA is angle at E, D, A. Wait, maybe I made a mistake. Wait, let's re-examine.

Wait, the next part: "Angle DAE is congruent to angle ____ because they are alternate interior angles." Since BC || AD, so transversal AC: angle DAE (at A, between D, A, E) and angle BCE (at C, between B, C, E) are alternate interior angles. So angle DAE ≅ angle BCE.

Then, "Triangle BEC is congruent to triangle ____ by the angle-side-angle triangle congruence theorem." We have angle CEB ≅ angle AED (vertical angles), AE ≅ CE (given, since AE and CE are marked congruent? Wait, the diagram has AE and CE with ticks, so AE = CE. Then angle DAE ≅ angle BCE (alternate interior angles). So triangle BEC: angles at E (CEB), at C (BCE), and side CE. Triangle AED: angles at E (AED, which is ≅ CEB), at A (DAE, which is ≅ BCE), and side AE (≅ CE). So by ASA, triangle BEC ≅ triangle AED? Wait, no, the options for the triangle congruent to BEC are AED, BCE, EBC, EDA, DEA. So triangle BEC ≅ triangle AED? Wait, but the options have AED as a triangle? Wait, triangle AED: vertices A, E, D. Triangle BEC: vertices B, E, C. So ASA: angle at E (CEB ≅ AED), side CE ≅ AE, angle at C (BCE ≅ DAE). So yes, triangle BEC ≅ triangle AED by ASA.

So let's fill the blanks:

1. Angle CEB is congruent to angle **AED** (vertical angles).
2. Angle DAE is congruent to angle **BCE** (alternate interior angles, BC || AD, transversal AC).
3. Triangle BEC is congruent to triangle **AED** (ASA: angle CEB ≅ AED, side CE ≅ AE, angle BCE ≅ DAE).

Then, "Therefore, segment AD is congruent to BC because they are corresponding parts to congruent triangles."

So let's go step by step:

First blank (angle congruent to CEB via vertical angles): angle AED? Wait, no, the options for the first angle: the first blank is "Angle CEB is congruent to angle ____ because they are vertical angles." The options are AED, BCE, EBC, EDA, DEA. So vertical angles: angle CEB and angle AED? Wait, DEA is angle at D, E, A, which is the same as angle AED (since angle AED is at A, E, D, and angle DEA is at D, E, A—same angle). Wait, maybe DEA? Wait, no, AED and DEA are the same angle (vertex E, sides AE and DE). So angle CEB ≅ angle AED (or DEA). But the options include AED and DEA. Wait, the first blank: let's check the options. The first blank's options: AED, BCE, EBC, EDA, DEA. So angle CEB and angle AED (or DEA) are vertical angles. So the first blank is DEA? Wait, no, AED is an option. Let's see the next blank: "Angle DAE is congruent to angle ____ because they are alternate interior angles." Since BC || AD, transversal AC: angle DAE (∠DAE) and angle BCE (∠BCE) are alternate interior angles. So that blank is BCE.

Then, "Triangle BEC is congruent to triangle ____ by ASA." We have ∠CEB ≅ ∠AED (vertical angles), CE ≅ AE (given), ∠BCE ≅ ∠DAE (alternate interior). So triangle BEC ≅ triangle AED (ASA: ∠CEB, CE, ∠BCE; ∠AED, AE, ∠DAE). So the triangle congruent to BEC is AED.

So putting it all together:

1. Angle CEB is congruent to angle **AED** (wait, no, DEA? Wait, maybe DEA? Wait, no, AED is the angle at E, A, D. DEA is angle at E, D, A. Same angle. But the options have AED. So first blank: AED? Wait, no, let's check the options again. The first blank's options: AED, BCE, EBC, EDA, DEA. So vertical angles: angle CEB and angle AED (or DEA). So the answer for the first blank is DEA? Wait, maybe I messed up. Wait, let's check the vertical angles again. If lines AC and BD intersect at E, then angle CEB (E between C and B) and angle AED (E between A and D) are vertical angles. So yes, angle CEB ≅ angle AED. So first blank: AED.

Second blank: angle DAE ≅ angle BCE (alternate interior angles, since BC || AD, transversal AC). So second blank: BCE.

Third blank: triangle BEC ≅ triangle AED (by ASA: ∠CEB ≅ ∠AED, CE ≅ AE, ∠BCE ≅ ∠DAE). So third blank: AED.

Step2: Verify Each Blank

- First blank (angle congruent to CEB via vertical angles): angle AED (since ∠CEB and ∠AED are vertical angles).
- Second blank (angle congruent to DAE via alternate interior angles): angle BCE (since BC || AD, transversal AC, so ∠DAE and ∠BCE are alternate interior).
- Third blank (triangle congruent to BEC via ASA): triangle AED (since ∠CEB ≅ ∠AED, CE ≅ AE, ∠BCE ≅ ∠DAE, so ASA applies to △BEC and △AED).

Answer:

First blank: AED
Second blank: BCE
Third blank: AED